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jdk/src/java.base/share/classes/java/lang/FdLibm.java
Joe Darcy fcaf871408 8302028: Port fdlibm atan2 to Java 
Reviewed-by: bpb
2023-02-22 22:49:59 +00:00

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/*
* Copyright (c) 1998, 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. 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.
*/
package java.lang;
/**
* Port of the "Freely Distributable Math Library", version 5.3, from
* C to Java.
*
* <p>The C version of fdlibm relied on the idiom of pointer aliasing
* a 64-bit double floating-point value as a two-element array of
* 32-bit integers and reading and writing the two halves of the
* double independently. This coding pattern was problematic to C
* optimizers and not directly expressible in Java. Therefore, rather
* than a memory level overlay, if portions of a double need to be
* operated on as integer values, the standard library methods for
* bitwise floating-point to integer conversion,
* Double.longBitsToDouble and Double.doubleToRawLongBits, are directly
* or indirectly used.
*
* <p>The C version of fdlibm also took some pains to signal the
* correct IEEE 754 exceptional conditions divide by zero, invalid,
* overflow and underflow. For example, overflow would be signaled by
* {@code huge * huge} where {@code huge} was a large constant that
* would overflow when squared. Since IEEE floating-point exceptional
* handling is not supported natively in the JVM, such coding patterns
* have been omitted from this port. For example, rather than {@code
* return huge * huge}, this port will use {@code return INFINITY}.
*
* <p>Various comparison and arithmetic operations in fdlibm could be
* done either based on the integer view of a value or directly on the
* floating-point representation. Which idiom is faster may depend on
* platform specific factors. However, for code clarity if no other
* reason, this port will favor expressing the semantics of those
* operations in terms of floating-point operations when convenient to
* do so.
*/
class FdLibm {
// Constants used by multiple algorithms
private static final double INFINITY = Double.POSITIVE_INFINITY;
private static final double TWO54 = 0x1.0p54; // 1.80143985094819840000e+16
private static final double HUGE = 1.0e+300;
private FdLibm() {
throw new UnsupportedOperationException("No FdLibm 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 );
}
/** Returns the arcsine of x.
*
* Method :
* Since asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
* we approximate asin(x) on [0,0.5] by
* asin(x) = x + x*x^2*R(x^2)
* where
* R(x^2) is a rational approximation of (asin(x)-x)/x^3
* and its remez error is bounded by
* |(asin(x)-x)/x^3 - R(x^2)| < 2^(-58.75)
*
* For x in [0.5,1]
* asin(x) = pi/2-2*asin(sqrt((1-x)/2))
* Let y = (1-x), z = y/2, s := sqrt(z), and pio2_hi+pio2_lo=pi/2;
* then for x>0.98
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio2_hi - (2*(s+s*z*R(z)) - pio2_lo)
* For x<=0.98, let pio4_hi = pio2_hi/2, then
* f = hi part of s;
* c = sqrt(z) - f = (z-f*f)/(s+f) ...f+c=sqrt(z)
* and
* asin(x) = pi/2 - 2*(s+s*z*R(z))
* = pio4_hi+(pio4-2s)-(2s*z*R(z)-pio2_lo)
* = pio4_hi+(pio4-2f)-(2s*z*R(z)-(pio2_lo+2c))
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
*/
static class Asin {
private Asin() {throw new UnsupportedOperationException();}
private static final double
pio2_hi = 0x1.921fb54442d18p0, // 1.57079632679489655800e+00
pio2_lo = 0x1.1a62633145c07p-54, // 6.12323399573676603587e-17
pio4_hi = 0x1.921fb54442d18p-1, // 7.85398163397448278999e-01
// coefficient for R(x^2)
pS0 = 0x1.5555555555555p-3, // 1.66666666666666657415e-01
pS1 = -0x1.4d61203eb6f7dp-2, // -3.25565818622400915405e-01
pS2 = 0x1.9c1550e884455p-3, // 2.01212532134862925881e-01
pS3 = -0x1.48228b5688f3bp-5, // -4.00555345006794114027e-02
pS4 = 0x1.9efe07501b288p-11, // 7.91534994289814532176e-04
pS5 = 0x1.23de10dfdf709p-15, // 3.47933107596021167570e-05
qS1 = -0x1.33a271c8a2d4bp1, // -2.40339491173441421878e+00
qS2 = 0x1.02ae59c598ac8p1, // 2.02094576023350569471e+00
qS3 = -0x1.6066c1b8d0159p-1, // -6.88283971605453293030e-01
qS4 = 0x1.3b8c5b12e9282p-4; // 7.70381505559019352791e-02
static double compute(double x) {
double t = 0, w, p, q, c, r, s;
int hx, ix;
hx = __HI(x);
ix = hx & 0x7fff_ffff;
if (ix >= 0x3ff0_0000) { // |x| >= 1
if(((ix - 0x3ff0_0000) | __LO(x)) == 0) {
// asin(1) = +-pi/2 with inexact
return x*pio2_hi + x*pio2_lo;
}
return (x - x)/(x - x); // asin(|x| > 1) is NaN
} else if (ix < 0x3fe0_0000) { // |x| < 0.5
if (ix < 0x3e40_0000) { // if |x| < 2**-27
if (HUGE + x > 1.0) {// return x with inexact if x != 0
return x;
}
} else {
t = x*x;
}
p = t*(pS0 + t*(pS1 + t*(pS2 + t*(pS3 + t*(pS4 + t*pS5)))));
q = 1.0 + t*(qS1 + t*(qS2 + t*(qS3 + t*qS4)));
w = p/q;
return x + x*w;
}
// 1 > |x| >= 0.5
w = 1.0 - Math.abs(x);
t = w*0.5;
p = t*(pS0 + t*(pS1 + t*(pS2 + t*(pS3 + t*(pS4 + t*pS5)))));
q = 1.0 + t*(qS1 + t*(qS2 + t*(qS3 + t*qS4)));
s = Math.sqrt(t);
if (ix >= 0x3FEF_3333) { // if |x| > 0.975
w = p/q;
t = pio2_hi - (2.0*(s + s*w) - pio2_lo);
} else {
w = s;
w = __LO(w, 0);
c = (t - w*w)/(s + w);
r = p/q;
p = 2.0*s*r - (pio2_lo - 2.0*c);
q = pio4_hi - 2.0*w;
t = pio4_hi - (p - q);
}
return (hx > 0) ? t : -t;
}
}
/** Returns the arccosine of x.
* Method :
* acos(x) = pi/2 - asin(x)
* acos(-x) = pi/2 + asin(x)
* For |x| <= 0.5
* acos(x) = pi/2 - (x + x*x^2*R(x^2)) (see asin.c)
* For x > 0.5
* acos(x) = pi/2 - (pi/2 - 2asin(sqrt((1-x)/2)))
* = 2asin(sqrt((1-x)/2))
* = 2s + 2s*z*R(z) ...z=(1-x)/2, s=sqrt(z)
* = 2f + (2c + 2s*z*R(z))
* where f=hi part of s, and c = (z-f*f)/(s+f) is the correction term
* for f so that f+c ~ sqrt(z).
* For x <- 0.5
* acos(x) = pi - 2asin(sqrt((1-|x|)/2))
* = pi - 0.5*(s+s*z*R(z)), where z=(1-|x|)/2,s=sqrt(z)
*
* Special cases:
* if x is NaN, return x itself;
* if |x|>1, return NaN with invalid signal.
*
* Function needed: sqrt
*/
static class Acos {
private Acos() {throw new UnsupportedOperationException();}
private static final double
pio2_hi = 0x1.921fb54442d18p0, // 1.57079632679489655800e+00
pio2_lo = 0x1.1a62633145c07p-54, // 6.12323399573676603587e-17
pS0 = 0x1.5555555555555p-3, // 1.66666666666666657415e-01
pS1 = -0x1.4d61203eb6f7dp-2, // -3.25565818622400915405e-01
pS2 = 0x1.9c1550e884455p-3, // 2.01212532134862925881e-01
pS3 = -0x1.48228b5688f3bp-5, // -4.00555345006794114027e-02
pS4 = 0x1.9efe07501b288p-11, // 7.91534994289814532176e-04
pS5 = 0x1.23de10dfdf709p-15, // 3.47933107596021167570e-05
qS1 = -0x1.33a271c8a2d4bp1, // -2.40339491173441421878e+00
qS2 = 0x1.02ae59c598ac8p1, // 2.02094576023350569471e+00
qS3 = -0x1.6066c1b8d0159p-1, // -6.88283971605453293030e-01
qS4 = 0x1.3b8c5b12e9282p-4; // 7.70381505559019352791e-02
static double compute(double x) {
double z, p, q, r, w, s, c, df;
int hx, ix;
hx = __HI(x);
ix = hx & 0x7fff_ffff;
if (ix >= 0x3ff0_0000) { // |x| >= 1
if (((ix - 0x3ff0_0000) | __LO(x)) == 0) { // |x| == 1
if (hx > 0) {// acos(1) = 0
return 0.0;
}else { // acos(-1)= pi
return Math.PI + 2.0*pio2_lo;
}
}
return (x-x)/(x-x); // acos(|x| > 1) is NaN
}
if (ix < 0x3fe0_0000) { // |x| < 0.5
if (ix <= 0x3c60_0000) { // if |x| < 2**-57
return pio2_hi + pio2_lo;
}
z = x*x;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p/q;
return pio2_hi - (x - (pio2_lo - x*r));
} else if (hx < 0) { // x < -0.5
z = (1.0 + x)*0.5;
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
s = Math.sqrt(z);
r = p/q;
w = r*s - pio2_lo;
return Math.PI - 2.0*(s+w);
} else { // x > 0.5
z = (1.0 - x)*0.5;
s = Math.sqrt(z);
df = s;
df = __LO(df, 0);
c = (z - df*df)/(s + df);
p = z*(pS0 + z*(pS1 + z*(pS2 + z*(pS3 + z*(pS4 + z*pS5)))));
q = 1.0 + z*(qS1 + z*(qS2 + z*(qS3 + z*qS4)));
r = p/q;
w = r*s + c;
return 2.0*(df + w);
}
}
}
/* Returns the arctangent of x.
* Method
* 1. Reduce x to positive by atan(x) = -atan(-x).
* 2. According to the integer k=4t+0.25 chopped, t=x, the argument
* is further reduced to one of the following intervals and the
* arctangent of t is evaluated by the corresponding formula:
*
* [0,7/16] atan(x) = t-t^3*(a1+t^2*(a2+...(a10+t^2*a11)...)
* [7/16,11/16] atan(x) = atan(1/2) + atan( (t-0.5)/(1+t/2) )
* [11/16.19/16] atan(x) = atan( 1 ) + atan( (t-1)/(1+t) )
* [19/16,39/16] atan(x) = atan(3/2) + atan( (t-1.5)/(1+1.5t) )
* [39/16,INF] atan(x) = atan(INF) + atan( -1/t )
*
* 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 Atan {
private Atan() {throw new UnsupportedOperationException();}
private static final double atanhi[] = {
0x1.dac670561bb4fp-2, // atan(0.5)hi 4.63647609000806093515e-01
0x1.921fb54442d18p-1, // atan(1.0)hi 7.85398163397448278999e-01
0x1.f730bd281f69bp-1, // atan(1.5)hi 9.82793723247329054082e-01
0x1.921fb54442d18p0, // atan(inf)hi 1.57079632679489655800e+00
};
private static final double atanlo[] = {
0x1.a2b7f222f65e2p-56, // atan(0.5)lo 2.26987774529616870924e-17
0x1.1a62633145c07p-55, // atan(1.0)lo 3.06161699786838301793e-17
0x1.007887af0cbbdp-56, // atan(1.5)lo 1.39033110312309984516e-17
0x1.1a62633145c07p-54, // atan(inf)lo 6.12323399573676603587e-17
};
private static final double aT[] = {
0x1.555555555550dp-2, // 3.33333333333329318027e-01
-0x1.999999998ebc4p-3, // -1.99999999998764832476e-01
0x1.24924920083ffp-3, // 1.42857142725034663711e-01
-0x1.c71c6fe231671p-4, // -1.11111104054623557880e-01
0x1.745cdc54c206ep-4, // 9.09088713343650656196e-02
-0x1.3b0f2af749a6dp-4, // -7.69187620504482999495e-02
0x1.10d66a0d03d51p-4, // 6.66107313738753120669e-02
-0x1.dde2d52defd9ap-5, // -5.83357013379057348645e-02
0x1.97b4b24760debp-5, // 4.97687799461593236017e-02
-0x1.2b4442c6a6c2fp-5, // -3.65315727442169155270e-02
0x1.0ad3ae322da11p-6, // 1.62858201153657823623e-02
};
static double compute(double x) {
double w, s1, s2, z;
int ix, hx, id;
hx = __HI(x);
ix = hx & 0x7fff_ffff;
if (ix >= 0x4410_0000) { // if |x| >= 2^66
if (ix > 0x7ff0_0000 ||
(ix == 0x7ff0_0000 && (__LO(x) != 0))) {
return x+x; // NaN
}
if (hx > 0) {
return atanhi[3] + atanlo[3];
} else {
return -atanhi[3] - atanlo[3];
}
} if (ix < 0x3fdc_0000) { // |x| < 0.4375
if (ix < 0x3e20_0000) { // |x| < 2^-29
if (HUGE + x > 1.0) { // raise inexact
return x;
}
}
id = -1;
} else {
x = Math.abs(x);
if (ix < 0x3ff3_0000) { // |x| < 1.1875
if (ix < 0x3fe60000) { // 7/16 <= |x| < 11/16
id = 0;
x = (2.0*x - 1.0)/(2.0 + x);
} else { // 11/16 <= |x| < 19/16
id = 1;
x = (x - 1.0)/(x + 1.0);
}
} else {
if (ix < 0x4003_8000) { // |x| < 2.4375
id = 2;
x = (x - 1.5)/(1.0 + 1.5*x);
} else { // 2.4375 <= |x| < 2^66
id = 3;
x = -1.0/x;
}
}
}
// end of argument reduction
z = x*x;
w = z*z;
// break sum from i=0 to 10 aT[i]z**(i+1) into odd and even poly
s1 = z*(aT[0] + w*(aT[2] + w*(aT[4] + w*(aT[6] + w*(aT[8] + w*aT[10])))));
s2 = w*(aT[1] + w*(aT[3] + w*(aT[5] + w*(aT[7] + w*aT[9]))));
if (id < 0) {
return x - x*(s1 + s2);
} else {
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return (hx < 0) ? -z: z;
}
}
}
/**
* Returns the angle theta from the conversion of rectangular
* coordinates (x, y) to polar coordinates (r, theta).
*
* Method :
* 1. Reduce y to positive by atan2(y,x)=-atan2(-y,x).
* 2. Reduce x to positive by (if x and y are unexceptional):
* ARG (x+iy) = arctan(y/x) ... if x > 0,
* ARG (x+iy) = pi - arctan[y/(-x)] ... if x < 0,
*
* Special cases:
*
* ATAN2((anything), NaN ) is NaN;
* ATAN2(NAN , (anything) ) is NaN;
* ATAN2(+-0, +(anything but NaN)) is +-0 ;
* ATAN2(+-0, -(anything but NaN)) is +-pi ;
* ATAN2(+-(anything but 0 and NaN), 0) is +-pi/2;
* ATAN2(+-(anything but INF and NaN), +INF) is +-0 ;
* ATAN2(+-(anything but INF and NaN), -INF) is +-pi;
* ATAN2(+-INF,+INF ) is +-pi/4 ;
* ATAN2(+-INF,-INF ) is +-3pi/4;
* ATAN2(+-INF, (anything but,0,NaN, and INF)) is +-pi/2;
*
* 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 Atan2 {
private Atan2() {throw new UnsupportedOperationException();}
private static final double
tiny = 1.0e-300,
pi_o_4 = 0x1.921fb54442d18p-1, // 7.8539816339744827900E-01
pi_o_2 = 0x1.921fb54442d18p0, // 1.5707963267948965580E+00
pi_lo = 0x1.1a62633145c07p-53; // 1.2246467991473531772E-16
static double compute(double y, double x) {
double z;
int k, m, hx, hy, ix, iy;
/*unsigned*/ int lx, ly;
hx = __HI(x);
ix = hx & 0x7fff_ffff;
lx = __LO(x);
hy = __HI(y);
iy = hy&0x7fff_ffff;
ly = __LO(y);
if (Double.isNaN(x) || Double.isNaN(y))
return x + y;
if (((hx - 0x3ff0_0000) | lx) == 0) // x = 1.0
return StrictMath.atan(y);
m = ((hy >> 31) & 1)|((hx >> 30) & 2); // 2*sign(x) + sign(y)
// when y = 0
if ((iy | ly) == 0) {
switch(m) {
case 0:
case 1: return y; // atan(+/-0, +anything) = +/-0
case 2: return Math.PI + tiny; // atan(+0, -anything) = pi
case 3: return -Math.PI - tiny; // atan(-0, -anything) = -pi
}
}
// when x = 0
if ((ix | lx) == 0) {
return (hy < 0)? -pi_o_2 - tiny : pi_o_2 + tiny;
}
// when x is INF
if (ix == 0x7ff0_0000) {
if (iy == 0x7ff0_0000) {
switch(m) {
case 0: return pi_o_4 + tiny; // atan(+INF, +INF)
case 1: return -pi_o_4 - tiny; // atan(-INF, +INF)
case 2: return 3.0*pi_o_4 + tiny; // atan(+INF, -INF)
case 3: return -3.0*pi_o_4 - tiny; // atan(-INF, -INF)
}
} else {
switch(m) {
case 0: return 0.0; // atan(+..., +INF)
case 1: return -0.0; // atan(-..., +INF)
case 2: return Math.PI + tiny; // atan(+..., -INF)
case 3: return -Math.PI - tiny; // atan(-..., -INF)
}
}
}
// when y is INF
if (iy == 0x7ff0_0000) {
return (hy < 0)? -pi_o_2 - tiny : pi_o_2 + tiny;
}
// compute y/x
k = (iy - ix) >> 20;
if (k > 60) { // |y/x| > 2**60
z = pi_o_2+0.5*pi_lo;
} else if (hx < 0 && k < -60) { // |y|/x < -2**60
z = 0.0;
} else { // safe to do y/x
z = StrictMath.atan(Math.abs(y/x));
}
switch (m) {
case 0: return z; // atan(+, +)
case 1: return -z; // atan(-, +)
case 2: return Math.PI - (z - pi_lo); // atan(+, -)
default: return (z - pi_lo) - Math.PI; // atan(-, -), case 3
}
}
}
/**
* 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 = 0x1.15f15f15f15f1p-1; // 19/35 ~= 5.42857142857142815906e-01
private static final double D = -0x1.691de2532c834p-1; // -864/1225 ~= 7.05306122448979611050e-01
private static final double E = 0x1.6a0ea0ea0ea0fp0; // 99/70 ~= 1.41428571428571436819e+00
private static final double F = 0x1.9b6db6db6db6ep0; // 45/28 ~= 1.60714285714285720630e+00
private static final double G = 0x1.6db6db6db6db7p-2; // 5/14 ~= 3.57142857142857150787e-01
private Cbrt() {
throw new UnsupportedOperationException();
}
public static double compute(double x) {
double t = 0.0;
double sign;
if (x == 0.0 || !Double.isFinite(x))
return x; // Handles signed zeros properly
sign = (x < 0.0) ? -1.0: 1.0;
x = Math.abs(x); // x <- |x|
// Rough cbrt to 5 bits
if (x < 0x1.0p-1022) { // subnormal number
t = 0x1.0p54; // set t= 2**54
t *= x;
t = __HI(t, __HI(t)/3 + B2);
} else {
int hx = __HI(x); // high word of x
t = __HI(t, hx/3 + B1);
}
// New cbrt to 23 bits, may be implemented in single precision
double r, s, w;
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;
// Restore the original sign bit
return sign * 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 ulp (unit in the last place)
*/
public static class Hypot {
public static final double TWO_MINUS_600 = 0x1.0p-600;
public static final double TWO_PLUS_600 = 0x1.0p+600;
private Hypot() {
throw new UnsupportedOperationException();
}
public static double compute(double x, double y) {
double a = Math.abs(x);
double b = Math.abs(y);
if (!Double.isFinite(a) || !Double.isFinite(b)) {
if (a == INFINITY || b == INFINITY)
return INFINITY;
else
return a + b; // Propagate NaN significand bits
}
if (b > a) {
double tmp = a;
a = b;
b = tmp;
}
assert a >= b;
// Doing bitwise conversion after screening for NaN allows
// the code to not worry about the possibility of
// "negative" NaN values.
// Note: the ha and hb variables are the high-order
// 32-bits of a and b stored as integer values. The ha and
// hb values are used first for a rough magnitude
// comparison of a and b and second for simulating higher
// precision by allowing a and b, respectively, to be
// decomposed into non-overlapping portions. Both of these
// uses could be eliminated. The magnitude comparison
// could be eliminated by extracting and comparing the
// exponents of a and b or just be performing a
// floating-point divide. Splitting a floating-point
// number into non-overlapping portions can be
// accomplished by judicious use of multiplies and
// additions. For details see T. J. Dekker, A Floating-Point
// Technique for Extending the Available Precision,
// Numerische Mathematik, vol. 18, 1971, pp.224-242 and
// subsequent work.
int ha = __HI(a); // high word of a
int hb = __HI(b); // high word of b
if ((ha - hb) > 0x3c00000) {
return a + b; // x / y > 2**60
}
int k = 0;
if (a > 0x1.00000_ffff_ffffp500) { // a > ~2**500
// scale a and b by 2**-600
ha -= 0x25800000;
hb -= 0x25800000;
a = a * TWO_MINUS_600;
b = b * TWO_MINUS_600;
k += 600;
}
double t1, t2;
if (b < 0x1.0p-500) { // b < 2**-500
if (b < Double.MIN_NORMAL) { // subnormal b or 0 */
if (b == 0.0)
return a;
t1 = 0x1.0p1022; // 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
a = a * TWO_PLUS_600;
b = b * TWO_PLUS_600;
k -= 600;
}
}
// medium size a and b
double 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 {
double y1, y2;
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) {
return Math.powerOfTwoD(k) * w;
} else
return w;
}
}
/**
* Compute x**y
* n
* Method: Let x = 2 * (1+f)
* 1. Compute and return log2(x) in two pieces:
* log2(x) = w1 + w2,
* where w1 has 53 - 24 = 29 bit trailing zeros.
* 2. Perform y*log2(x) = n+y' by simulating multi-precision
* arithmetic, where |y'| <= 0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 1. (anything) ** 0 is 1
* 2. (anything) ** 1 is itself
* 3. (anything) ** NAN is NAN
* 4. NAN ** (anything except 0) is NAN
* 5. +-(|x| > 1) ** +INF is +INF
* 6. +-(|x| > 1) ** -INF is +0
* 7. +-(|x| < 1) ** +INF is +0
* 8. +-(|x| < 1) ** -INF is +INF
* 9. +-1 ** +-INF is NAN
* 10. +0 ** (+anything except 0, NAN) is +0
* 11. -0 ** (+anything except 0, NAN, odd integer) is +0
* 12. +0 ** (-anything except 0, NAN) is +INF
* 13. -0 ** (-anything except 0, NAN, odd integer) is +INF
* 14. -0 ** (odd integer) = -( +0 ** (odd integer) )
* 15. +INF ** (+anything except 0,NAN) is +INF
* 16. +INF ** (-anything except 0,NAN) is +0
* 17. -INF ** (anything) = -0 ** (-anything)
* 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
* 19. (-anything except 0 and inf) ** (non-integer) is NAN
*
* Accuracy:
* pow(x,y) returns x**y nearly rounded. In particular
* pow(integer,integer)
* always returns the correct integer provided it is
* representable.
*/
public static class Pow {
private Pow() {
throw new UnsupportedOperationException();
}
public static double compute(final double x, final double y) {
double z;
double r, s, t, u, v, w;
int i, j, k, n;
// y == zero: x**0 = 1
if (y == 0.0)
return 1.0;
// +/-NaN return x + y to propagate NaN significands
if (Double.isNaN(x) || Double.isNaN(y))
return x + y;
final double y_abs = Math.abs(y);
double x_abs = Math.abs(x);
// Special values of y
if (y == 2.0) {
return x * x;
} else if (y == 0.5) {
if (x >= -Double.MAX_VALUE) // Handle x == -infinity later
return Math.sqrt(x + 0.0); // Add 0.0 to properly handle x == -0.0
} else if (y_abs == 1.0) { // y is +/-1
return (y == 1.0) ? x : 1.0 / x;
} else if (y_abs == INFINITY) { // y is +/-infinity
if (x_abs == 1.0)
return y - y; // inf**+/-1 is NaN
else if (x_abs > 1.0) // (|x| > 1)**+/-inf = inf, 0
return (y >= 0) ? y : 0.0;
else // (|x| < 1)**-/+inf = inf, 0
return (y < 0) ? -y : 0.0;
}
final int hx = __HI(x);
int ix = hx & 0x7fffffff;
/*
* When x < 0, determine if y is an odd integer:
* y_is_int = 0 ... y is not an integer
* y_is_int = 1 ... y is an odd int
* y_is_int = 2 ... y is an even int
*/
int y_is_int = 0;
if (hx < 0) {
if (y_abs >= 0x1.0p53) // |y| >= 2^53 = 9.007199254740992E15
y_is_int = 2; // y is an even integer since ulp(2^53) = 2.0
else if (y_abs >= 1.0) { // |y| >= 1.0
long y_abs_as_long = (long) y_abs;
if ( ((double) y_abs_as_long) == y_abs) {
y_is_int = 2 - (int)(y_abs_as_long & 0x1L);
}
}
}
// Special value of x
if (x_abs == 0.0 ||
x_abs == INFINITY ||
x_abs == 1.0) {
z = x_abs; // x is +/-0, +/-inf, +/-1
if (y < 0.0)
z = 1.0/z; // z = (1/|x|)
if (hx < 0) {
if (((ix - 0x3ff00000) | y_is_int) == 0) {
z = (z-z)/(z-z); // (-1)**non-int is NaN
} else if (y_is_int == 1)
z = -1.0 * z; // (x < 0)**odd = -(|x|**odd)
}
return z;
}
n = (hx >> 31) + 1;
// (x < 0)**(non-int) is NaN
if ((n | y_is_int) == 0)
return (x-x)/(x-x);
s = 1.0; // s (sign of result -ve**odd) = -1 else = 1
if ( (n | (y_is_int - 1)) == 0)
s = -1.0; // (-ve)**(odd int)
double p_h, p_l, t1, t2;
// |y| is huge
if (y_abs > 0x1.00000_ffff_ffffp31) { // if |y| > ~2**31
final double INV_LN2 = 0x1.7154_7652_b82fep0; // 1.44269504088896338700e+00 = 1/ln2
final double INV_LN2_H = 0x1.715476p0; // 1.44269502162933349609e+00 = 24 bits of 1/ln2
final double INV_LN2_L = 0x1.4ae0_bf85_ddf44p-26; // 1.92596299112661746887e-08 = 1/ln2 tail
// Over/underflow if x is not close to one
if (x_abs < 0x1.fffff_0000_0000p-1) // |x| < ~0.9999995231628418
return (y < 0.0) ? s * INFINITY : s * 0.0;
if (x_abs > 0x1.00000_ffff_ffffp0) // |x| > ~1.0
return (y > 0.0) ? s * INFINITY : s * 0.0;
/*
* now |1-x| is tiny <= 2**-20, sufficient to compute
* log(x) by x - x^2/2 + x^3/3 - x^4/4
*/
t = x_abs - 1.0; // t has 20 trailing zeros
w = (t * t) * (0.5 - t * (0.3333333333333333333333 - t * 0.25));
u = INV_LN2_H * t; // INV_LN2_H has 21 sig. bits
v = t * INV_LN2_L - w * INV_LN2;
t1 = u + v;
t1 =__LO(t1, 0);
t2 = v - (t1 - u);
} else {
final double CP = 0x1.ec70_9dc3_a03fdp-1; // 9.61796693925975554329e-01 = 2/(3ln2)
final double CP_H = 0x1.ec709ep-1; // 9.61796700954437255859e-01 = (float)cp
final double CP_L = -0x1.e2fe_0145_b01f5p-28; // -7.02846165095275826516e-09 = tail of CP_H
double z_h, z_l, ss, s2, s_h, s_l, t_h, t_l;
n = 0;
// Take care of subnormal numbers
if (ix < 0x00100000) {
x_abs *= 0x1.0p53; // 2^53 = 9007199254740992.0
n -= 53;
ix = __HI(x_abs);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
// Determine interval
ix = j | 0x3ff00000; // Normalize ix
if (j <= 0x3988E)
k = 0; // |x| <sqrt(3/2)
else if (j < 0xBB67A)
k = 1; // |x| <sqrt(3)
else {
k = 0;
n += 1;
ix -= 0x00100000;
}
x_abs = __HI(x_abs, ix);
// Compute ss = s_h + s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5)
final double BP[] = {1.0,
1.5};
final double DP_H[] = {0.0,
0x1.2b80_34p-1}; // 5.84962487220764160156e-01
final double DP_L[] = {0.0,
0x1.cfde_b43c_fd006p-27};// 1.35003920212974897128e-08
// Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
final double L1 = 0x1.3333_3333_33303p-1; // 5.99999999999994648725e-01
final double L2 = 0x1.b6db_6db6_fabffp-2; // 4.28571428578550184252e-01
final double L3 = 0x1.5555_5518_f264dp-2; // 3.33333329818377432918e-01
final double L4 = 0x1.1746_0a91_d4101p-2; // 2.72728123808534006489e-01
final double L5 = 0x1.d864_a93c_9db65p-3; // 2.30660745775561754067e-01
final double L6 = 0x1.a7e2_84a4_54eefp-3; // 2.06975017800338417784e-01
u = x_abs - BP[k]; // BP[0]=1.0, BP[1]=1.5
v = 1.0 / (x_abs + BP[k]);
ss = u * v;
s_h = ss;
s_h = __LO(s_h, 0);
// t_h=x_abs + BP[k] High
t_h = 0.0;
t_h = __HI(t_h, ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18) );
t_l = x_abs - (t_h - BP[k]);
s_l = v * ((u - s_h * t_h) - s_h * t_l);
// Compute log(x_abs)
s2 = ss * ss;
r = s2 * s2* (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6)))));
r += s_l * (s_h + ss);
s2 = s_h * s_h;
t_h = 3.0 + s2 + r;
t_h = __LO(t_h, 0);
t_l = r - ((t_h - 3.0) - s2);
// u+v = ss*(1+...)
u = s_h * t_h;
v = s_l * t_h + t_l * ss;
// 2/(3log2)*(ss + ...)
p_h = u + v;
p_h = __LO(p_h, 0);
p_l = v - (p_h - u);
z_h = CP_H * p_h; // CP_H + CP_L = 2/(3*log2)
z_l = CP_L * p_h + p_l * CP + DP_L[k];
// log2(x_abs) = (ss + ..)*2/(3*log2) = n + DP_H + z_h + z_l
t = (double)n;
t1 = (((z_h + z_l) + DP_H[k]) + t);
t1 = __LO(t1, 0);
t2 = z_l - (((t1 - t) - DP_H[k]) - z_h);
}
// Split up y into (y1 + y2) and compute (y1 + y2) * (t1 + t2)
double y1 = y;
y1 = __LO(y1, 0);
p_l = (y - y1) * t1 + y * t2;
p_h = y1 * t1;
z = p_l + p_h;
j = __HI(z);
i = __LO(z);
if (j >= 0x40900000) { // z >= 1024
if (((j - 0x40900000) | i)!=0) // if z > 1024
return s * INFINITY; // Overflow
else {
final double OVT = 8.0085662595372944372e-0017; // -(1024-log2(ovfl+.5ulp))
if (p_l + OVT > z - p_h)
return s * INFINITY; // Overflow
}
} else if ((j & 0x7fffffff) >= 0x4090cc00 ) { // z <= -1075
if (((j - 0xc090cc00) | i)!=0) // z < -1075
return s * 0.0; // Underflow
else {
if (p_l <= z - p_h)
return s * 0.0; // Underflow
}
}
/*
* Compute 2**(p_h+p_l)
*/
// Poly coefs for (3/2)*(log(x)-2s-2/3*s**3
final double P1 = 0x1.5555_5555_5553ep-3; // 1.66666666666666019037e-01
final double P2 = -0x1.6c16_c16b_ebd93p-9; // -2.77777777770155933842e-03
final double P3 = 0x1.1566_aaf2_5de2cp-14; // 6.61375632143793436117e-05
final double P4 = -0x1.bbd4_1c5d_26bf1p-20; // -1.65339022054652515390e-06
final double P5 = 0x1.6376_972b_ea4d0p-25; // 4.13813679705723846039e-08
final double LG2 = 0x1.62e4_2fef_a39efp-1; // 6.93147180559945286227e-01
final double LG2_H = 0x1.62e43p-1; // 6.93147182464599609375e-01
final double LG2_L = -0x1.05c6_10ca_86c39p-29; // -1.90465429995776804525e-09
i = j & 0x7fffffff;
k = (i >> 20) - 0x3ff;
n = 0;
if (i > 0x3fe00000) { // if |z| > 0.5, set n = [z + 0.5]
n = j + (0x00100000 >> (k + 1));
k = ((n & 0x7fffffff) >> 20) - 0x3ff; // new k for n
t = 0.0;
t = __HI(t, (n & ~(0x000fffff >> k)) );
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if (j < 0)
n = -n;
p_h -= t;
}
t = p_l + p_h;
t = __LO(t, 0);
u = t * LG2_H;
v = (p_l - (t - p_h)) * LG2 + t * LG2_L;
z = u + v;
w = v - (z - u);
t = z * z;
t1 = z - t * (P1 + t * (P2 + t * (P3 + t * (P4 + t * P5))));
r = (z * t1)/(t1 - 2.0) - (w + z * w);
z = 1.0 - (r - z);
j = __HI(z);
j += (n << 20);
if ((j >> 20) <= 0)
z = Math.scalb(z, n); // subnormal output
else {
int z_hi = __HI(z);
z_hi += (n << 20);
z = __HI(z, z_hi);
}
return s * z;
}
}
/**
* 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 final class Exp {
private Exp() {throw new UnsupportedOperationException();}
private static final double[] half = {0.5, -0.5,};
private static final double huge = 1.0e+300;
private static final double twom1000= 0x1.0p-1000; // 9.33263618503218878990e-302 = 2^-1000
private static final double o_threshold= 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
private static final double u_threshold= -0x1.74910d52d3051p9; // -7.45133219101941108420e+02;
private static final double[] ln2HI ={ 0x1.62e42feep-1, // 6.93147180369123816490e-01
-0x1.62e42feep-1}; // -6.93147180369123816490e-01
private static final double[] ln2LO ={ 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10
-0x1.a39ef35793c76p-33}; // -1.90821492927058770002e-10
private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
private static final double P1 = 0x1.555555555553ep-3; // 1.66666666666666019037e-01
private static final double P2 = -0x1.6c16c16bebd93p-9; // -2.77777777770155933842e-03
private static final double P3 = 0x1.1566aaf25de2cp-14; // 6.61375632143793436117e-05
private static final double P4 = -0x1.bbd41c5d26bf1p-20; // -1.65339022054652515390e-06
private static final double P5 = 0x1.6376972bea4d0p-25; // 4.13813679705723846039e-08
public static double compute(double x) {
double y;
double hi = 0.0;
double lo = 0.0;
double c;
double t;
int k = 0;
int 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) // unsigned compare needed here?
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 > 1.0)
return 1.0 + 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 1.0 - ((x*c)/(c - 2.0) - x);
else
y = 1.0 - ((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;
}
}
}
/**
* Return the (natural) logarithm of x
*
* Method :
* 1. Argument Reduction: find k and f such that
* x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* 2. Approximation of log(1+f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Reme algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s +Lg6*s +Lg7*s
* (the values of Lg1 to Lg7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lg1*s +...+Lg7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log(1+f) = f - s*(f - R) (if f is not too large)
* log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
*
* 3. Finally, log(x) = k*ln2 + log(1+f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log(x) is NaN with signal if x < 0 (including -INF) ;
* log(+INF) is +INF; log(0) is -INF with signal;
* log(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* 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 final class Log {
private Log() {throw new UnsupportedOperationException();}
private static final double
ln2_hi = 0x1.62e42feep-1, // 6.93147180369123816490e-01
ln2_lo = 0x1.a39ef35793c76p-33, // 1.90821492927058770002e-10
Lg1 = 0x1.5555555555593p-1, // 6.666666666666735130e-01
Lg2 = 0x1.999999997fa04p-2, // 3.999999999940941908e-01
Lg3 = 0x1.2492494229359p-2, // 2.857142874366239149e-01
Lg4 = 0x1.c71c51d8e78afp-3, // 2.222219843214978396e-01
Lg5 = 0x1.7466496cb03dep-3, // 1.818357216161805012e-01
Lg6 = 0x1.39a09d078c69fp-3, // 1.531383769920937332e-01
Lg7 = 0x1.2f112df3e5244p-3; // 1.479819860511658591e-01
private static final double zero = 0.0;
static double compute(double x) {
double hfsq, f, s, z, R, w, t1, t2, dk;
int k, hx, i, j;
/*unsigned*/ int lx;
hx = __HI(x); // high word of x
lx = __LO(x); // low word of x
k=0;
if (hx < 0x0010_0000) { // x < 2**-1022
if (((hx & 0x7fff_ffff) | lx) == 0) { // log(+-0) = -inf
return -TWO54/zero;
}
if (hx < 0) { // log(-#) = NaN
return (x - x)/zero;
}
k -= 54;
x *= TWO54; // subnormal number, scale up x
hx = __HI(x); // high word of x
}
if (hx >= 0x7ff0_0000) {
return x + x;
}
k += (hx >> 20) - 1023;
hx &= 0x000f_ffff;
i = (hx + 0x9_5f64) & 0x10_0000;
x =__HI(x, hx | (i ^ 0x3ff0_0000)); // normalize x or x/2
k += (i >> 20);
f = x - 1.0;
if ((0x000f_ffff & (2 + hx)) < 3) {// |f| < 2**-20
if (f == zero) {
if (k == 0) {
return zero;
} else {
dk = (double)k;
return dk*ln2_hi + dk*ln2_lo;
}
}
R = f*f*(0.5 - 0.33333333333333333*f);
if (k == 0) {
return f - R;
} else {
dk = (double)k;
return dk*ln2_hi - ((R - dk*ln2_lo) - f);
}
}
s = f/(2.0 + f);
dk = (double)k;
z = s*s;
i = hx - 0x6_147a;
w = z*z;
j = 0x6b851 - hx;
t1= w*(Lg2 + w*(Lg4 + w*Lg6));
t2= z*(Lg1 + w*(Lg3 + w*(Lg5 + w*Lg7)));
i |= j;
R = t2 + t1;
if (i > 0) {
hfsq = 0.5*f*f;
if (k == 0) {
return f-(hfsq - s*(hfsq + R));
} else {
return dk*ln2_hi - ((hfsq - (s*(hfsq + R) + dk*ln2_lo)) - f);
}
} else {
if (k == 0) {
return f - s*(f - R);
} else {
return dk*ln2_hi - ((s*(f - R) - dk*ln2_lo) - f);
}
}
}
}
/**
* Return the base 10 logarithm of x
*
* Method :
* Let log10_2hi = leading 40 bits of log10(2) and
* log10_2lo = log10(2) - log10_2hi,
* ivln10 = 1/log(10) rounded.
* Then
* n = ilogb(x),
* if(n<0) n = n+1;
* x = scalbn(x,-n);
* log10(x) := n*log10_2hi + (n*log10_2lo + ivln10*log(x))
*
* Note 1:
* To guarantee log10(10**n)=n, where 10**n is normal, the rounding
* mode must set to Round-to-Nearest.
* Note 2:
* [1/log(10)] rounded to 53 bits has error .198 ulps;
* log10 is monotonic at all binary break points.
*
* Special cases:
* log10(x) is NaN with signal if x < 0;
* log10(+INF) is +INF with no signal; log10(0) is -INF with signal;
* log10(NaN) is that NaN with no signal;
* log10(10**N) = N for N=0,1,...,22.
*
* 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 Log10 {
private static final double ivln10 = 0x1.bcb7b1526e50ep-2; // 4.34294481903251816668e-01
private static final double log10_2hi = 0x1.34413509f6p-2; // 3.01029995663611771306e-01;
private static final double log10_2lo = 0x1.9fef311f12b36p-42; // 3.69423907715893078616e-13;
private Log10() {
throw new UnsupportedOperationException();
}
public static double compute(double x) {
double y, z;
int i, k;
int hx = __HI(x); // high word of x
int lx = __LO(x); // low word of x
k=0;
if (hx < 0x0010_0000) { /* x < 2**-1022 */
if (((hx & 0x7fff_ffff) | lx) == 0) {
return -TWO54/0.0; /* log(+-0)=-inf */
}
if (hx < 0) {
return (x - x)/0.0; /* log(-#) = NaN */
}
k -= 54;
x *= TWO54; /* subnormal number, scale up x */
hx = __HI(x);
}
if (hx >= 0x7ff0_0000) {
return x + x;
}
k += (hx >> 20) - 1023;
i = (k & 0x8000_0000) >>> 31; // unsigned shift
hx = (hx & 0x000f_ffff) | ((0x3ff - i) << 20);
y = (double)(k + i);
x = __HI(x, hx); // replace high word of x with hx
z = y * log10_2lo + ivln10 * StrictMath.log(x);
return z + y * log10_2hi;
}
}
/**
* Returns the natural logarithm of the sum of the argument and 1.
*
* Method :
* 1. Argument Reduction: find k and f such that
* 1+x = 2^k * (1+f),
* where sqrt(2)/2 < 1+f < sqrt(2) .
*
* Note. If k=0, then f=x is exact. However, if k!=0, then f
* may not be representable exactly. In that case, a correction
* term is need. Let u=1+x rounded. Let c = (1+x)-u, then
* log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
* and add back the correction term c/u.
* (Note: when x > 2**53, one can simply return log(x))
*
* 2. Approximation of log1p(f).
* Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
* = 2s + 2/3 s**3 + 2/5 s**5 + .....,
* = 2s + s*R
* We use a special Reme algorithm on [0,0.1716] to generate
* a polynomial of degree 14 to approximate R The maximum error
* of this polynomial approximation is bounded by 2**-58.45. In
* other words,
* 2 4 6 8 10 12 14
* R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s +Lp6*s +Lp7*s
* (the values of Lp1 to Lp7 are listed in the program)
* and
* | 2 14 | -58.45
* | Lp1*s +...+Lp7*s - R(z) | <= 2
* | |
* Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
* In order to guarantee error in log below 1ulp, we compute log
* by
* log1p(f) = f - (hfsq - s*(hfsq+R)).
*
* 3. Finally, log1p(x) = k*ln2 + log1p(f).
* = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
* Here ln2 is split into two floating point number:
* ln2_hi + ln2_lo,
* where n*ln2_hi is always exact for |n| < 2000.
*
* Special cases:
* log1p(x) is NaN with signal if x < -1 (including -INF) ;
* log1p(+INF) is +INF; log1p(-1) is -INF with signal;
* log1p(NaN) is that NaN with no signal.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* 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.
*
* Note: Assuming log() return accurate answer, the following
* algorithm can be used to compute log1p(x) to within a few ULP:
*
* u = 1+x;
* if(u==1.0) return x ; else
* return log(u)*(x/(u-1.0));
*
* See HP-15C Advanced Functions Handbook, p.193.
*/
static class Log1p {
private static final double ln2_hi = 0x1.62e42feep-1; // 6.93147180369123816490e-01
private static final double ln2_lo = 0x1.a39ef35793c76p-33; // 1.90821492927058770002e-10
private static final double Lp1 = 0x1.5555555555593p-1; // 6.666666666666735130e-01
private static final double Lp2 = 0x1.999999997fa04p-2; // 3.999999999940941908e-01
private static final double Lp3 = 0x1.2492494229359p-2; // 2.857142874366239149e-01
private static final double Lp4 = 0x1.c71c51d8e78afp-3; // 2.222219843214978396e-01
private static final double Lp5 = 0x1.7466496cb03dep-3; // 1.818357216161805012e-01
private static final double Lp6 = 0x1.39a09d078c69fp-3; // 1.531383769920937332e-01
private static final double Lp7 = 0x1.2f112df3e5244p-3; // 1.479819860511658591e-01
public static double compute(double x) {
double hfsq, f=0, c=0, s, z, R, u;
int k, hx, hu=0, ax;
hx = __HI(x); /* high word of x */
ax = hx & 0x7fff_ffff;
k = 1;
if (hx < 0x3FDA_827A) { /* x < 0.41422 */
if (ax >= 0x3ff0_0000) { /* x <= -1.0 */
if (x == -1.0) /* log1p(-1)=-inf */
return -INFINITY;
else
return Double.NaN; /* log1p(x < -1) = NaN */
}
if (ax < 0x3e20_0000) { /* |x| < 2**-29 */
if (TWO54 + x > 0.0 /* raise inexact */
&& ax < 0x3c90_0000) /* |x| < 2**-54 */
return x;
else
return x - x*x*0.5;
}
if (hx > 0 || hx <= 0xbfd2_bec3) { /* -0.2929 < x < 0.41422 */
k=0;
f=x;
hu=1;
}
}
if (hx >= 0x7ff0_0000) {
return x + x;
}
if (k != 0) {
if (hx < 0x4340_0000) {
u = 1.0 + x;
hu = __HI(u); /* high word of u */
k = (hu >> 20) - 1023;
c = (k > 0)? 1.0 - (u-x) : x-(u-1.0); /* correction term */
c /= u;
} else {
u = x;
hu = __HI(u); /* high word of u */
k = (hu >> 20) - 1023;
c = 0;
}
hu &= 0x000f_ffff;
if (hu < 0x6_a09e) {
u = __HI(u, hu | 0x3ff0_0000); /* normalize u */
} else {
k += 1;
u = __HI(u, hu | 0x3fe0_0000); /* normalize u/2 */
hu = (0x0010_0000 - hu) >> 2;
}
f = u - 1.0;
}
hfsq = 0.5*f*f;
if (hu == 0) { /* |f| < 2**-20 */
if (f == 0.0) {
if (k == 0) {
return 0.0;
} else {
c += k * ln2_lo;
return k * ln2_hi + c;
}
}
R = hfsq * (1.0 - 0.66666666666666666*f);
if (k == 0) {
return f - R;
} else {
return k * ln2_hi - ((R-(k * ln2_lo+c)) - f);
}
}
s = f/(2.0 + f);
z = s * s;
R = z * (Lp1 + z * (Lp2 + z * (Lp3 + z * (Lp4 + z * (Lp5 + z * (Lp6 + z*Lp7))))));
if (k == 0) {
return f - (hfsq - s*(hfsq + R));
} else {
return k * ln2_hi - ((hfsq - (s*(hfsq + R) + (k * ln2_lo+c))) - f);
}
}
}
/* expm1(x)
* Returns exp(x)-1, the exponential of x minus 1.
*
* Method
* 1. Argument reduction:
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658
*
* Here a correction term c will be computed to compensate
* the error in r when rounded to a floating-point number.
*
* 2. Approximating expm1(r) by a special rational function on
* the interval [0,0.34658]:
* Since
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ...
* we define R1(r*r) by
* r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r)
* That is,
* R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r)
* = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r))
* = 1 - r^2/60 + r^4/2520 - r^6/100800 + ...
* We use a special Reme algorithm on [0,0.347] to generate
* a polynomial of degree 5 in r*r to approximate R1. The
* maximum error of this polynomial approximation is bounded
* by 2**-61. In other words,
* R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5
* where Q1 = -1.6666666666666567384E-2,
* Q2 = 3.9682539681370365873E-4,
* Q3 = -9.9206344733435987357E-6,
* Q4 = 2.5051361420808517002E-7,
* Q5 = -6.2843505682382617102E-9;
* (where z=r*r, and the values of Q1 to Q5 are listed below)
* with error bounded by
* | 5 | -61
* | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2
* | |
*
* expm1(r) = exp(r)-1 is then computed by the following
* specific way which minimize the accumulation rounding error:
* 2 3
* r r [ 3 - (R1 + R1*r/2) ]
* expm1(r) = r + --- + --- * [--------------------]
* 2 2 [ 6 - r*(3 - R1*r/2) ]
*
* To compensate the error in the argument reduction, we use
* expm1(r+c) = expm1(r) + c + expm1(r)*c
* ~ expm1(r) + c + r*c
* Thus c+r*c will be added in as the correction terms for
* expm1(r+c). Now rearrange the term to avoid optimization
* screw up:
* ( 2 2 )
* ({ ( r [ R1 - (3 - R1*r/2) ] ) } r )
* expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- )
* ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 )
* ( )
*
* = r - E
* 3. Scale back to obtain expm1(x):
* From step 1, we have
* expm1(x) = either 2^k*[expm1(r)+1] - 1
* = or 2^k*[expm1(r) + (1-2^-k)]
* 4. Implementation notes:
* (A). To save one multiplication, we scale the coefficient Qi
* to Qi*2^i, and replace z by (x^2)/2.
* (B). To achieve maximum accuracy, we compute expm1(x) by
* (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf)
* (ii) if k=0, return r-E
* (iii) if k=-1, return 0.5*(r-E)-0.5
* (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E)
* else return 1.0+2.0*(r-E);
* (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1)
* (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else
* (vii) return 2^k(1-((E+2^-k)-r))
*
* Special cases:
* expm1(INF) is INF, expm1(NaN) is NaN;
* expm1(-INF) is -1, and
* for finite argument, only expm1(0)=0 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 expm1(x) overflow
*
* 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 Expm1 {
private static final double one = 1.0;
private static final double huge = 1.0e+300;
private static final double tiny = 1.0e-300;
private static final double o_threshold = 0x1.62e42fefa39efp9; // 7.09782712893383973096e+02
private static final double ln2_hi = 0x1.62e42feep-1; // 6.93147180369123816490e-01
private static final double ln2_lo = 0x1.a39ef35793c76p-33; // 1.90821492927058770002e-10
private static final double invln2 = 0x1.71547652b82fep0; // 1.44269504088896338700e+00
// scaled coefficients related to expm1
private static final double Q1 = -0x1.11111111110f4p-5; // -3.33333333333331316428e-02
private static final double Q2 = 0x1.a01a019fe5585p-10; // 1.58730158725481460165e-03
private static final double Q3 = -0x1.4ce199eaadbb7p-14; // -7.93650757867487942473e-05
private static final double Q4 = 0x1.0cfca86e65239p-18; // 4.00821782732936239552e-06
private static final double Q5 = -0x1.afdb76e09c32dp-23; // -2.01099218183624371326e-07
static double compute(double x) {
double y, hi, lo, c=0, t, e, hxs, hfx, r1;
int k, xsb;
/*unsigned*/ int hx;
hx = __HI(x); // high word of x
xsb = hx & 0x8000_0000; // sign bit of x
y = Math.abs(x);
hx &= 0x7fff_ffff; // high word of |x|
// filter out huge and non-finite argument
if (hx >= 0x4043_687A) { // if |x| >= 56*ln2
if (hx >= 0x4086_2E42) { // if |x| >= 709.78...
if (hx >= 0x7ff_00000) {
if (((hx & 0xf_ffff) | __LO(x)) != 0) {
return x + x; // NaN
} else {
return (xsb == 0)? x : -1.0; // exp(+-inf)={inf,-1}
}
}
if (x > o_threshold) {
return huge*huge; // overflow
}
}
if (xsb != 0) { // x < -56*ln2, return -1.0 with inexact
if (x + tiny < 0.0) { // raise inexact
return tiny - one; // return -1
}
}
}
// argument reduction
if (hx > 0x3fd6_2e42) { // if |x| > 0.5 ln2
if (hx < 0x3FF0_A2B2) { // and |x| < 1.5 ln2
if (xsb == 0) {
hi = x - ln2_hi;
lo = ln2_lo;
k = 1;
} else {
hi = x + ln2_hi;
lo = -ln2_lo;
k = -1;
}
} else {
k = (int)(invln2*x + ((xsb == 0) ? 0.5 : -0.5));
t = k;
hi = x - t*ln2_hi; // t*ln2_hi is exact here
lo = t*ln2_lo;
}
x = hi - lo;
c = (hi - x) - lo;
} else if (hx < 0x3c90_0000) { // when |x| < 2**-54, return x
t = huge + x; // return x with inexact flags when x != 0
return x - (t - (huge + x));
} else {
k = 0;
}
// x is now in primary range
hfx = 0.5*x;
hxs = x*hfx;
r1 = one + hxs*(Q1 + hxs*(Q2 + hxs*(Q3 + hxs*(Q4 + hxs*Q5))));
t = 3.0 - r1*hfx;
e = hxs *((r1 - t)/(6.0 - x*t));
if (k == 0) {
return x - (x*e - hxs); // c is 0
} else {
e = (x*(e - c) - c);
e -= hxs;
if (k == -1) {
return 0.5*(x - e) - 0.5;
}
if (k == 1) {
if (x < -0.25) {
return -2.0*(e - (x + 0.5));
} else {
return one + 2.0*(x - e);
}
}
if (k <= -2 || k > 56) { // suffice to return exp(x) - 1
y = one - (e - x);
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
return y - one;
}
t = one;
if (k < 20) {
t = __HI(t, 0x3ff0_0000 - (0x2_00000 >> k)); // t = 1-2^-k
y = t - ( e - x);
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
} else {
t = __HI(t, ((0x3ff - k) << 20)); // 2^-k
y = x - (e + t);
y += one;
y = __HI(y, __HI(y) + (k << 20)); // add k to y's exponent
}
}
return y;
}
}
/**
* Method :
* mathematically sinh(x) if defined to be (exp(x)-exp(-x))/2
* 1. Replace x by |x| (sinh(-x) = -sinh(x)).
* 2.
* E + E/(E+1)
* 0 <= x <= 22 : sinh(x) := --------------, E=expm1(x)
* 2
*
* 22 <= x <= lnovft : sinh(x) := exp(x)/2
* lnovft <= x <= ln2ovft: sinh(x) := exp(x/2)/2 * exp(x/2)
* ln2ovft < x : sinh(x) := x*shuge (overflow)
*
* Special cases:
* sinh(x) is |x| if x is +INF, -INF, or NaN.
* only sinh(0)=0 is exact for finite x.
*/
static final class Sinh {
private Sinh() {throw new UnsupportedOperationException();}
private static final double shuge = 1.0e307;
static double compute(double x) {
double t, w, h;
int ix, jx;
/* unsigned */ int lx;
// High word of |x|
jx = __HI(x);
ix = jx & 0x7fff_ffff;
// x is INF or NaN
if (ix >= 0x7ff0_0000) {
return x + x;
}
h = 0.5;
if (jx < 0) {
h = -h;
}
// |x| in [0,22], return sign(x)*0.5*(E+E/(E+1)))
if (ix < 0x4036_0000) { // |x| < 22
if (ix < 0x3e30_0000) // |x| < 2**-28
if (shuge + x > 1.0) { // sinh(tiny) = tiny with inexact
return x;
}
t = StrictMath.expm1(Math.abs(x));
if (ix < 0x3ff0_0000) {
return h*(2.0 * t - t*t/(t + 1.0));
}
return h*(t + t/(t + 1.0));
}
// |x| in [22, log(maxdouble)] return 0.5*exp(|x|)
if (ix < 0x4086_2E42) {
return h*StrictMath.exp(Math.abs(x));
}
// |x| in [log(maxdouble), overflowthresold]
lx = __LO(x);
if (ix < 0x4086_33CE ||
((ix == 0x4086_33ce) &&
(Long.compareUnsigned(lx, 0x8fb9_f87d) <= 0 ))) {
w = StrictMath.exp(0.5 * Math.abs(x));
t = h * w;
return t * w;
}
// |x| > overflowthresold, sinh(x) overflow
return x * shuge;
}
}
/**
* Method :
* mathematically cosh(x) if defined to be (exp(x)+exp(-x))/2
* 1. Replace x by |x| (cosh(x) = cosh(-x)).
* 2.
* [ exp(x) - 1 ]^2
* 0 <= x <= ln2/2 : cosh(x) := 1 + -------------------
* 2*exp(x)
*
* exp(x) + 1/exp(x)
* ln2/2 <= x <= 22 : cosh(x) := -------------------
* 2
* 22 <= x <= lnovft : cosh(x) := exp(x)/2
* lnovft <= x <= ln2ovft: cosh(x) := exp(x/2)/2 * exp(x/2)
* ln2ovft < x : cosh(x) := huge*huge (overflow)
*
* Special cases:
* cosh(x) is |x| if x is +INF, -INF, or NaN.
* only cosh(0)=1 is exact for finite x.
*/
static final class Cosh {
private Cosh() {throw new UnsupportedOperationException();}
private static final double huge = 1.0e300;
static double compute(double x) {
double t, w;
int ix;
/*unsigned*/ int lx;
// High word of |x|
ix = __HI(x);
ix &= 0x7fff_ffff;
// x is INF or NaN
if (ix >= 0x7ff0_0000) {
return x*x;
}
// |x| in [0,0.5*ln2], return 1+expm1(|x|)^2/(2*exp(|x|))
if (ix < 0x3fd6_2e43) {
t = StrictMath.expm1(Math.abs(x));
w = 1.0 + t;
if (ix < 0x3c80_0000) { // cosh(tiny) = 1
return w;
}
return 1.0 + (t * t)/(w + w);
}
// |x| in [0.5*ln2, 22], return (exp(|x|) + 1/exp(|x|)/2
if (ix < 0x4036_0000) {
t = StrictMath.exp(Math.abs(x));
return 0.5*t + 0.5/t;
}
// |x| in [22, log(maxdouble)] return 0.5*exp(|x|)
if (ix < 0x4086_2E42) {
return 0.5*StrictMath.exp(Math.abs(x));
}
// |x| in [log(maxdouble), overflowthresold]
lx = __LO(x);
if (ix<0x4086_33CE ||
((ix == 0x4086_33ce) &&
(Integer.compareUnsigned(lx, 0x8fb9_f87d) <= 0))) {
w = StrictMath.exp(0.5*Math.abs(x));
t = 0.5*w;
return t*w;
}
// |x| > overflowthresold, cosh(x) overflow
return huge*huge;
}
}
/**
* Return the Hyperbolic Tangent of x
*
* Method :
* x -x
* e - e
* 0. tanh(x) is defined to be -----------
* x -x
* e + e
* 1. reduce x to non-negative by tanh(-x) = -tanh(x).
* 2. 0 <= x <= 2**-55 : tanh(x) := x*(one+x)
* -t
* 2**-55 < x <= 1 : tanh(x) := -----; t = expm1(-2x)
* t + 2
* 2
* 1 <= x <= 22.0 : tanh(x) := 1- ----- ; t=expm1(2x)
* t + 2
* 22.0 < x <= INF : tanh(x) := 1.
*
* Special cases:
* tanh(NaN) is NaN;
* only tanh(0)=0 is exact for finite argument.
*/
static final class Tanh {
private Tanh() {throw new UnsupportedOperationException();}
private static final double tiny = 1.0e-300;
static double compute(double x) {
double t, z;
int jx, ix;
// High word of |x|.
jx = __HI(x);
ix = jx & 0x7fff_ffff;
// x is INF or NaN
if (ix >= 0x7ff0_0000) {
if (jx >= 0) { // tanh(+-inf)=+-1
return 1.0/x + 1.0;
} else { // tanh(NaN) = NaN
return 1.0/x - 1.0;
}
}
// |x| < 22
if (ix < 0x4036_0000) { // |x| < 22
if (ix<0x3c80_0000) // |x| < 2**-55
return x*(1.0 + x); // tanh(small) = small
if (ix>=0x3ff0_0000) { // |x| >= 1
t = StrictMath.expm1(2.0*Math.abs(x));
z = 1.0 - 2.0/(t + 2.0);
} else {
t = StrictMath.expm1(-2.0*Math.abs(x));
z= -t/(t + 2.0);
}
} else { // |x| > 22, return +-1
z = 1.0 - tiny; // raised inexact flag
}
return (jx >= 0)? z: -z;
}
}
}
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