/*
* Copyright (c) 1998, 2016, 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. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
* by Oracle in the LICENSE file that accompanied this code.
*
* 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.
*/
/**
* A transliteration of the "Freely Distributable Math Library"
* algorithms from C into Java. That is, this port of the algorithms
* is as close to the C originals as possible while still being
* readable legal Java.
*/
public class FdlibmTranslit {
private FdlibmTranslit() {
throw new UnsupportedOperationException("No FdLibmTranslit instances for you.");
}
/**
* Return the low-order 32 bits of the double argument as an int.
*/
private static int __LO(double x) {
long transducer = Double.doubleToRawLongBits(x);
return (int)transducer;
}
/**
* Return a double with its low-order bits of the second argument
* and the high-order bits of the first argument..
*/
private static double __LO(double x, int low) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0xFFFF_FFFF_0000_0000L) |
(low & 0x0000_0000_FFFF_FFFFL));
}
/**
* Return the high-order 32 bits of the double argument as an int.
*/
private static int __HI(double x) {
long transducer = Double.doubleToRawLongBits(x);
return (int)(transducer >> 32);
}
/**
* Return a double with its high-order bits of the second argument
* and the low-order bits of the first argument..
*/
private static double __HI(double x, int high) {
long transX = Double.doubleToRawLongBits(x);
return Double.longBitsToDouble((transX & 0x0000_0000_FFFF_FFFFL) |
( ((long)high)) << 32 );
}
public static double hypot(double x, double y) {
return Hypot.compute(x, y);
}
/**
* cbrt(x)
* Return cube root of x
*/
public static class Cbrt {
// unsigned
private static final int B1 = 715094163; /* B1 = (682-0.03306235651)*2**20 */
private static final int B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
private static final double C = 5.42857142857142815906e-01; /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
private static final double D = -7.05306122448979611050e-01; /* -864/1225 = 0xBFE691DE, 0x2532C834 */
private static final double E = 1.41428571428571436819e+00; /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
private static final double F = 1.60714285714285720630e+00; /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
private static final double G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
public static strictfp double compute(double x) {
int hx;
double r, s, t=0.0, w;
int sign; // unsigned
hx = __HI(x); // high word of x
sign = hx & 0x80000000; // sign= sign(x)
hx ^= sign;
if (hx >= 0x7ff00000)
return (x+x); // cbrt(NaN,INF) is itself
if ((hx | __LO(x)) == 0)
return(x); // cbrt(0) is itself
x = __HI(x, hx); // x <- |x|
// rough cbrt to 5 bits
if (hx < 0x00100000) { // subnormal number
t = __HI(t, 0x43500000); // set t= 2**54
t *= x;
t = __HI(t, __HI(t)/3+B2);
} else {
t = __HI(t, hx/3+B1);
}
// new cbrt to 23 bits, may be implemented in single precision
r = t * t/x;
s = C + r*t;
t *= G + F/(s + E + D/s);
// chopped to 20 bits and make it larger than cbrt(x)
t = __LO(t, 0);
t = __HI(t, __HI(t)+0x00000001);
// one step newton iteration to 53 bits with error less than 0.667 ulps
s = t * t; // t*t is exact
r = x / s;
w = t + t;
r= (r - t)/(w + r); // r-s is exact
t= t + t*r;
// retore the sign bit
t = __HI(t, __HI(t) | sign);
return(t);
}
}
/**
* hypot(x,y)
*
* Method :
* If (assume round-to-nearest) z = x*x + y*y
* has error less than sqrt(2)/2 ulp, than
* sqrt(z) has error less than 1 ulp (exercise).
*
* So, compute sqrt(x*x + y*y) with some care as
* follows to get the error below 1 ulp:
*
* Assume x > y > 0;
* (if possible, set rounding to round-to-nearest)
* 1. if x > 2y use
* x1*x1 + (y*y + (x2*(x + x1))) for x*x + y*y
* where x1 = x with lower 32 bits cleared, x2 = x - x1; else
* 2. if x <= 2y use
* t1*y1 + ((x-y) * (x-y) + (t1*y2 + t2*y))
* where t1 = 2x with lower 32 bits cleared, t2 = 2x - t1,
* y1= y with lower 32 bits chopped, y2 = y - y1.
*
* NOTE: scaling may be necessary if some argument is too
* large or too tiny
*
* Special cases:
* hypot(x,y) is INF if x or y is +INF or -INF; else
* hypot(x,y) is NAN if x or y is NAN.
*
* Accuracy:
* hypot(x,y) returns sqrt(x^2 + y^2) with error less
* than 1 ulps (units in the last place)
*/
static class Hypot {
public static double compute(double x, double y) {
double a = x;
double b = y;
double t1, t2, y1, y2, w;
int j, k, ha, hb;
ha = __HI(x) & 0x7fffffff; // high word of x
hb = __HI(y) & 0x7fffffff; // high word of y
if(hb > ha) {
a = y;
b = x;
j = ha;
ha = hb;
hb = j;
} else {
a = x;
b = y;
}
a = __HI(a, ha); // a <- |a|
b = __HI(b, hb); // b <- |b|
if ((ha - hb) > 0x3c00000) {
return a + b; // x / y > 2**60
}
k=0;
if (ha > 0x5f300000) { // a>2**500
if (ha >= 0x7ff00000) { // Inf or NaN
w = a + b; // for sNaN
if (((ha & 0xfffff) | __LO(a)) == 0)
w = a;
if (((hb ^ 0x7ff00000) | __LO(b)) == 0)
w = b;
return w;
}
// scale a and b by 2**-600
ha -= 0x25800000;
hb -= 0x25800000;
k += 600;
a = __HI(a, ha);
b = __HI(b, hb);
}
if (hb < 0x20b00000) { // b < 2**-500
if (hb <= 0x000fffff) { // subnormal b or 0 */
if ((hb | (__LO(b))) == 0)
return a;
t1 = 0;
t1 = __HI(t1, 0x7fd00000); // t1=2^1022
b *= t1;
a *= t1;
k -= 1022;
} else { // scale a and b by 2^600
ha += 0x25800000; // a *= 2^600
hb += 0x25800000; // b *= 2^600
k -= 600;
a = __HI(a, ha);
b = __HI(b, hb);
}
}
// medium size a and b
w = a - b;
if (w > b) {
t1 = 0;
t1 = __HI(t1, ha);
t2 = a - t1;
w = Math.sqrt(t1*t1 - (b*(-b) - t2 * (a + t1)));
} else {
a = a + a;
y1 = 0;
y1 = __HI(y1, hb);
y2 = b - y1;
t1 = 0;
t1 = __HI(t1, ha + 0x00100000);
t2 = a - t1;
w = Math.sqrt(t1*y1 - (w*(-w) - (t1*y2 + t2*b)));
}
if (k != 0) {
t1 = 1.0;
int t1_hi = __HI(t1);
t1_hi += (k << 20);
t1 = __HI(t1, t1_hi);
return t1 * w;
} else
return w;
}
}
/**
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Reme algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | <= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + -------
* R - r
* r*R1(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - R1(r)
* where
* 2 4 10
* R1(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
* if x > 7.09782712893383973096e+02 then exp(x) overflow
* if x < -7.45133219101941108420e+02 then exp(x) underflow
*
* 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.
*/
static class Exp {
private static final double one = 1.0;
private static final double[] halF = {0.5,-0.5,};
private static final double huge = 1.0e+300;
private static final double twom1000= 9.33263618503218878990e-302; /* 2**-1000=0x01700000,0*/
private static final double o_threshold= 7.09782712893383973096e+02; /* 0x40862E42, 0xFEFA39EF */
private static final double u_threshold= -7.45133219101941108420e+02; /* 0xc0874910, 0xD52D3051 */
private static final double[] ln2HI ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-6.93147180369123816490e-01}; /* 0xbfe62e42, 0xfee00000 */
private static final double[] ln2LO ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-1.90821492927058770002e-10,}; /* 0xbdea39ef, 0x35793c76 */
private static final double invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
private static final double P1 = 1.66666666666666019037e-01; /* 0x3FC55555, 0x5555553E */
private static final double P2 = -2.77777777770155933842e-03; /* 0xBF66C16C, 0x16BEBD93 */
private static final double P3 = 6.61375632143793436117e-05; /* 0x3F11566A, 0xAF25DE2C */
private static final double P4 = -1.65339022054652515390e-06; /* 0xBEBBBD41, 0xC5D26BF1 */
private static final double P5 = 4.13813679705723846039e-08; /* 0x3E663769, 0x72BEA4D0 */
public static strictfp double compute(double x) {
double y,hi=0,lo=0,c,t;
int k=0,xsb;
/*unsigned*/ int hx;
hx = __HI(x); /* high word of x */
xsb = (hx>>31)&1; /* sign bit of x */
hx &= 0x7fffffff; /* high word of |x| */
/* filter out non-finite argument */
if(hx >= 0x40862E42) { /* if |x|>=709.78... */
if(hx>=0x7ff00000) {
if(((hx&0xfffff)|__LO(x))!=0)
return x+x; /* NaN */
else return (xsb==0)? x:0.0; /* exp(+-inf)={inf,0} */
}
if(x > o_threshold) return huge*huge; /* overflow */
if(x < u_threshold) return twom1000*twom1000; /* underflow */
}
/* argument reduction */
if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */
if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */
hi = x-ln2HI[xsb]; lo=ln2LO[xsb]; k = 1-xsb-xsb;
} else {
k = (int)(invln2*x+halF[xsb]);
t = k;
hi = x - t*ln2HI[0]; /* t*ln2HI is exact here */
lo = t*ln2LO[0];
}
x = hi - lo;
}
else if(hx < 0x3e300000) { /* when |x|<2**-28 */
if(huge+x>one) return one+x;/* trigger inexact */
}
else k = 0;
/* x is now in primary range */
t = x*x;
c = x - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
if(k==0) return one-((x*c)/(c-2.0)-x);
else y = one-((lo-(x*c)/(2.0-c))-hi);
if(k >= -1021) {
y = __HI(y, __HI(y) + (k<<20)); /* add k to y's exponent */
return y;
} else {
y = __HI(y, __HI(y) + ((k+1000)<<20));/* add k to y's exponent */
return y*twom1000;
}
}
}
}