diff --git a/src/java.base/share/classes/java/lang/FdLibm.java b/src/java.base/share/classes/java/lang/FdLibm.java index af0a331e8e6..4c4dd417027 100644 --- a/src/java.base/share/classes/java/lang/FdLibm.java +++ b/src/java.base/share/classes/java/lang/FdLibm.java @@ -60,6 +60,7 @@ package java.lang; class FdLibm { // Constants used by multiple algorithms private static final double INFINITY = Double.POSITIVE_INFINITY; + private static final double TWO24 = 0x1.0p24; // 1.67772160000000000000e+07 private static final double TWO54 = 0x1.0p54; // 1.80143985094819840000e+16 private static final double HUGE = 1.0e+300; @@ -113,6 +114,910 @@ class FdLibm { (low & 0xffff_ffffL)); } + /** sin(x) + * Return sine function of x. + * + * kernel function: + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cose function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Sin { + private Sin() {throw new UnsupportedOperationException();} + + static double compute(double x) { + double[] y = new double[2]; + double z = 0.0; + int n, ix; + + // High word of x. + ix = __HI(x); + + // |x| ~< pi/4 + ix &= 0x7fff_ffff; + if (ix <= 0x3fe9_21fb) { + return __kernel_sin(x, z, 0); + } else if (ix>=0x7ff0_0000) { // sin(Inf or NaN) is NaN + return x - x; + } else { // argument reduction needed + n = RemPio2.__ieee754_rem_pio2(x, y); + switch(n & 3) { + case 0: return Sin.__kernel_sin(y[0], y[1], 1); + case 1: return Cos.__kernel_cos(y[0], y[1]); + case 2: return -Sin.__kernel_sin(y[0], y[1], 1); + default: + return -Cos.__kernel_cos(y[0], y[1]); + } + } + } + + /** __kernel_sin( x, y, iy) + * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). + * + * Algorithm + * 1. Since sin(-x) = -sin(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. + * 3. sin(x) is approximated by a polynomial of degree 13 on + * [0,pi/4] + * 3 13 + * sin(x) ~ x + S1*x + ... + S6*x + * where + * + * |sin(x) 2 4 6 8 10 12 | -58 + * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 + * | x | + * + * 4. sin(x+y) = sin(x) + sin'(x')*y + * ~ sin(x) + (1-x*x/2)*y + * For better accuracy, let + * 3 2 2 2 2 + * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) + * then 3 2 + * sin(x) = x + (S1*x + (x *(r-y/2)+y)) + */ + private static final double + S1 = -0x1.5555555555549p-3, // -1.66666666666666324348e-01 + S2 = 0x1.111111110f8a6p-7, // 8.33333333332248946124e-03 + S3 = -0x1.a01a019c161d5p-13, // -1.98412698298579493134e-04 + S4 = 0x1.71de357b1fe7dp-19, // 2.75573137070700676789e-06 + S5 = -0x1.ae5e68a2b9cebp-26, // -2.50507602534068634195e-08 + S6 = 0x1.5d93a5acfd57cp-33; // 1.58969099521155010221e-10 + + static double __kernel_sin(double x, double y, int iy) { + double z, r, v; + int ix; + ix = __HI(x) & 0x7fff_ffff; // high word of x + if (ix < 0x3e40_0000) { // |x| < 2**-27 + if ((int)x == 0) // generate inexact + return x; + } + z = x*x; + v = z*x; + r = S2 + z*(S3 + z*(S4 + z*(S5 + z*S6))); + if (iy == 0) { + return x + v*(S1 + z*r); + } else { + return x - ((z*(0.5*y - v*r) - y) - v*S1); + } + } + } + + /** cos(x) + * Return cosine function of x. + * + * kernel function: + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cosine function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Cos { + private Cos() {throw new UnsupportedOperationException();} + + static double compute(double x) { + double[] y = new double[2]; + double z = 0.0; + int n, ix; + + // High word of x. + ix = __HI(x); + + // |x| ~< pi/4 + ix &= 0x7fff_ffff; + if (ix <= 0x3fe9_21fb) { + return __kernel_cos(x, z); + } else if (ix >= 0x7ff0_0000) { // cos(Inf or NaN) is NaN + return x-x; + } else { // argument reduction needed + n = RemPio2.__ieee754_rem_pio2(x,y); + switch (n & 3) { + case 0: return Cos.__kernel_cos(y[0], y[1]); + case 1: return -Sin.__kernel_sin(y[0], y[1],1); + case 2: return -Cos.__kernel_cos(y[0], y[1]); + default: + return Sin.__kernel_sin(y[0], y[1], 1); + } + } + } + + /** + * __kernel_cos( x, y ) + * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * + * Algorithm + * 1. Since cos(-x) = cos(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx < 0x3e4000000), return 1 with inexact if x != 0. + * 3. cos(x) is approximated by a polynomial of degree 14 on + * [0,pi/4] + * 4 14 + * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x + * where the remez error is + * + * | 2 4 6 8 10 12 14 | -58 + * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 + * | | + * + * 4 6 8 10 12 14 + * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then + * cos(x) = 1 - x*x/2 + r + * since cos(x+y) ~ cos(x) - sin(x)*y + * ~ cos(x) - x*y, + * a correction term is necessary in cos(x) and hence + * cos(x+y) = 1 - (x*x/2 - (r - x*y)) + * For better accuracy when x > 0.3, let qx = |x|/4 with + * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. + * Then + * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). + * Note that 1-qx and (x*x/2-qx) is EXACT here, and the + * magnitude of the latter is at least a quarter of x*x/2, + * thus, reducing the rounding error in the subtraction. + */ + private static final double + C1 = 0x1.555555555554cp-5, // 4.16666666666666019037e-02 + C2 = -0x1.6c16c16c15177p-10, // -1.38888888888741095749e-03 + C3 = 0x1.a01a019cb159p-16, // 2.48015872894767294178e-05 + C4 = -0x1.27e4f809c52adp-22, // -2.75573143513906633035e-07 + C5 = 0x1.1ee9ebdb4b1c4p-29, // 2.08757232129817482790e-09 + C6 = -0x1.8fae9be8838d4p-37; // -1.13596475577881948265e-11 + + static double __kernel_cos(double x, double y) { + double a, hz, z, r, qx = 0.0; + int ix; + ix = __HI(x) & 0x7fff_ffff; // ix = |x|'s high word + if (ix < 0x3e40_0000) { // if x < 2**27 + if (((int)x) == 0) { // generate inexact + return 1.0; + } + } + z = x*x; + r = z*(C1 + z*(C2 + z*(C3 + z*(C4 + z*(C5 + z*C6))))); + if (ix < 0x3FD3_3333) { // if |x| < 0.3 + return 1.0 - (0.5*z - (z*r - x*y)); + } else { + if (ix > 0x3fe9_0000) { // x > 0.78125 + qx = 0.28125; + } else { + qx = __HI_LO(ix - 0x0020_0000, 0); + } + hz = 0.5*z - qx; + a = 1.0 - qx; + return a - (hz - (z*r - x*y)); + } + } + } + + /** tan(x) + * Return tangent function of x. + * + * kernel function: + * __kernel_tan ... tangent function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Tan { + private Tan() {throw new UnsupportedOperationException();} + + static double compute(double x) { + double[] y = new double[2]; + double z = 0.0; + int n, ix; + + // High word of x. + ix = __HI(x); + + // |x| ~< pi/4 + ix &= 0x7fff_ffff; + if (ix <= 0x3fe9_21fb) { + return __kernel_tan(x, z, 1); + } else if (ix >= 0x7ff0_0000) { // tan(Inf or NaN) is NaN + return x-x; // NaN + } else { // argument reduction needed + n = RemPio2.__ieee754_rem_pio2(x, y); + return __kernel_tan(y[0], y[1], 1 - ((n & 1) << 1)); // 1 -- n even; -1 -- n odd + } + } + + /** __kernel_tan( x, y, k ) + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input k indicates whether tan (if k=1) or + * -1/tan (if k= -1) is returned. + * + * Algorithm + * 1. Since tan(-x) = -tan(x), we need only to consider positive x. + * 2. if x < 2^-28 (hx < 0x3e300000 0), return x with inexact if x != 0. + * 3. tan(x) is approximated by a odd polynomial of degree 27 on + * [0, 0.67434] + * 3 27 + * tan(x) ~ x + T1*x + ... + T13*x + * where + * + * |tan(x) 2 4 26 | -59.2 + * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 + * | x | + * + * Note: tan(x+y) = tan(x) + tan'(x)*y + * ~ tan(x) + (1+x*x)*y + * Therefore, for better accuracy in computing tan(x+y), let + * 3 2 2 2 2 + * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) + * then + * 3 2 + * tan(x+y) = x + (T1*x + (x *(r+y)+y)) + * + * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then + * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) + * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) + */ + private static final double + pio4 = 0x1.921fb54442d18p-1, // 7.85398163397448278999e-01 + pio4lo= 0x1.1a62633145c07p-55, // 3.06161699786838301793e-17 + T[] = { + 0x1.5555555555563p-2, // 3.33333333333334091986e-01 + 0x1.111111110fe7ap-3, // 1.33333333333201242699e-01 + 0x1.ba1ba1bb341fep-5, // 5.39682539762260521377e-02 + 0x1.664f48406d637p-6, // 2.18694882948595424599e-02 + 0x1.226e3e96e8493p-7, // 8.86323982359930005737e-03 + 0x1.d6d22c9560328p-9, // 3.59207910759131235356e-03 + 0x1.7dbc8fee08315p-10, // 1.45620945432529025516e-03 + 0x1.344d8f2f26501p-11, // 5.88041240820264096874e-04 + 0x1.026f71a8d1068p-12, // 2.46463134818469906812e-04 + 0x1.47e88a03792a6p-14, // 7.81794442939557092300e-05 + 0x1.2b80f32f0a7e9p-14, // 7.14072491382608190305e-05 + -0x1.375cbdb605373p-16, // -1.85586374855275456654e-05 + 0x1.b2a7074bf7ad4p-16, // 2.59073051863633712884e-05 + }; + + static double __kernel_tan(double x, double y, int iy) { + double z, r, v, w, s; + int ix, hx; + hx = __HI(x); // high word of x + ix = hx&0x7fff_ffff; // high word of |x| + if (ix < 0x3e30_0000) { // x < 2**-28 + if ((int)x == 0) { // generate inexact + if (((ix | __LO(x)) | (iy + 1)) == 0) { + return 1.0 / Math.abs(x); + } else { + if (iy == 1) { + return x; + } else { // compute -1 / (x+y) carefully + double a, t; + + z = w = x + y; + z= __LO(z, 0); + v = y - (z - x); + t = a = -1.0 / w; + t = __LO(t, 0); + s = 1.0 + t * z; + return t + a * (s + t * v); + } + } + } + } + if (ix >= 0x3FE5_9428) { // |x| >= 0.6744 + if ( hx < 0) { + x = -x; + y = -y; + } + z = pio4 - x; + w = pio4lo - y; + x = z + w; + y = 0.0; + } + z = x*x; + w = z*z; + /* Break x^5*(T[1]+x^2*T[2]+...) into + * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) + */ + r = T[1] + w*(T[3] + w*(T[5] + w*(T[7] + w*(T[9] + w*T[11])))); + v = z*(T[2] + w*(T[4] + w*(T[6] + w*(T[8] + w*(T[10] + w*T[12]))))); + s = z*x; + r = y + z*(s*(r + v) + y); + r += T[0]*s; + w = x + r; + if (ix >= 0x3FE5_9428) { + v = (double)iy; + return (double)(1-((hx >> 30) & 2))*(v - 2.0*(x - (w*w/(w + v) - r))); + } + if (iy == 1) { + return w; + } else { /* if were to allow error up to 2 ulp, + could simply return -1.0/(x + r) here */ + // compute -1.0/(x + r) accurately + double a,t; + z = w; + z = __LO(z, 0); + v = r - (z - x); // z + v = r + x + t = a = -1.0/w; // a = -1.0/w + t = __LO(t, 0); + s = 1.0 + t*z; + return t + a*(s + t*v); + } + } + } + + /** __ieee754_rem_pio2(x,y) + * + * return the remainder of x rem pi/2 in y[0]+y[1] + * use __kernel_rem_pio2() + */ + static class RemPio2 { + /* + * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi + */ + private static final int[] two_over_pi = { + 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, + 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, + 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, + 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, + 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, + 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, + 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, + 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, + 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, + 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, + 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, + }; + + private static final int[] npio2_hw = { + 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, + 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, + 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, + 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, + 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, + 0x404858EB, 0x404921FB, + }; + + /* + * invpio2: 53 bits of 2/pi + * pio2_1: first 33 bit of pi/2 + * pio2_1t: pi/2 - pio2_1 + * pio2_2: second 33 bit of pi/2 + * pio2_2t: pi/2 - (pio2_1+pio2_2) + * pio2_3: third 33 bit of pi/2 + * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) + */ + + private static final double + invpio2 = 0x1.45f306dc9c883p-1, // 6.36619772367581382433e-01 + pio2_1 = 0x1.921fb544p0, // 1.57079632673412561417e+00 + pio2_1t = 0x1.0b4611a626331p-34, // 6.07710050650619224932e-11 + pio2_2 = 0x1.0b4611a6p-34, // 6.07710050630396597660e-11 + pio2_2t = 0x1.3198a2e037073p-69, // 2.02226624879595063154e-21 + pio2_3 = 0x1.3198a2ep-69, // 2.02226624871116645580e-21 + pio2_3t = 0x1.b839a252049c1p-104; // 8.47842766036889956997e-32 + + static int __ieee754_rem_pio2(double x, double[] y) { + double z = 0.0, w, t, r, fn; + double[] tx = new double[3]; + int e0, i, j, nx, n, ix, hx; + + hx = __HI(x); // high word of x + ix = hx & 0x7fff_ffff; + if (ix <= 0x3fe9_21fb) { // |x| ~<= pi/4 , no need for reduction + y[0] = x; + y[1] = 0; + return 0; + } + if (ix < 0x4002_d97c) { // |x| < 3pi/4, special case with n=+-1 + if (hx > 0) { + z = x - pio2_1; + if (ix != 0x3ff9_21fb) { // 33+53 bit pi is good enough + y[0] = z - pio2_1t; + y[1] = (z - y[0]) - pio2_1t; + } else { // near pi/2, use 33+33+53 bit pi + z -= pio2_2; + y[0] = z - pio2_2t; + y[1] = (z-y[0])-pio2_2t; + } + return 1; + } else { // negative x + z = x + pio2_1; + if (ix != 0x3ff_921fb) { // 33+53 bit pi is good enough + y[0] = z + pio2_1t; + y[1] = (z - y[0]) + pio2_1t; + } else { // near pi/2, use 33+33+53 bit pi + z += pio2_2; + y[0] = z + pio2_2t; + y[1] = (z - y[0]) + pio2_2t; + } + return -1; + } + } + if (ix <= 0x4139_21fb) { // |x| ~<= 2^19*(pi/2), medium size + t = Math.abs(x); + n = (int) (t*invpio2 + 0.5); + fn = (double)n; + r = t - fn*pio2_1; + w = fn*pio2_1t; // 1st round good to 85 bit + if (n < 32 && ix != npio2_hw[n - 1]) { + y[0] = r - w; // quick check no cancellation + } else { + j = ix >> 20; + y[0] = r - w; + i = j - (((__HI(y[0])) >> 20) & 0x7ff); + if (i > 16) { // 2nd iteration needed, good to 118 + t = r; + w = fn*pio2_2; + r = t - w; + w = fn*pio2_2t - ((t - r) - w); + y[0] = r - w; + i = j - (((__HI(y[0])) >> 20) & 0x7ff); + if (i > 49) { // 3rd iteration need, 151 bits acc + t = r; // will cover all possible cases + w = fn*pio2_3; + r = t - w; + w = fn*pio2_3t - ((t - r) - w); + y[0] = r - w; + } + } + } + y[1] = (r - y[0]) - w; + if (hx < 0) { + y[0] = -y[0]; + y[1] = -y[1]; + return -n; + } else { + return n; + } + } + /* + * all other (large) arguments + */ + if (ix >= 0x7ff0_0000) { // x is inf or NaN + y[0] = y[1] = x - x; + return 0; + } + // set z = scalbn(|x|,ilogb(x)-23) + z = __LO(z, __LO(x)); + e0 = (ix >> 20) - 1046; /* e0 = ilogb(z)-23; */ + z = __HI(z, ix - (e0 << 20)); + for (i=0; i < 2; i++) { + tx[i] = (double)((int)(z)); + z = (z - tx[i])*TWO24; + } + tx[2] = z; + nx = 3; + while (tx[nx - 1] == 0.0) { // skip zero term + nx--; + } + n = KernelRemPio2.__kernel_rem_pio2(tx, y, e0, nx, 2, two_over_pi); + if (hx < 0) { + y[0] = -y[0]; + y[1] = -y[1]; + return -n; + } + return n; + } + } + + /** + * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) + * double x[],y[]; int e0,nx,prec; int ipio2[]; + * + * __kernel_rem_pio2 return the last three digits of N with + * y = x - N*pi/2 + * so that |y| < pi/2. + * + * The method is to compute the integer (mod 8) and fraction parts of + * (2/pi)*x without doing the full multiplication. In general we + * skip the part of the product that are known to be a huge integer ( + * more accurately, = 0 mod 8 ). Thus the number of operations are + * independent of the exponent of the input. + * + * (2/pi) is represented by an array of 24-bit integers in ipio2[]. + * + * Input parameters: + * x[] The input value (must be positive) is broken into nx + * pieces of 24-bit integers in double precision format. + * x[i] will be the i-th 24 bit of x. The scaled exponent + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 + * match x's up to 24 bits. + * + * Example of breaking a double positive z into x[0]+x[1]+x[2]: + * e0 = ilogb(z)-23 + * z = scalbn(z,-e0) + * for i = 0,1,2 + * x[i] = floor(z) + * z = (z-x[i])*2**24 + * + * + * y[] output result in an array of double precision numbers. + * The dimension of y[] is: + * 24-bit precision 1 + * 53-bit precision 2 + * 64-bit precision 2 + * 113-bit precision 3 + * The actual value is the sum of them. Thus for 113-bit + * precision, one may have to do something like: + * + * long double t,w,r_head, r_tail; + * t = (long double)y[2] + (long double)y[1]; + * w = (long double)y[0]; + * r_head = t+w; + * r_tail = w - (r_head - t); + * + * e0 The exponent of x[0] + * + * nx dimension of x[] + * + * prec an integer indicating the precision: + * 0 24 bits (single) + * 1 53 bits (double) + * 2 64 bits (extended) + * 3 113 bits (quad) + * + * ipio2[] + * integer array, contains the (24*i)-th to (24*i+23)-th + * bit of 2/pi after binary point. The corresponding + * floating value is + * + * ipio2[i] * 2^(-24(i+1)). + * + * External function: + * double scalbn(), floor(); + * + * + * Here is the description of some local variables: + * + * jk jk+1 is the initial number of terms of ipio2[] needed + * in the computation. The recommended value is 2,3,4, + * 6 for single, double, extended,and quad. + * + * jz local integer variable indicating the number of + * terms of ipio2[] used. + * + * jx nx - 1 + * + * jv index for pointing to the suitable ipio2[] for the + * computation. In general, we want + * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 + * is an integer. Thus + * e0-3-24*jv >= 0 or (e0-3)/24 >= jv + * Hence jv = max(0,(e0-3)/24). + * + * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. + * + * q[] double array with integral value, representing the + * 24-bits chunk of the product of x and 2/pi. + * + * q0 the corresponding exponent of q[0]. Note that the + * exponent for q[i] would be q0-24*i. + * + * PIo2[] double precision array, obtained by cutting pi/2 + * into 24 bits chunks. + * + * f[] ipio2[] in floating point + * + * iq[] integer array by breaking up q[] in 24-bits chunk. + * + * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] + * + * ih integer. If >0 it indicates q[] is >= 0.5, hence + * it also indicates the *sign* of the result. + * + */ + static class KernelRemPio2 { + /* + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + + private static final int init_jk[] = {2, 3, 4, 6}; // initial value for jk + + private static final double PIo2[] = { + 0x1.921fb4p0, // 1.57079625129699707031e+00 + 0x1.4442dp-24, // 7.54978941586159635335e-08 + 0x1.846988p-48, // 5.39030252995776476554e-15 + 0x1.8cc516p-72, // 3.28200341580791294123e-22 + 0x1.01b838p-96, // 1.27065575308067607349e-29 + 0x1.a25204p-120, // 1.22933308981111328932e-36 + 0x1.382228p-145, // 2.73370053816464559624e-44 + 0x1.9f31dp-169, // 2.16741683877804819444e-51 + }; + + static final double + twon24 = 0x1.0p-24; // 5.96046447753906250000e-08 + + static int __kernel_rem_pio2(double[] x, double[] y, int e0, int nx, int prec, final int[] ipio2) { + int jz, jx, jv, jp, jk, carry, n, i, j, k, m, q0, ih; + int[] iq = new int[20]; + double z,fw; + double [] f = new double[20]; + double [] fq= new double[20]; + double [] q = new double[20]; + + // initialize jk + jk = init_jk[prec]; + jp = jk; + + // determine jx, jv, q0, note that 3 > q0 + jx = nx - 1; + jv = (e0 - 3)/24; + if (jv < 0) { + jv = 0; + } + q0 = e0 - 24*(jv + 1); + + // set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] + j = jv - jx; + m = jx + jk; + for (i = 0; i <= m; i++, j++) { + f[i] = (j < 0) ? 0.0 : (double) ipio2[j]; + } + + // compute q[0],q[1],...q[jk] + for (i=0; i <= jk; i++) { + for(j = 0, fw = 0.0; j <= jx; j++) { + fw += x[j]*f[jx + i - j]; + } + q[i] = fw; + } + + jz = jk; + while(true) { + // distill q[] into iq[] reversingly + for(i=0, j=jz, z=q[jz]; j > 0; i++, j--) { + fw = (double)((int)(twon24* z)); + iq[i] = (int)(z - TWO24*fw); + z = q[j - 1] + fw; + } + + // compute n + z = Math.scalb(z, q0); // actual value of z + z -= 8.0*Math.floor(z*0.125); // trim off integer >= 8 + n = (int) z; + z -= (double)n; + ih = 0; + if (q0 > 0) { // need iq[jz - 1] to determine n + i = (iq[jz - 1] >> (24 - q0)); + n += i; + iq[jz - 1] -= i << (24 - q0); + ih = iq[jz - 1] >> (23 - q0); + } else if (q0 == 0) { + ih = iq[jz-1]>>23; + } else if (z >= 0.5) { + ih=2; + } + + if (ih > 0) { // q > 0.5 + n += 1; + carry = 0; + for (i=0; i < jz; i++) { // compute 1-q + j = iq[i]; + if (carry == 0) { + if (j != 0) { + carry = 1; + iq[i] = 0x100_0000 - j; + } + } else { + iq[i] = 0xff_ffff - j; + } + } + if (q0 > 0) { // rare case: chance is 1 in 12 + switch(q0) { + case 1: + iq[jz-1] &= 0x7f_ffff; + break; + case 2: + iq[jz-1] &= 0x3f_ffff; + break; + } + } + if (ih == 2) { + z = 1.0 - z; + if (carry != 0) { + z -= Math.scalb(1.0, q0); + } + } + } + + // check if recomputation is needed + if (z == 0.0) { + j = 0; + for (i = jz - 1; i >= jk; i--) { + j |= iq[i]; + } + if (j == 0) { // need recomputation + for(k=1; iq[jk - k] == 0; k++); // k = no. of terms needed + + for(i = jz + 1; i <= jz + k; i++) { // add q[jz+1] to q[jz+k] + f[jx + i] = (double) ipio2[jv + i]; + for (j=0, fw = 0.0; j <= jx; j++) { + fw += x[j]*f[jx + i - j]; + } + q[i] = fw; + } + jz += k; + continue; + } else { + break; + } + } else { + break; + } + } + + // chop off zero terms + if (z == 0.0) { + jz -= 1; + q0 -= 24; + while (iq[jz] == 0) { + jz--; + q0-=24; + } + } else { // break z into 24-bit if necessary + z = Math.scalb(z, -q0); + if (z >= TWO24) { + fw = (double)((int)(twon24*z)); + iq[jz] = (int)(z - TWO24*fw); + jz += 1; + q0 += 24; + iq[jz] = (int) fw; + } else { + iq[jz] = (int) z; + } + } + + // convert integer "bit" chunk to floating-point value + fw = Math.scalb(1.0, q0); + for(i = jz; i>=0; i--) { + q[i] = fw*(double)iq[i]; + fw *= twon24; + } + + // compute PIo2[0,...,jp]*q[jz,...,0] + for(i = jz; i>=0; i--) { + for (fw = 0.0, k = 0; k <= jp && k <= jz-i; k++) { + fw += PIo2[k]*q[i + k]; + } + fq[jz - i] = fw; + } + + // compress fq[] into y[] + switch(prec) { + case 0: + fw = 0.0; + for (i = jz; i >=0; i--) { + fw += fq[i]; + } + y[0] = (ih == 0)? fw: -fw; + break; + case 1: + case 2: + fw = 0.0; + for (i = jz; i>=0; i--) { + fw += fq[i]; + } + y[0] = (ih == 0) ? fw: -fw; + fw = fq[0] - fw; + for (i = 1; i <= jz; i++) { + fw += fq[i]; + } + y[1] = (ih == 0)? fw: -fw; + break; + case 3: // painful + for (i = jz; i > 0; i--) { + fw = fq[i - 1] + fq[i]; + fq[i] += fq[i - 1] - fw; + fq[i - 1] = fw; + } + for (i = jz; i>1; i--) { + fw = fq[i - 1] + fq[i]; + fq[i] += fq[i - 1] - fw; + fq[i-1] = fw; + } + for (fw = 0.0, i = jz; i >= 2; i--) { + fw += fq[i]; + } + if (ih == 0) { + y[0] = fq[0]; + y[1] = fq[1]; + y[2] = fw; + } else { + y[0] = -fq[0]; + y[1] = -fq[1]; + y[2] = -fw; + } + } + return n & 7; + } + } + /** Returns the arcsine of x. * * Method : diff --git a/src/java.base/share/classes/java/lang/StrictMath.java b/src/java.base/share/classes/java/lang/StrictMath.java index 9e3fa9e0db6..0df14e41fb5 100644 --- a/src/java.base/share/classes/java/lang/StrictMath.java +++ b/src/java.base/share/classes/java/lang/StrictMath.java @@ -127,7 +127,9 @@ public final class StrictMath { * @param a an angle, in radians. * @return the sine of the argument. */ - public static native double sin(double a); + public static double sin(double a) { + return FdLibm.Sin.compute(a); + } /** * Returns the trigonometric cosine of an angle. Special cases: @@ -139,7 +141,9 @@ public final class StrictMath { * @param a an angle, in radians. * @return the cosine of the argument. */ - public static native double cos(double a); + public static double cos(double a) { + return FdLibm.Cos.compute(a); + } /** * Returns the trigonometric tangent of an angle. Special cases: @@ -151,7 +155,9 @@ public final class StrictMath { * @param a an angle, in radians. * @return the tangent of the argument. */ - public static native double tan(double a); + public static double tan(double a) { + return FdLibm.Tan.compute(a); + } /** * Returns the arc sine of a value; the returned angle is in the diff --git a/test/jdk/java/lang/Math/SinCosTests.java b/test/jdk/java/lang/Math/SinCosTests.java new file mode 100644 index 00000000000..0a8a803ba30 --- /dev/null +++ b/test/jdk/java/lang/Math/SinCosTests.java @@ -0,0 +1,129 @@ +/* + * Copyright (c) 2003, 2023, Oracle and/or its affiliates. All rights reserved. + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. + * + * This code is free software; you can redistribute it and/or modify it + * under the terms of the GNU General Public License version 2 only, as + * published by the Free Software Foundation. + * + * This code is distributed in the hope that it will be useful, but WITHOUT + * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or + * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License + * version 2 for more details (a copy is included in the LICENSE file that + * accompanied this code). + * + * You should have received a copy of the GNU General Public License version + * 2 along with this work; if not, write to the Free Software Foundation, + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. + * + * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA + * or visit www.oracle.com if you need additional information or have any + * questions. + */ + +/* + * @test + * @library /test/lib + * @build Tests + * @run main SinCosTests + * @bug 8302040 + * @summary Tests for {Math, StrictMath}.sqrt + */ + +public class SinCosTests { + private SinCosTests(){} + + public static void main(String... argv) { + int failures = 0; + + failures += testSin(); + failures += testCos(); + + if (failures > 0) { + System.err.println("Testing sin and cos incurred " + + failures + " failures."); + throw new RuntimeException(); + } + } + + private static final double InfinityD = Double.POSITIVE_INFINITY; + private static final double NaNd = Double.NaN; + + /** + * "Special cases: + * + * If the argument is NaN or an infinity, then the result is NaN. + * + * If the argument is zero, then the result is a zero with the + * same sign as the argument." + */ + private static int testSin() { + int failures = 0; + + for(double nan : Tests.NaNs) { + failures += testSinCase(nan, NaNd); + } + + double [][] testCases = { + {+InfinityD, NaNd}, + {-InfinityD, NaNd}, + + {+0.0, +0.0}, + {-0.0, -0.0}, + + }; + + for(int i = 0; i < testCases.length; i++) { + failures += testSinCase(testCases[i][0], testCases[i][1]); + } + + return failures; + } + + /** + * "Special cases: + * + * If the argument is NaN or an infinity, then the result is NaN. + * If the argument is zero, then the result is 1.0." + */ + private static int testCos() { + int failures = 0; + + for(double nan : Tests.NaNs) { + failures += testCosCase(nan, NaNd); + } + + double [][] testCases = { + {+InfinityD, NaNd}, + {-InfinityD, NaNd}, + + {+0.0, +1.0}, + {-0.0, +1.0}, + + }; + + for(int i = 0; i < testCases.length; i++) { + failures += testCosCase(testCases[i][0], testCases[i][1]); + } + + return failures; + } + + private static int testSinCase(double input, double expected) { + int failures=0; + + failures+=Tests.test("Math.sin", input, Math::sin, expected); + failures+=Tests.test("StrictMath.sin", input, StrictMath::sin, expected); + + return failures; + } + + private static int testCosCase(double input, double expected) { + int failures=0; + + failures+=Tests.test("Math.cos", input, Math::cos, expected); + failures+=Tests.test("StrictMath.cos", input, StrictMath::cos, expected); + + return failures; + } +} diff --git a/test/jdk/java/lang/Math/TanTests.java b/test/jdk/java/lang/Math/TanTests.java index 04ed7d47ed1..cafa284c3ec 100644 --- a/test/jdk/java/lang/Math/TanTests.java +++ b/test/jdk/java/lang/Math/TanTests.java @@ -23,21 +23,74 @@ /* * @test - * @bug 5033578 + * @bug 5033578 8302027 + * @build Tests + * @run main TanTests * @summary Tests for {Math, StrictMath}.tan */ public class TanTests { private TanTests(){} - static int testTanCase(double input, double expected, double ulps) { + private static final double NaNd = Double.NaN; + private static final double InfinityD = Double.POSITIVE_INFINITY; + + public static void main(String... argv) { + int failures = 0; + + failures += testTanNaN(); + failures += testTanCardinal(); + failures += testTan(); + + if (failures > 0) { + System.err.println("Testing tan incurred " + + failures + " failures."); + throw new RuntimeException(); + } + } + + private static int testTanCase(double input, double expected, double ulps) { int failures = 0; failures += Tests.testUlpDiff("StrictMath.tan", input, StrictMath::tan, expected, ulps); failures += Tests.testUlpDiff("Math.tan", input, Math::tan, expected, ulps); return failures; } - static int testTan() { + private static int testTanNaN() { + int failures = 0; + + // "If the argument is NaN or an infinity, then the result is NaN." + for(double nan : Tests.NaNs) { + failures += Tests.test("StrictMath.tan", nan, StrictMath::tan, NaNd); + failures += Tests.test("Math.tan", nan, Math::tan, NaNd); + } + + return failures; + } + + private static int testTanCardinal() { + int failures = 0; + + double [][] testCases = { + // "If the argument is NaN or an infinity, then the result is NaN." + { InfinityD, NaNd}, + {-InfinityD, NaNd}, + + // "If the argument is zero, then the result is a zero + // with the same sign as the argument." + {-0.0, -0.0}, + {+0.0, +0.0}, + }; + + for(double[] testCase : testCases) { + failures += Tests.test("StrictMath.tan", testCase[0], StrictMath::tan, testCase[1]); + failures += Tests.test("Math.tan", testCase[0], Math::tan, testCase[1]); + } + + return failures; + } + + private static int testTan() { int failures = 0; double [][] testCases = { @@ -169,16 +222,4 @@ public class TanTests { return failures; } - - public static void main(String... argv) { - int failures = 0; - - failures += testTan(); - - if (failures > 0) { - System.err.println("Testing tan incurred " - + failures + " failures."); - throw new RuntimeException(); - } - } } diff --git a/test/jdk/java/lang/StrictMath/ExhaustingTests.java b/test/jdk/java/lang/StrictMath/ExhaustingTests.java index a6bb47e1402..e7657306a3c 100644 --- a/test/jdk/java/lang/StrictMath/ExhaustingTests.java +++ b/test/jdk/java/lang/StrictMath/ExhaustingTests.java @@ -23,7 +23,7 @@ /* * @test - * @bug 8301833 8302026 8301444 8302028 8302040 + * @bug 8301833 8302026 8301444 8302028 8302040 8302027 * @build Tests * @build FdlibmTranslit * @build ExhaustingTests @@ -84,9 +84,9 @@ public class ExhaustingTests { new UnaryTestCase("cosh", FdlibmTranslit::cosh, StrictMath::cosh, DEFAULT_SHIFT), new UnaryTestCase("tanh", FdlibmTranslit::tanh, StrictMath::tanh, DEFAULT_SHIFT), - // new UnaryTestCase("sin", FdlibmTranslit::sin, StrictMath::sin, DEFAULT_SHIFT), - // new UnaryTestCase("cos", FdlibmTranslit::cos, StrictMath::cos, DEFAULT_SHIFT), - // new UnaryTestCase("tan", FdlibmTranslit::tan, StrictMath::tan, DEFAULT_SHIFT), + new UnaryTestCase("sin", FdlibmTranslit::sin, StrictMath::sin, DEFAULT_SHIFT), + new UnaryTestCase("cos", FdlibmTranslit::cos, StrictMath::cos, DEFAULT_SHIFT), + new UnaryTestCase("tan", FdlibmTranslit::tan, StrictMath::tan, DEFAULT_SHIFT), new UnaryTestCase("asin", FdlibmTranslit::asin, StrictMath::asin, DEFAULT_SHIFT), new UnaryTestCase("acos", FdlibmTranslit::acos, StrictMath::acos, DEFAULT_SHIFT), diff --git a/test/jdk/java/lang/StrictMath/FdlibmTranslit.java b/test/jdk/java/lang/StrictMath/FdlibmTranslit.java index b0c79c145ae..ed3c9840201 100644 --- a/test/jdk/java/lang/StrictMath/FdlibmTranslit.java +++ b/test/jdk/java/lang/StrictMath/FdlibmTranslit.java @@ -70,6 +70,18 @@ public class FdlibmTranslit { ( ((long)high)) << 32 ); } + public static double sin(double x) { + return Sin.compute(x); + } + + public static double cos(double x) { + return Cos.compute(x); + } + + public static double tan(double x) { + return Tan.compute(x); + } + public static double asin(double x) { return Asin.compute(x); } @@ -130,6 +142,866 @@ public class FdlibmTranslit { return Tanh.compute(x); } + // ----------------------------------------------------------------------------------------- + + /** sin(x) + * Return sine function of x. + * + * kernel function: + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cose function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Sin { + static double compute(double x) { + double[] y = new double[2]; + double z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); + + /* sin(Inf or NaN) is NaN */ + else if (ix>=0x7ff00000) return x-x; + + /* argument reduction needed */ + else { + n = RemPio2.__ieee754_rem_pio2(x,y); + switch(n&3) { + case 0: return Sin.__kernel_sin(y[0],y[1],1); + case 1: return Cos.__kernel_cos(y[0],y[1]); + case 2: return -Sin.__kernel_sin(y[0],y[1],1); + default: + return -Cos.__kernel_cos(y[0],y[1]); + } + } + } + + /** __kernel_sin( x, y, iy) + * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). + * + * Algorithm + * 1. Since sin(-x) = -sin(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. + * 3. sin(x) is approximated by a polynomial of degree 13 on + * [0,pi/4] + * 3 13 + * sin(x) ~ x + S1*x + ... + S6*x + * where + * + * |sin(x) 2 4 6 8 10 12 | -58 + * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 + * | x | + * + * 4. sin(x+y) = sin(x) + sin'(x')*y + * ~ sin(x) + (1-x*x/2)*y + * For better accuracy, let + * 3 2 2 2 2 + * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) + * then 3 2 + * sin(x) = x + (S1*x + (x *(r-y/2)+y)) + */ + private static final double + half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ + S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ + S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ + S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ + S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ + S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ + S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ + + static double __kernel_sin(double x, double y, int iy) { + double z,r,v; + int ix; + ix = __HI(x)&0x7fffffff; /* high word of x */ + if(ix<0x3e400000) /* |x| < 2**-27 */ + {if((int)x==0) return x;} /* generate inexact */ + z = x*x; + v = z*x; + r = S2+z*(S3+z*(S4+z*(S5+z*S6))); + if(iy==0) return x+v*(S1+z*r); + else return x-((z*(half*y-v*r)-y)-v*S1); + } + } + + + /** cos(x) + * Return cosine function of x. + * + * kernel function: + * __kernel_sin ... sine function on [-pi/4,pi/4] + * __kernel_cos ... cosine function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Cos { + static double compute(double x) { + double[] y = new double[2]; + double z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if(ix <= 0x3fe921fb) return __kernel_cos(x,z); + + /* cos(Inf or NaN) is NaN */ + else if (ix>=0x7ff00000) return x-x; + + /* argument reduction needed */ + else { + n = RemPio2.__ieee754_rem_pio2(x,y); + switch(n&3) { + case 0: return Cos.__kernel_cos(y[0],y[1]); + case 1: return -Sin.__kernel_sin(y[0],y[1],1); + case 2: return -Cos.__kernel_cos(y[0],y[1]); + default: + return Sin.__kernel_sin(y[0],y[1],1); + } + } + } + + /** + * __kernel_cos( x, y ) + * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * + * Algorithm + * 1. Since cos(-x) = cos(x), we need only to consider positive x. + * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. + * 3. cos(x) is approximated by a polynomial of degree 14 on + * [0,pi/4] + * 4 14 + * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x + * where the remez error is + * + * | 2 4 6 8 10 12 14 | -58 + * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 + * | | + * + * 4 6 8 10 12 14 + * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then + * cos(x) = 1 - x*x/2 + r + * since cos(x+y) ~ cos(x) - sin(x)*y + * ~ cos(x) - x*y, + * a correction term is necessary in cos(x) and hence + * cos(x+y) = 1 - (x*x/2 - (r - x*y)) + * For better accuracy when x > 0.3, let qx = |x|/4 with + * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. + * Then + * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). + * Note that 1-qx and (x*x/2-qx) is EXACT here, and the + * magnitude of the latter is at least a quarter of x*x/2, + * thus, reducing the rounding error in the subtraction. + */ + private static final double + one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ + C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ + C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ + C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ + C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ + C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ + C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ + + static double __kernel_cos(double x, double y) { + double a,hz,z,r,qx = 0.0; + int ix; + ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ + if(ix<0x3e400000) { /* if x < 2**27 */ + if(((int)x)==0) return one; /* generate inexact */ + } + z = x*x; + r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); + if(ix < 0x3FD33333) /* if |x| < 0.3 */ + return one - (0.5*z - (z*r - x*y)); + else { + if(ix > 0x3fe90000) { /* x > 0.78125 */ + qx = 0.28125; + } else { + //__HI(qx) = ix-0x00200000; /* x/4 */ + qx = __HI(qx, ix-0x00200000); + // __LO(qx) = 0; + qx = __LO(qx, 0); + } + hz = 0.5*z-qx; + a = one-qx; + return a - (hz - (z*r-x*y)); + } + } + } + + /** tan(x) + * Return tangent function of x. + * + * kernel function: + * __kernel_tan ... tangent function on [-pi/4,pi/4] + * __ieee754_rem_pio2 ... argument reduction routine + * + * Method. + * Let S,C and T denote the sin, cos and tan respectively on + * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 + * in [-pi/4 , +pi/4], and let n = k mod 4. + * We have + * + * n sin(x) cos(x) tan(x) + * ---------------------------------------------------------- + * 0 S C T + * 1 C -S -1/T + * 2 -S -C T + * 3 -C S -1/T + * ---------------------------------------------------------- + * + * Special cases: + * Let trig be any of sin, cos, or tan. + * trig(+-INF) is NaN, with signals; + * trig(NaN) is that NaN; + * + * Accuracy: + * TRIG(x) returns trig(x) nearly rounded + */ + static class Tan { + static double compute(double x) { + double[] y= new double[2]; + double z=0.0; + int n, ix; + + /* High word of x. */ + ix = __HI(x); + + /* |x| ~< pi/4 */ + ix &= 0x7fffffff; + if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); + + /* tan(Inf or NaN) is NaN */ + else if (ix>=0x7ff00000) return x-x; /* NaN */ + + /* argument reduction needed */ + else { + n = RemPio2.__ieee754_rem_pio2(x,y); + return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even + -1 -- n odd */ + } + } + + /** __kernel_tan( x, y, k ) + * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 + * Input x is assumed to be bounded by ~pi/4 in magnitude. + * Input y is the tail of x. + * Input k indicates whether tan (if k=1) or + * -1/tan (if k= -1) is returned. + * + * Algorithm + * 1. Since tan(-x) = -tan(x), we need only to consider positive x. + * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. + * 3. tan(x) is approximated by a odd polynomial of degree 27 on + * [0,0.67434] + * 3 27 + * tan(x) ~ x + T1*x + ... + T13*x + * where + * + * |tan(x) 2 4 26 | -59.2 + * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 + * | x | + * + * Note: tan(x+y) = tan(x) + tan'(x)*y + * ~ tan(x) + (1+x*x)*y + * Therefore, for better accuracy in computing tan(x+y), let + * 3 2 2 2 2 + * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) + * then + * 3 2 + * tan(x+y) = x + (T1*x + (x *(r+y)+y)) + * + * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then + * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) + * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) + */ + private static final double + one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ + pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ + pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ + T[] = { + 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ + 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ + 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ + 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ + 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ + 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ + 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ + 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ + 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ + 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ + 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ + -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ + 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ + }; + + static double __kernel_tan(double x, double y, int iy) { + double z,r,v,w,s; + int ix,hx; + hx = __HI(x); /* high word of x */ + ix = hx&0x7fffffff; /* high word of |x| */ + if(ix<0x3e300000) { /* x < 2**-28 */ + if((int)x==0) { /* generate inexact */ + if (((ix | __LO(x)) | (iy + 1)) == 0) + return one / Math.abs(x); + else { + if (iy == 1) + return x; + else { /* compute -1 / (x+y) carefully */ + double a, t; + + z = w = x + y; + // __LO(z) = 0; + z= __LO(z, 0); + v = y - (z - x); + t = a = -one / w; + //__LO(t) = 0; + t = __LO(t, 0); + s = one + t * z; + return t + a * (s + t * v); + } + } + } + } + if(ix>=0x3FE59428) { /* |x|>=0.6744 */ + if(hx<0) {x = -x; y = -y;} + z = pio4-x; + w = pio4lo-y; + x = z+w; y = 0.0; + } + z = x*x; + w = z*z; + /* Break x^5*(T[1]+x^2*T[2]+...) into + * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + + * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) + */ + r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); + v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); + s = z*x; + r = y + z*(s*(r+v)+y); + r += T[0]*s; + w = x+r; + if(ix>=0x3FE59428) { + v = (double)iy; + return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); + } + if(iy==1) return w; + else { /* if allow error up to 2 ulp, + simply return -1.0/(x+r) here */ + /* compute -1.0/(x+r) accurately */ + double a,t; + z = w; + // __LO(z) = 0; + z = __LO(z, 0); + v = r-(z - x); /* z+v = r+x */ + t = a = -1.0/w; /* a = -1.0/w */ + // __LO(t) = 0; + t = __LO(t, 0); + s = 1.0+t*z; + return t+a*(s+t*v); + } + } + } + + /** __ieee754_rem_pio2(x,y) + * + * return the remainder of x rem pi/2 in y[0]+y[1] + * use __kernel_rem_pio2() + */ + static class RemPio2 { + /* + * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi + */ + private static final int[] two_over_pi = { + 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, + 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, + 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, + 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, + 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, + 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, + 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, + 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, + 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, + 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, + 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, + }; + + private static final int[] npio2_hw = { + 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, + 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, + 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, + 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, + 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, + 0x404858EB, 0x404921FB, + }; + + /* + * invpio2: 53 bits of 2/pi + * pio2_1: first 33 bit of pi/2 + * pio2_1t: pi/2 - pio2_1 + * pio2_2: second 33 bit of pi/2 + * pio2_2t: pi/2 - (pio2_1+pio2_2) + * pio2_3: third 33 bit of pi/2 + * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) + */ + + private static final double + zero = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ + half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ + two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ + invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ + pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ + pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ + pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ + pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ + pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ + pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ + + static int __ieee754_rem_pio2(double x, double[] y) { + double z = 0.0,w,t,r,fn; + double[] tx = new double[3]; + int e0,i,j,nx,n,ix,hx; + + hx = __HI(x); /* high word of x */ + ix = hx&0x7fffffff; + if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ + {y[0] = x; y[1] = 0; return 0;} + if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ + if(hx>0) { + z = x - pio2_1; + if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ + y[0] = z - pio2_1t; + y[1] = (z-y[0])-pio2_1t; + } else { /* near pi/2, use 33+33+53 bit pi */ + z -= pio2_2; + y[0] = z - pio2_2t; + y[1] = (z-y[0])-pio2_2t; + } + return 1; + } else { /* negative x */ + z = x + pio2_1; + if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ + y[0] = z + pio2_1t; + y[1] = (z-y[0])+pio2_1t; + } else { /* near pi/2, use 33+33+53 bit pi */ + z += pio2_2; + y[0] = z + pio2_2t; + y[1] = (z-y[0])+pio2_2t; + } + return -1; + } + } + if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ + t = Math.abs(x); + n = (int) (t*invpio2+half); + fn = (double)n; + r = t-fn*pio2_1; + w = fn*pio2_1t; /* 1st round good to 85 bit */ + if(n<32&&ix!=npio2_hw[n-1]) { + y[0] = r-w; /* quick check no cancellation */ + } else { + j = ix>>20; + y[0] = r-w; + i = j-(((__HI(y[0]))>>20)&0x7ff); + if(i>16) { /* 2nd iteration needed, good to 118 */ + t = r; + w = fn*pio2_2; + r = t-w; + w = fn*pio2_2t-((t-r)-w); + y[0] = r-w; + i = j-(((__HI(y[0]))>>20)&0x7ff); + if(i>49) { /* 3rd iteration need, 151 bits acc */ + t = r; /* will cover all possible cases */ + w = fn*pio2_3; + r = t-w; + w = fn*pio2_3t-((t-r)-w); + y[0] = r-w; + } + } + } + y[1] = (r-y[0])-w; + if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} + else return n; + } + /* + * all other (large) arguments + */ + if(ix>=0x7ff00000) { /* x is inf or NaN */ + y[0]=y[1]=x-x; return 0; + } + /* set z = scalbn(|x|,ilogb(x)-23) */ + // __LO(z) = __LO(x); + z = __LO(z, __LO(x)); + e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ + // __HI(z) = ix - (e0<<20); + z = __HI(z, ix - (e0<<20)); + for(i=0;i<2;i++) { + tx[i] = (double)((int)(z)); + z = (z-tx[i])*two24; + } + tx[2] = z; + nx = 3; + while(tx[nx-1]==zero) nx--; /* skip zero term */ + n = KernelRemPio2.__kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); + if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} + return n; + } + + } + + /** + * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) + * double x[],y[]; int e0,nx,prec; int ipio2[]; + * + * __kernel_rem_pio2 return the last three digits of N with + * y = x - N*pi/2 + * so that |y| < pi/2. + * + * The method is to compute the integer (mod 8) and fraction parts of + * (2/pi)*x without doing the full multiplication. In general we + * skip the part of the product that are known to be a huge integer ( + * more accurately, = 0 mod 8 ). Thus the number of operations are + * independent of the exponent of the input. + * + * (2/pi) is represented by an array of 24-bit integers in ipio2[]. + * + * Input parameters: + * x[] The input value (must be positive) is broken into nx + * pieces of 24-bit integers in double precision format. + * x[i] will be the i-th 24 bit of x. The scaled exponent + * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 + * match x's up to 24 bits. + * + * Example of breaking a double positive z into x[0]+x[1]+x[2]: + * e0 = ilogb(z)-23 + * z = scalbn(z,-e0) + * for i = 0,1,2 + * x[i] = floor(z) + * z = (z-x[i])*2**24 + * + * + * y[] output result in an array of double precision numbers. + * The dimension of y[] is: + * 24-bit precision 1 + * 53-bit precision 2 + * 64-bit precision 2 + * 113-bit precision 3 + * The actual value is the sum of them. Thus for 113-bit + * precision, one may have to do something like: + * + * long double t,w,r_head, r_tail; + * t = (long double)y[2] + (long double)y[1]; + * w = (long double)y[0]; + * r_head = t+w; + * r_tail = w - (r_head - t); + * + * e0 The exponent of x[0] + * + * nx dimension of x[] + * + * prec an integer indicating the precision: + * 0 24 bits (single) + * 1 53 bits (double) + * 2 64 bits (extended) + * 3 113 bits (quad) + * + * ipio2[] + * integer array, contains the (24*i)-th to (24*i+23)-th + * bit of 2/pi after binary point. The corresponding + * floating value is + * + * ipio2[i] * 2^(-24(i+1)). + * + * External function: + * double scalbn(), floor(); + * + * + * Here is the description of some local variables: + * + * jk jk+1 is the initial number of terms of ipio2[] needed + * in the computation. The recommended value is 2,3,4, + * 6 for single, double, extended,and quad. + * + * jz local integer variable indicating the number of + * terms of ipio2[] used. + * + * jx nx - 1 + * + * jv index for pointing to the suitable ipio2[] for the + * computation. In general, we want + * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 + * is an integer. Thus + * e0-3-24*jv >= 0 or (e0-3)/24 >= jv + * Hence jv = max(0,(e0-3)/24). + * + * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. + * + * q[] double array with integral value, representing the + * 24-bits chunk of the product of x and 2/pi. + * + * q0 the corresponding exponent of q[0]. Note that the + * exponent for q[i] would be q0-24*i. + * + * PIo2[] double precision array, obtained by cutting pi/2 + * into 24 bits chunks. + * + * f[] ipio2[] in floating point + * + * iq[] integer array by breaking up q[] in 24-bits chunk. + * + * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] + * + * ih integer. If >0 it indicates q[] is >= 0.5, hence + * it also indicates the *sign* of the result. + * + */ + static class KernelRemPio2 { + /* + * Constants: + * The hexadecimal values are the intended ones for the following + * constants. The decimal values may be used, provided that the + * compiler will convert from decimal to binary accurately enough + * to produce the hexadecimal values shown. + */ + + private static final int[] init_jk = {2,3,4,6}; /* initial value for jk */ + + private static final double[] PIo2 = { + 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ + 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ + 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ + 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ + 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ + 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ + 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ + 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ + }; + static final double + zero = 0.0, + one = 1.0, + two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ + twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ + + static int __kernel_rem_pio2(double[] x, double[] y, int e0, int nx, int prec, final int[] ipio2) { + int jz,jx,jv,jp,jk,carry,n,i,j,k,m,q0,ih; + int[] iq = new int[20]; + double z,fw; + double [] f = new double[20]; + double [] fq= new double[20]; + double [] q = new double[20]; + + /* initialize jk*/ + jk = init_jk[prec]; + jp = jk; + + /* determine jx,jv,q0, note that 3>q0 */ + jx = nx-1; + jv = (e0-3)/24; if(jv<0) jv=0; + q0 = e0-24*(jv+1); + + /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ + j = jv-jx; m = jx+jk; + for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j]; + + /* compute q[0],q[1],...q[jk] */ + for (i=0;i<=jk;i++) { + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; + q[i] = fw; + } + + jz = jk; + /* + * Transliteration note: the FDLIBM C sources have a + * "recompute:" label at this point and a "goto + * recompute;" later on at the indicated point. This + * structure was replaced by wrapping the code in the + * while(true){...} loop below, replacing the goto with + * the continue to re-execute the loop and by adding + * breaks to exit the loop on the other control flow + * paths. + */ + while(true) { + /* distill q[] into iq[] reversingly */ + for(i=0,j=jz,z=q[jz];j>0;i++,j--) { + fw = (double)((int)(twon24* z)); + iq[i] = (int)(z-two24*fw); + z = q[j-1]+fw; + } + + /* compute n */ + z = Math.scalb(z,q0); /* actual value of z */ + z -= 8.0*Math.floor(z*0.125); /* trim off integer >= 8 */ + n = (int) z; + z -= (double)n; + ih = 0; + if(q0>0) { /* need iq[jz-1] to determine n */ + i = (iq[jz-1]>>(24-q0)); n += i; + iq[jz-1] -= i<<(24-q0); + ih = iq[jz-1]>>(23-q0); + } + else if(q0==0) ih = iq[jz-1]>>23; + else if(z>=0.5) ih=2; + + if(ih>0) { /* q > 0.5 */ + n += 1; carry = 0; + for(i=0;i0) { /* rare case: chance is 1 in 12 */ + switch(q0) { + case 1: + iq[jz-1] &= 0x7fffff; break; + case 2: + iq[jz-1] &= 0x3fffff; break; + } + } + if(ih==2) { + z = one - z; + if(carry!=0) z -= Math.scalb(one,q0); + } + } + + /* check if recomputation is needed */ + if(z==zero) { + j = 0; + for (i=jz-1;i>=jk;i--) j |= iq[i]; + if(j==0) { /* need recomputation */ + for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ + + for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ + f[jx+i] = (double) ipio2[jv+i]; + for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; + q[i] = fw; + } + jz += k; + // At this point "goto recompute;" in the original C sources. + continue; + } else { break;} + } else {break;} + } + + /* chop off zero terms */ + if(z==0.0) { + jz -= 1; q0 -= 24; + while(iq[jz]==0) { jz--; q0-=24;} + } else { /* break z into 24-bit if necessary */ + z = Math.scalb(z,-q0); + if(z>=two24) { + fw = (double)((int)(twon24*z)); + iq[jz] = (int)(z-two24*fw); + jz += 1; q0 += 24; + iq[jz] = (int) fw; + } else iq[jz] = (int) z ; + } + + /* convert integer "bit" chunk to floating-point value */ + fw = Math.scalb(one,q0); + for(i=jz;i>=0;i--) { + q[i] = fw*(double)iq[i]; fw*=twon24; + } + + /* compute PIo2[0,...,jp]*q[jz,...,0] */ + for(i=jz;i>=0;i--) { + for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; + fq[jz-i] = fw; + } + + /* compress fq[] into y[] */ + switch(prec) { + case 0: + fw = 0.0; + for (i=jz;i>=0;i--) fw += fq[i]; + y[0] = (ih==0)? fw: -fw; + break; + case 1: + case 2: + fw = 0.0; + for (i=jz;i>=0;i--) fw += fq[i]; + y[0] = (ih==0)? fw: -fw; + fw = fq[0]-fw; + for (i=1;i<=jz;i++) fw += fq[i]; + y[1] = (ih==0)? fw: -fw; + break; + case 3: /* painful */ + for (i=jz;i>0;i--) { + fw = fq[i-1]+fq[i]; + fq[i] += fq[i-1]-fw; + fq[i-1] = fw; + } + for (i=jz;i>1;i--) { + fw = fq[i-1]+fq[i]; + fq[i] += fq[i-1]-fw; + fq[i-1] = fw; + } + for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; + if(ih==0) { + y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; + } else { + y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; + } + } + return n&7; + } + } /** Returns the arcsine of x. * diff --git a/test/jdk/java/lang/StrictMath/TrigTests.java b/test/jdk/java/lang/StrictMath/TrigTests.java new file mode 100644 index 00000000000..369b1c2aef9 --- /dev/null +++ b/test/jdk/java/lang/StrictMath/TrigTests.java @@ -0,0 +1,318 @@ +/* + * Copyright (c) 2003, 2023, Oracle and/or its affiliates. All rights reserved. + * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. + * + * This code is free software; you can redistribute it and/or modify it + * under the terms of the GNU General Public License version 2 only, as + * published by the Free Software Foundation. + * + * This code is distributed in the hope that it will be useful, but WITHOUT + * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or + * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License + * version 2 for more details (a copy is included in the LICENSE file that + * accompanied this code). + * + * You should have received a copy of the GNU General Public License version + * 2 along with this work; if not, write to the Free Software Foundation, + * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. + * + * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA + * or visit www.oracle.com if you need additional information or have any + * questions. + */ +import jdk.test.lib.RandomFactory; +import java.util.function.DoubleUnaryOperator; + +/* + * @test + * @bug 8302027 + * @key randomness + * @library /test/lib + * @build jdk.test.lib.RandomFactory + * @build Tests + * @build FdlibmTranslit + * @build TrigTests + * @run main TrigTests + * @summary Tests for StrictMath.{sin, cos, tan} + */ + +/** + * The tests in ../Math/{TanTests.java, SinCosTests.java} test + * properties that should hold for any implementation of the trig + * functions sin, cos, and tan, including the FDLIBM-based ones + * required by the StrictMath class. Therefore, the test cases in + * ../Math/{TanTests.java, SinCosTests.java} are run against both the + * Math and StrictMath versions of the trig methods. The role of this + * test is to verify that the FDLIBM algorithms are being used by + * running golden file tests on values that may vary from one + * conforming implementation of the trig functions to another. + */ + +public class TrigTests { + private TrigTests(){} + + public static void main(String... args) { + int failures = 0; + + failures += testAgainstTranslitCommon(); + + failures += testAgainstTranslitSin(); + failures += testAgainstTranslitCos(); + failures += testAgainstTranslitTan(); + + if (failures > 0) { + System.err.println("Testing the trig functions incurred " + + failures + " failures."); + throw new RuntimeException(); + } + } + + /** + * Bundle together groups of testing methods. + */ + private static enum TrigTest { + SIN(TrigTests::testSinCase, FdlibmTranslit::sin), + COS(TrigTests::testCosCase, FdlibmTranslit::cos), + TAN(TrigTests::testTanCase, FdlibmTranslit::tan); + + private DoubleDoubleToInt testCase; + private DoubleUnaryOperator transliteration; + + TrigTest(DoubleDoubleToInt testCase, DoubleUnaryOperator transliteration) { + this.testCase = testCase; + this.transliteration = transliteration; + } + + public DoubleDoubleToInt testCase() {return testCase;} + public DoubleUnaryOperator transliteration() {return transliteration;} + } + + // Initialize shared random number generator + private static java.util.Random random = RandomFactory.getRandom(); + + /** + * Test against shared points of interest. + */ + private static int testAgainstTranslitCommon() { + int failures = 0; + double[] pointsOfInterest = { + Math.PI/4.0, + -Math.PI/4.0, + + Math.PI/2.0, + -Math.PI/2.0, + + 3.0*Math.PI/2.0, + -3.0*Math.PI/2.0, + + Math.PI, + -Math.PI, + + 2.0*Math.PI, + -2.0*Math.PI, + + Double.MIN_NORMAL, + 1.0, + Tests.createRandomDouble(random), + }; + + for (var testMethods : TrigTest.values()) { + for (double testPoint : pointsOfInterest) { + failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, testMethods); + } + } + + return failures; + } + + /** + * Test StrictMath.sin against transliteration port of sin. + */ + private static int testAgainstTranslitSin() { + int failures = 0; + + // Probe near decision points in the FDLIBM algorithm. + double[] decisionPoints = { + 0x1.0p-27, + -0x1.0p-27, + }; + + for (double testPoint : decisionPoints) { + failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.SIN); + } + + // Inputs where Math.sin and StrictMath.sin differ for at least + // one Math.sin implementation. + double [][] testCases = { + {0x1.00000006eeeefp-12, 0x1.ffffffb888889p-13}, + {0x1.00000006eeefp-12, 0x1.ffffffb88888bp-13}, + {0x1.00000006eeef1p-12, 0x1.ffffffb88888dp-13}, + {0x1.000000001bba2p-9, 0x1.ffffeaaae2633p-10}, + {0x1.000000000013p-1, 0x1.eaee8744b0806p-2}, + {0x1.0000000000012p0, 0x1.aed548f090d02p-1}, + {0x1.00000000004e1p9, 0x1.45b52f29ac36p-4}, + {0x1.00000000000cp10, -0x1.44ad26136ce5fp-3}, + {0x1.000000000020bp11, -0x1.4092047afcd2p-2}, + {0x1.0000000000003p12, -0x1.3074ea23314dep-1}, + {0x1.0000000000174p50, -0x1.54cd5e7e9e3d2p-1}, + {0x1.0000000000005p51, -0x1.8c35b0d728faep-2}, + {0x1.0000000000101p113, -0x1.69e9ed300b1dcp-1}, + {0x1.0000000000017p114, 0x1.f6b44aa2a1c9cp-1}, + {0x1.00000000001abp128, -0x1.ecaddc1136bb2p-1}, + {0x1.000000000001bp129, -0x1.682ccb977e4dp-1}, + {0x1.0p233, 0x1.7c54e75ed6077p-1}, + {0x1.00000000000fcp299, 0x1.78ad2fd7aef78p-1}, + {0x1.0000000000002p300, -0x1.1adaf3550facp-1}, + {0x1.00000000001afp1023, 0x1.d1c804ef2eeccp-1}, + }; + + for (double[] testCase: testCases) { + failures+=testSinCase(testCase[0], testCase[1]); + } + + return failures; + } + + /** + * Test StrictMath.cos against transliteration port of cos. + */ + private static int testAgainstTranslitCos() { + int failures = 0; + + // Probe near decision points in the FDLIBM algorithm. + double[] decisionPoints = { + 0x1.0p27, + -0x1.0p27, + + 0.78125, + -0.78125, + }; + + for (double testPoint : decisionPoints) { + failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.COS); + } + + // Inputs where Math.cos and StrictMath.cos differ for at least + // one Math.cos implementation. + double [][] testCases = { + {0x1.000000076aaa6p-10, 0x1.fffff00000147p-1}, + {0x1.000000002e4fbp-8, 0x1.ffff00001554fp-1}, + {0x1.0000000000318p-2, 0x1.f01549f7dee4p-1}, + {0x1.000000000011ep-1, 0x1.c1528065b7cc6p-1}, + {0x1.0000000000174p0, 0x1.14a280fb50419p-1}, + {0x1.0000000000019p1, -0x1.aa226575372bbp-2}, + {0x1.00000000018c9p9, -0x1.fe60f23b0016ap-1}, + {0x1.0000000000022p10, 0x1.f98669d7b18d6p-1}, + {0x1.0000000000281p11, 0x1.e6439428b217p-1}, + {0x1.0000000000001p12, 0x1.9ba4a85e6e173p-1}, + {0x1.0000000000211p20, 0x1.e33ad93554beep-1}, + {0x1.0000000000006p21, 0x1.9027223f77694p-1}, + {0x1.00000000000b8p95, 0x1.8315138968a66p-1}, + {0x1.0000000000043p96, 0x1.5b302d1c86cbcp-4}, + {0x1.000000000013ap127, -0x1.740d46d7821f4p-1}, + {0x1.0000000000002p128, -0x1.e050345cf2161p-1}, + {0x1.000000000014p299, 0x1.6c5f3c84352fep-1}, + {0x1.0000000000007p300, -0x1.55109bfdf1c5cp-1}, + {0x1.000000000010ep400, 0x1.e725637029938p-2}, + {0x1.0000000000007p401, 0x1.1f89e14e29ccep-1}, + {0x1.0p402, 0x1.be2d53c4560dcp-1}, + {0x1.000000000015fp1023, -0x1.2f2596c42735cp-1}, + }; + + for (double[] testCase: testCases) { + failures+=testCosCase(testCase[0], testCase[1]); + } + + return failures; + } + + /** + * Test StrictMath.tan against transliteration port of tan + */ + private static int testAgainstTranslitTan() { + int failures = 0; + + // Probe near decision points in the FDLIBM algorithm. + double[] decisionPoints = { + 0x1.0p-28, + -0x1.0p-28, + + 0.6744, + -0.6744, + }; + + for (double testPoint : decisionPoints) { + failures += testRangeMidpoint(testPoint, Math.ulp(testPoint), 1000, TrigTest.TAN); + } + + // Inputs where Math.tan and StrictMath.tan differ for at least + // one Math.tan implementation. + double [][] testCases = { + {0x1.00000002221fep-13, 0x1.0000001777753p-13}, + {0x1.0000000088859p-12, 0x1.00000055dddbp-12}, + {0x1.0000000008787p-10, 0x1.000005555defep-10}, + {0x1.0000000001423p-9, 0x1.0000155558b9ap-9}, + {0x1.00000000005d9p-2, 0x1.05785a43c529p-2}, + {0x1.000000000001fp-1, 0x1.17b4f5bf34772p-1}, + {0x1.000000000006ep0, 0x1.8eb245cbee51ep0}, + {0x1.0000000000032p1, -0x1.17af62e094fd7p1}, + {0x1.00000000006a7p9, -0x1.46be0efd0f8cp-4}, + {0x1.0p10, -0x1.48d5be43ada01p-3}, + {0x1.00000000000c3p32, 0x1.0ad3757181cbap-1}, + {0x1.0000000000005p33, 0x1.6e07fbf43d47p0}, + {0x1.0000000000124p127, -0x1.3baa73a93958p0}, + {0x1.000000000002p128, -0x1.bf05a77a8df0cp-1}, + {0x1.000000000011cp299, 0x1.8a6f42eaa3d1fp0}, + {0x1.000000000001cp300, -0x1.b30fc9f73002cp-1}, + {0x1.0000000000013p500, -0x1.c4e46751be12cp-1}, + {0x1.00000000000ep1023, -0x1.d52c4ec04f108p-2} + }; + + for (double[] testCase: testCases) { + failures+=testTanCase(testCase[0], testCase[1]); + } + + return failures; + } + + private interface DoubleDoubleToInt { + int apply(double x, double y); + } + + private static int testRange(double start, double increment, int count, + TrigTest testMethods) { + int failures = 0; + double x = start; + for (int i = 0; i < count; i++, x += increment) { + failures += + testMethods.testCase().apply(x, testMethods.transliteration().applyAsDouble(x)); + } + return failures; + } + + private static int testRangeMidpoint(double midpoint, double increment, int count, + TrigTest testMethods) { + int failures = 0; + double x = midpoint - increment*(count / 2) ; + for (int i = 0; i < count; i++, x += increment) { + failures += + testMethods.testCase().apply(x, testMethods.transliteration().applyAsDouble(x)); + } + return failures; + } + + private static int testSinCase(double input, double expected) { + return Tests.test("StrictMath.sin(double)", input, + StrictMath::sin, expected); + } + + private static int testCosCase(double input, double expected) { + return Tests.test("StrictMath.cos(double)", input, + StrictMath::cos, expected); + } + + private static int testTanCase(double input, double expected) { + return Tests.test("StrictMath.tan(double)", input, + StrictMath::tan, expected); + } +}