mquickjs-hs-0.1.2.4: mquickjs/libm.c
/*
* Tiny Math Library
*
* Copyright (c) 2024 Fabrice Bellard
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
* THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include <stdio.h>
#include <inttypes.h>
#include <math.h>
#define NDEBUG
#include <assert.h>
#include "cutils.h"
#include "libm.h"
/* define to enable softfloat support */
//#define USE_SOFTFLOAT
/* use less code for tan() but currently less precise */
#define USE_TAN_SHORTCUT
/*
TODO:
- smaller scalbn implementation ?
- add all ES6 math functions
*/
/*
tc32:
- base: size libm+libgcc: 21368
- size libm+libgcc: 11832
x86:
- size libm softfp: 18510
- size libm hardfp: 10051
TODO:
- unify i32 bit and i64 bit conversions
- unify comparisons operations
*/
typedef enum {
RM_RNE, /* Round to Nearest, ties to Even */
RM_RTZ, /* Round towards Zero */
RM_RDN, /* Round Down (must be even) */
RM_RUP, /* Round Up (must be odd) */
RM_RMM, /* Round to Nearest, ties to Max Magnitude */
RM_RMMUP, /* only for rint_sf64(): round to nearest, ties to +inf (must be odd) */
} RoundingModeEnum;
#define FFLAG_INVALID_OP (1 << 4)
#define FFLAG_DIVIDE_ZERO (1 << 3)
#define FFLAG_OVERFLOW (1 << 2)
#define FFLAG_UNDERFLOW (1 << 1)
#define FFLAG_INEXACT (1 << 0)
typedef enum {
FMINMAX_PROP, /* min(1, qNaN/sNaN) -> qNaN */
FMINMAX_IEEE754_2008, /* min(1, qNaN) -> 1, min(1, sNaN) -> qNaN */
FMINMAX_IEEE754_201X, /* min(1, qNaN/sNaN) -> 1 */
} SoftFPMinMaxTypeEnum;
typedef uint32_t sfloat32;
typedef uint64_t sfloat64;
#define F_STATIC static __maybe_unused
#define F_USE_FFLAGS 0
#define F_SIZE 32
#define F_NORMALIZE_ONLY
#include "softfp_template.h"
#define F_SIZE 64
#include "softfp_template.h"
#ifdef USE_SOFTFLOAT
/* wrappers */
double __adddf3(double a, double b)
{
return uint64_as_float64(add_sf64(float64_as_uint64(a),
float64_as_uint64(b), RM_RNE));
}
double __subdf3(double a, double b)
{
return uint64_as_float64(sub_sf64(float64_as_uint64(a),
float64_as_uint64(b), RM_RNE));
}
double __muldf3(double a, double b)
{
return uint64_as_float64(mul_sf64(float64_as_uint64(a),
float64_as_uint64(b), RM_RNE));
}
double __divdf3(double a, double b)
{
return uint64_as_float64(div_sf64(float64_as_uint64(a),
float64_as_uint64(b), RM_RNE));
}
/* comparisons */
int __eqdf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
return ret;
}
/* NaN: return 0 */
int __nedf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
if (unlikely(ret == 2))
return 0;
else
return ret;
}
int __ledf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
return ret;
}
int __ltdf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
return ret;
}
int __gedf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
if (unlikely(ret == 2))
return -1;
else
return ret;
}
int __gtdf2(double a, double b)
{
int ret = cmp_sf64(float64_as_uint64(a),
float64_as_uint64(b));
if (unlikely(ret == 2))
return -1;
else
return ret;
}
int __unorddf2(double a, double b)
{
return isnan_sf64(float64_as_uint64(a)) ||
isnan_sf64(float64_as_uint64(b));
}
/* conversions */
double __floatsidf(int32_t a)
{
return uint64_as_float64(cvt_i32_sf64(a, RM_RNE));
}
double __floatdidf(int64_t a)
{
return uint64_as_float64(cvt_i64_sf64(a, RM_RNE));
}
double __floatunsidf(unsigned int a)
{
return uint64_as_float64(cvt_u32_sf64(a, RM_RNE));
}
int32_t __fixdfsi(double a)
{
return cvt_sf64_i32(float64_as_uint64(a), RM_RTZ);
}
double __extendsfdf2(float a)
{
return uint64_as_float64(cvt_sf32_sf64(float_as_uint(a)));
}
float __truncdfsf2(double a)
{
return uint_as_float(cvt_sf64_sf32(float64_as_uint64(a), RM_RNE));
}
double js_sqrt(double a)
{
return uint64_as_float64(sqrt_sf64(float64_as_uint64(a), RM_RNE));
}
#if defined(__tc32__)
/* XXX: check */
int __fpclassifyd(double a)
{
uint64_t u = float64_as_uint64(a);
uint32_t h = u >> 32;
uint32_t l = u;
h &= 0x7fffffff;
if (h >= 0x7ff00000) {
if (h == 0x7ff00000 && l == 0)
return FP_INFINITE;
else
return FP_NAN;
} else if (h < 0x00100000) {
if (h == 0 && l == 0)
return FP_ZERO;
else
return FP_SUBNORMAL;
} else {
return FP_NORMAL;
}
}
#endif
#endif /* USE_SOFTFLOAT */
int32_t js_lrint(double a)
{
return cvt_sf64_i32(float64_as_uint64(a), RM_RNE);
}
double js_fmod(double a, double b)
{
return uint64_as_float64(fmod_sf64(float64_as_uint64(a), float64_as_uint64(b)));
}
/* supported rounding modes: RM_UP, RM_DN, RM_RTZ, RM_RMMUP, RM_RMM */
static double rint_sf64(double a, RoundingModeEnum rm)
{
uint64_t u = float64_as_uint64(a);
uint64_t frac_mask, one, m, addend;
int e;
unsigned int s;
e = ((u >> 52) & 0x7ff) - 0x3ff;
s = u >> 63;
if (e < 0) {
m = u & (((uint64_t)1 << 52) - 1);
if (e == -0x3ff && m == 0) {
/* zero: nothing to do */
} else {
/* abs(a) < 1 */
s = u >> 63;
one = (uint64_t)0x3ff << 52;
u = 0;
switch(rm) {
case RM_RUP:
case RM_RDN:
if (s ^ (rm & 1))
u = one;
break;
default:
case RM_RMM:
case RM_RMMUP:
if (e == -1 && (m != 0 || (m == 0 && (!s || rm == RM_RMM))))
u = one;
break;
case RM_RTZ:
break;
}
u |= (uint64_t)s << 63;
}
} else if (e < 52) {
one = (uint64_t)1 << (52 - e);
frac_mask = one - 1;
addend = 0;
switch(rm) {
case RM_RMMUP:
addend = (one >> 1) - s;
break;
default:
case RM_RMM:
addend = (one >> 1);
break;
case RM_RTZ:
break;
case RM_RUP:
case RM_RDN:
if (s ^ (rm & 1))
addend = one - 1;
break;
}
u += addend;
u &= ~frac_mask; /* truncate to an integer */
}
/* otherwise: abs(a) >= 2^52, or NaN, +/-Infinity: no change */
return uint64_as_float64(u);
}
double js_floor(double x)
{
return rint_sf64(x, RM_RDN);
}
double js_ceil(double x)
{
return rint_sf64(x, RM_RUP);
}
double js_trunc(double x)
{
return rint_sf64(x, RM_RTZ);
}
double js_round_inf(double x)
{
return rint_sf64(x, RM_RMMUP);
}
double js_fabs(double x)
{
uint64_t a = float64_as_uint64(x);
return uint64_as_float64(a & 0x7fffffffffffffff);
}
/************************************************************/
/* libm */
#define EXTRACT_WORDS(ix0,ix1,d) \
do { \
uint64_t __u = float64_as_uint64(d); \
(ix0) = (uint32_t)(__u >> 32); \
(ix1) = (uint32_t)__u; \
} while (0)
static uint32_t get_high_word(double d)
{
return float64_as_uint64(d) >> 32;
}
static double set_high_word(double d, uint32_t h)
{
uint64_t u = float64_as_uint64(d);
u = (u & 0xffffffff) | ((uint64_t)h << 32);
return uint64_as_float64(u);
}
static uint32_t get_low_word(double d)
{
return float64_as_uint64(d);
}
/* set the low 32 bits to zero */
static double zero_low(double x)
{
uint64_t u = float64_as_uint64(x);
u &= 0xffffffff00000000;
return uint64_as_float64(u);
}
static double float64_from_u32(uint32_t h, uint32_t l)
{
return uint64_as_float64(((uint64_t)h << 32) | l);
}
static const double zero = 0.0;
static const double one = 1.0;
static const double half = 5.00000000000000000000e-01;
static const double tiny = 1.0e-300;
static const double huge = 1.0e300;
/* @(#)s_scalbn.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* scalbn (double x, int n)
* scalbn(x,n) returns x* 2**n computed by exponent
* manipulation rather than by actually performing an
* exponentiation or a multiplication.
*/
static const double
two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */
twom54 = 5.55111512312578270212e-17; /* 0x3C900000, 0x00000000 */
double js_scalbn(double x, int n)
{
int k,hx,lx;
EXTRACT_WORDS(hx, lx, x);
k = (hx&0x7ff00000)>>20; /* extract exponent */
if (k==0) { /* 0 or subnormal x */
if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */
x *= two54;
hx = get_high_word(x);
k = ((hx&0x7ff00000)>>20) - 54;
if (n< -50000) return tiny*x; /*underflow*/
}
if (k==0x7ff) return x+x; /* NaN or Inf */
k = k+n;
if (k > 0x7fe) return huge*copysign(huge,x); /* overflow */
if (k > 0) /* normal result */
{x = set_high_word(x, (hx&0x800fffff)|(k<<20)); return x;}
if (k <= -54) {
if (n > 50000) /* in case integer overflow in n+k */
return huge*copysign(huge,x); /*overflow*/
else
return tiny*copysign(tiny,x); /*underflow*/
}
k += 54; /* subnormal result */
x = set_high_word(x, (hx&0x800fffff)|(k<<20));
return x*twom54;
}
#ifndef USE_SOFTFLOAT
/* @(#)e_sqrt.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_sqrt(x)
* Return correctly rounded sqrt.
* ------------------------------------------
* | Use the hardware sqrt if you have one |
* ------------------------------------------
* Method:
* Bit by bit method using integer arithmetic. (Slow, but portable)
* 1. Normalization
* Scale x to y in [1,4) with even powers of 2:
* find an integer k such that 1 <= (y=x*2^(2k)) < 4, then
* sqrt(x) = 2^k * sqrt(y)
* 2. Bit by bit computation
* Let q = sqrt(y) truncated to i bit after binary point (q = 1),
* i 0
* i+1 2
* s = 2*q , and y = 2 * ( y - q ). (1)
* i i i i
*
* To compute q from q , one checks whether
* i+1 i
*
* -(i+1) 2
* (q + 2 ) <= y. (2)
* i
* -(i+1)
* If (2) is false, then q = q ; otherwise q = q + 2 .
* i+1 i i+1 i
*
* With some algebraic manipulation, it is not difficult to see
* that (2) is equivalent to
* -(i+1)
* s + 2 <= y (3)
* i i
*
* The advantage of (3) is that s and y can be computed by
* i i
* the following recurrence formula:
* if (3) is false
*
* s = s , y = y ; (4)
* i+1 i i+1 i
*
* otherwise,
* -i -(i+1)
* s = s + 2 , y = y - s - 2 (5)
* i+1 i i+1 i i
*
* One may easily use induction to prove (4) and (5).
* Note. Since the left hand side of (3) contain only i+2 bits,
* it does not necessary to do a full (53-bit) comparison
* in (3).
* 3. Final rounding
* After generating the 53 bits result, we compute one more bit.
* Together with the remainder, we can decide whether the
* result is exact, bigger than 1/2ulp, or less than 1/2ulp
* (it will never equal to 1/2ulp).
* The rounding mode can be detected by checking whether
* huge + tiny is equal to huge, and whether huge - tiny is
* equal to huge for some floating point number "huge" and "tiny".
*
* Special cases:
* sqrt(+-0) = +-0 ... exact
* sqrt(inf) = inf
* sqrt(-ve) = NaN ... with invalid signal
* sqrt(NaN) = NaN ... with invalid signal for signaling NaN
*
* Other methods : see the appended file at the end of the program below.
*---------------
*/
#if defined(__aarch64__) || defined(__x86_64__) || defined(__i386__)
/* hardware sqrt is available */
double js_sqrt(double x)
{
return sqrt(x);
}
#else
double js_sqrt(double x)
{
double z;
int sign = (int)0x80000000;
unsigned r,t1,s1,ix1,q1;
int ix0,s0,q,m,t,i;
EXTRACT_WORDS(ix0, ix1, x);
/* take care of Inf and NaN */
if((ix0&0x7ff00000)==0x7ff00000) {
return x*x+x; /* sqrt(NaN)=NaN, sqrt(+inf)=+inf
sqrt(-inf)=sNaN */
}
/* take care of zero */
if(ix0<=0) {
if(((ix0&(~sign))|ix1)==0) return x;/* sqrt(+-0) = +-0 */
else if(ix0<0)
return (x-x)/(x-x); /* sqrt(-ve) = sNaN */
}
/* normalize x */
m = (ix0>>20);
if(m==0) { /* subnormal x */
while(ix0==0) {
m -= 21;
ix0 |= (ix1>>11); ix1 <<= 21;
}
for(i=0;(ix0&0x00100000)==0;i++) ix0<<=1;
m -= i-1;
ix0 |= (ix1>>(32-i));
ix1 <<= i;
}
m -= 1023; /* unbias exponent */
ix0 = (ix0&0x000fffff)|0x00100000;
if(m&1){ /* odd m, double x to make it even */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
}
m >>= 1; /* m = [m/2] */
/* generate sqrt(x) bit by bit */
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
q = q1 = s0 = s1 = 0; /* [q,q1] = sqrt(x) */
r = 0x00200000; /* r = moving bit from right to left */
while(r!=0) {
t = s0+r;
if(t<=ix0) {
s0 = t+r;
ix0 -= t;
q += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r>>=1;
}
r = sign;
while(r!=0) {
t1 = s1+r;
t = s0;
if((t<ix0)||((t==ix0)&&(t1<=ix1))) {
s1 = t1+r;
if(((t1&sign)==sign)&&(s1&sign)==0) s0 += 1;
ix0 -= t;
if (ix1 < t1) ix0 -= 1;
ix1 -= t1;
q1 += r;
}
ix0 += ix0 + ((ix1&sign)>>31);
ix1 += ix1;
r>>=1;
}
/* use floating add to find out rounding direction */
if((ix0|ix1)!=0) {
z = one-tiny; /* trigger inexact flag */
if (z>=one) {
z = one+tiny;
if (q1==(unsigned)0xffffffff) { q1=0; q += 1;}
else if (z>one) {
if (q1==(unsigned)0xfffffffe) q+=1;
q1+=2;
} else
q1 += (q1&1);
}
}
ix0 = (q>>1)+0x3fe00000;
ix1 = q1>>1;
if ((q&1)==1) ix1 |= sign;
ix0 += (m <<20);
return float64_from_u32(ix0, ix1);
}
#endif /* !hardware sqrt */
#endif /* USE_SOFTFLOAT */
/* to have smaller code */
/* n >= 1 */
/* return sum(x^i*coefs[i] with i = 0 ... n - 1 and n >= 1 using
Horner algorithm. */
static double eval_poly(double x, const double *coefs, int n)
{
double r;
int i;
r = coefs[n - 1];
for(i = n - 2; i >= 0; i--) {
r = r * x + coefs[i];
}
return r;
}
/* @(#)k_sin.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __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))
*/
static const double
S1 = -1.66666666666666324348e-01; /* 0xBFC55555, 0x55555549 */
static const double S_tab[] = {
/* 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 */
};
/* iy=0 if y is zero */
static double __kernel_sin(double x, double y, int iy)
{
double z,r,v;
int ix;
ix = get_high_word(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 = eval_poly(z, S_tab, 5);
if(iy==0) return x+v*(S1+z*r);
else return x-((z*(half*y-v*r)-y)-v*S1);
}
/* @(#)k_cos.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* __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.
*/
static const double C_tab[] = {
/* 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;
int ix;
ix = get_high_word(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 * eval_poly(z, C_tab, 6);
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 {
qx = float64_from_u32(ix-0x00200000, 0); /* x/4 */
}
hz = 0.5*z-qx;
a = one-qx;
return a - (hz - (z*r-x*y));
}
}
/* rem_pio2 */
#define T_LEN 19
/* T[i] = floor(2^(64*(T_LEN - i))/2pi) mod 2^64 */
static const uint64_t T[T_LEN] = {
0x1580cc11bf1edaea,
0x9afed7ec47e35742,
0xcf41ce7de294a4ba,
0x5d49eeb1faf97c5e,
0xd3d18fd9a797fa8b,
0xdb4d9fb3c9f2c26d,
0xfbcbc462d6829b47,
0xc7fe25fff7816603,
0x272117e2ef7e4a0e,
0x4e64758e60d4ce7d,
0x3a671c09ad17df90,
0xba208d7d4baed121,
0x3f877ac72c4a69cf,
0x01924bba82746487,
0x6dc91b8e909374b8,
0x7f9458eaf7aef158,
0x36d8a5664f10e410,
0x7f09d5f47d4d3770,
0x28be60db9391054a, /* high part */
};
/* PIO2[i] = floor(2^(64*(2 - i))*PI/4) mod 2^64 */
static const uint64_t PIO4[2] = {
0xc4c6628b80dc1cd1,
0xc90fdaa22168c234,
};
static uint64_t get_u64_at_bit(const uint64_t *tab, uint32_t tab_len,
uint32_t pos)
{
uint64_t v;
uint32_t p = pos / 64;
int shift = pos % 64;
v = tab[p] >> shift;
if (shift != 0 && (p + 1) < tab_len)
v |= tab[p + 1] << (64 - shift);
return v;
}
/* return n = round(x/(pi/2)) (only low 2 bits are valid) and
(y[0], y[1]) = x - (pi/2) * n.
'x' must be finite and such as abs(x) >= PI/4.
The initial algorithm comes from the CORE-MATH project.
*/
static int rem_pio2_large(double x, double *y)
{
uint64_t m;
int e, sgn, n, rnd, j, i, y_sgn;
uint64_t c[2], d[3];
uint64_t r0, r1;
uint32_t carry, carry1;
m = float64_as_uint64(x);
sgn = m >> 63;
e = (m >> 52) & 0x7ff;
/* 1022 <= e <= 2047 */
m = (m & (((uint64_t)1 << 52) - 1)) | ((uint64_t)1 << 52);
/* multiply m by T[j:j+192] */
j = T_LEN * 64 - (e - 1075) - 192;
/* 53 <= j <= 1077 */
// printf("m=0x%016" PRIx64 " e=%d j=%d\n", m, e, j);
for(i = 0; i < 3; i++) {
d[i] = get_u64_at_bit(T, T_LEN, j + i * 64);
}
r1 = mul_u64(&r0, m, d[0]);
c[0] = r1;
r1 = mul_u64(&r0, m, d[1]);
c[0] += r0;
carry = c[0] < r0;
c[1] = r1 + carry;
mul_u64(&r0, m, d[2]);
c[1] += r0;
// printf("c0=%016" PRIx64 " %016" PRIx64 "\n", c[1], c[0]);
/* n = round(c[1]/2^62) */
n = c[1] >> 62;
rnd = (c[1] >> 61) & 1;
n += rnd;
/* c = c * 4 - n */
c[1] = (c[1] << 2) | (c[0] >> 62);
c[0] = (c[0] << 2);
y_sgn = sgn;
if (rnd) {
/* 'y' sign change */
y_sgn ^= 1;
c[0] = ~c[0];
c[1] = ~c[1];
if (++c[0] == 0)
c[1]++;
}
// printf("c1=%016" PRIx64 " %016" PRIx64 " n=%d sgn=%d\n", c[1], c[0], n, sgn);
/* c = c * (PI/2) (high 128 bits of the product) */
r1 = mul_u64(&r0, c[0], PIO4[1]);
d[0] = r0;
d[1] = r1;
r1 = mul_u64(&r0, c[1], PIO4[0]);
d[0] += r0;
carry = d[0] < r0;
d[1] += r1;
carry1 = d[1] < r1;
d[1] += carry;
carry1 |= (d[1] < carry);
d[2] = carry1;
r1 = mul_u64(&r0, c[1], PIO4[1]);
d[1] += r0;
carry = d[1] < r0;
d[2] += r1 + carry;
/* convert d to two float64 */
// printf("d=%016" PRIx64 " %016" PRIx64 "\n", d[2], d[1]);
if (d[2] == 0) {
/* should never happen (see ARGUMENT REDUCTION FOR HUGE
ARGUMENTS: Good to the Last Bit, K. C. Ng and the members
of the FP group of SunPro */
y[0] = y[1] = 0;
} else {
uint64_t m0, m1;
int e1;
e = clz64(d[2]);
d[2] = (d[2] << e) | (d[1] >> (64 - e));
d[1] = (d[1] << e);
// printf("d=%016" PRIx64 " %016" PRIx64 " e=%d\n", d[2], d[1], e);
m0 = (d[2] >> 11) & (((uint64_t)1 << 52) - 1);
m1 = ((d[2] & 0x7ff) << 42) | (d[1] >> (64 - 42));
y[0] = uint64_as_float64(((uint64_t)y_sgn << 63) |
((uint64_t)(1023 - e) << 52) |
m0);
if (m1 == 0) {
y[1] = 0;
} else {
e1 = clz64(m1) - 11;
m1 = (m1 << e1) & (((uint64_t)1 << 52) - 1);
y[1] = uint64_as_float64(((uint64_t)y_sgn << 63) |
((uint64_t)(1023 - e - 53 - e1) << 52) |
m1);
}
}
if (sgn)
n = -n;
return n;
}
#ifdef USE_SOFTFLOAT
/* when using softfloat, the FP reduction should be not much faster
than the generic one */
int js_rem_pio2(double x, double *y)
{
int ix,hx;
hx = get_high_word(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;
}
/*
* all other (large) arguments
*/
if(ix>=0x7ff00000) { /* x is inf or NaN */
y[0]=y[1]=x-x;
return 0;
}
return rem_pio2_large(x, y);
}
#else
/*
* 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)
*/
static const double
invpio2 = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
static const double pio2_tab[3] = {
/* pio2_1 */ 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
/* pio2_2 */ 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
/* pio2_3 */ 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
};
static const double pio2_t_tab[3] = {
/* pio2_1t */ 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
/* pio2_2t */ 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
/* pio2_3t */ 8.47842766036889956997e-32, /* 0x397B839A, 0x252049C1 */
};
static uint8_t rem_pio2_emax[2] = { 16, 49 };
int js_rem_pio2(double x, double *y)
{
double w,t,r,fn;
int i,j,n,ix,hx,it;
hx = get_high_word(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<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
t = fabs(x);
if (ix<0x4002d97c) {
/* |x| < 3pi/4, special case with n=+-1 */
n = 1;
fn = 1;
} else {
n = (int) (t*invpio2+half);
fn = (double)n;
}
it = 0;
for(;;) {
/* 1st round good to 85 bit */
/* 2nd iteration needed, good to 118 */
/* 3rd iteration need, 151 bits acc */
r = t-fn*pio2_tab[it];
w = fn*pio2_t_tab[it];
y[0] = r-w;
j = ix>>20;
i = j-(((get_high_word(y[0]))>>20)&0x7ff);
if (it == 2 || i <= rem_pio2_emax[it])
break;
t = r;
it++;
}
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;
}
return rem_pio2_large(x, y);
}
#endif /* !USE_SOFTFLOAT */
/* @(#)s_sin.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* 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
*/
/* flag = 0: sin()
flag = 1: cos()
flag = 3: tan()
*/
static double js_sin_cos(double x, int flag)
{
double y[2], z, s, c;
int ix;
uint32_t n;
/* High word of x. */
ix = get_high_word(x);
/* sin(Inf or NaN) is NaN */
if (ix>=0x7ff00000)
return x-x;
n = js_rem_pio2(x,y);
s = c = 0; /* avoid warning */
if (flag == 3 || (n & 1) == flag) {
s = __kernel_sin(y[0],y[1],1);
if (flag != 3)
goto done;
}
if (flag == 3 || (n & 1) != flag) {
c = __kernel_cos(y[0],y[1]);
if (flag != 3) {
s = c;
goto done;
}
}
if (n & 1)
z = -c / s;
else
z = s / c;
return z;
done:
if ((n + flag) & 2)
s = -s;
return s;
}
double js_sin(double x)
{
return js_sin_cos(x, 0);
}
double js_cos(double x)
{
return js_sin_cos(x, 1);
}
#ifdef USE_TAN_SHORTCUT
double js_tan(double x)
{
return js_sin_cos(x, 3);
}
#endif
#ifndef USE_TAN_SHORTCUT
/*
* ====================================================
* Copyright 2004 Sun Microsystems, Inc. All Rights Reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* INDENT OFF */
/* __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)))
*/
static const double T0 = 3.33333333333334091986e-01; /* 3FD55555, 55555563 */
static const double T_even[] = {
5.39682539762260521377e-02, /* 3FABA1BA, 1BB341FE */
8.86323982359930005737e-03, /* 3F8226E3, E96E8493 */
1.45620945432529025516e-03, /* 3F57DBC8, FEE08315 */
2.46463134818469906812e-04, /* 3F3026F7, 1A8D1068 */
7.14072491382608190305e-05, /* 3F12B80F, 32F0A7E9 */
2.59073051863633712884e-05, /* 3EFB2A70, 74BF7AD4 */
};
static const double T_odd[] = {
1.33333333333201242699e-01, /* 3FC11111, 1110FE7A */
2.18694882948595424599e-02, /* 3F9664F4, 8406D637 */
3.59207910759131235356e-03, /* 3F6D6D22, C9560328 */
5.88041240820264096874e-04, /* 3F4344D8, F2F26501 */
7.81794442939557092300e-05, /* 3F147E88, A03792A6 */
-1.85586374855275456654e-05, /* BEF375CB, DB605373 */
};
static const double pio4 = 7.85398163397448278999e-01, /* 3FE921FB, 54442D18 */
pio4lo = 3.06161699786838301793e-17; /* 3C81A626, 33145C07 */
/* compute -1 / (x+y) carefully */
static double minus_inv(double x, double y)
{
double a, t, z, v, s, w;
w = x + y;
z = zero_low(w);
v = y - (z - x);
a = -one / w;
t = zero_low(a);
s = one + t * z;
return t + a * (s + t * v);
}
static double
__kernel_tan(double x, double y, int iy) {
double z, r, v, w, s;
int ix, hx;
hx = get_high_word(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 | get_low_word(x)) | (iy + 1)) == 0)
return one / fabs(x);
else {
if (iy == 1)
return x;
else
return minus_inv(x, y);
}
}
}
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 = eval_poly(w, T_odd, 6);
v = z * eval_poly(w, T_even, 6);
s = z * x;
r = y + z * (s * (r + v) + y);
r += T0 * 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
*/
return minus_inv(x, r);
}
}
/* @(#)s_tan.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* 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
*/
double js_tan(double x)
{
double y[2],z=0.0;
int n, ix;
/* High word of x. */
ix = get_high_word(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 = js_rem_pio2(x,y);
return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
-1 -- n odd */
}
}
#endif
/* @(#)e_asin.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_asin(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 const double
pio2_hi = 1.57079632679489655800e+00, /* 0x3FF921FB, 0x54442D18 */
pio2_lo = 6.12323399573676603587e-17, /* 0x3C91A626, 0x33145C07 */
pio4_hi = 7.85398163397448278999e-01; /* 0x3FE921FB, 0x54442D18 */
/* coefficient for R(x^2) */
static const double pS[] = {
/* pS0 */ 1.66666666666666657415e-01, /* 0x3FC55555, 0x55555555 */
/* pS1 */-3.25565818622400915405e-01, /* 0xBFD4D612, 0x03EB6F7D */
/* pS2 */ 2.01212532134862925881e-01, /* 0x3FC9C155, 0x0E884455 */
/* pS3 */ -4.00555345006794114027e-02, /* 0xBFA48228, 0xB5688F3B */
/* pS4 */ 7.91534994289814532176e-04, /* 0x3F49EFE0, 0x7501B288 */
/* pS5 */ 3.47933107596021167570e-05, /* 0x3F023DE1, 0x0DFDF709 */
};
static const double qS[] = {
/* qS1 */ -2.40339491173441421878e+00, /* 0xC0033A27, 0x1C8A2D4B */
/* qS2 */ 2.02094576023350569471e+00, /* 0x40002AE5, 0x9C598AC8 */
/* qS3 */ -6.88283971605453293030e-01, /* 0xBFE6066C, 0x1B8D0159 */
/* qS4 */ 7.70381505559019352791e-02, /* 0x3FB3B8C5, 0xB12E9282 */
};
static double R(double t)
{
double p, q, w;
p = t * eval_poly(t, pS, 6);
q = one + t * eval_poly(t, qS, 4);
w = p/q;
return w;
}
double js_asin(double x)
{
double t,w,p,q,c,r,s;
int hx,ix;
hx = get_high_word(x);
ix = hx&0x7fffffff;
if(ix>= 0x3ff00000) { /* |x|>= 1 */
if(((ix-0x3ff00000)|get_low_word(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<0x3fe00000) { /* |x|<0.5 */
if(ix<0x3e400000) { /* if |x| < 2**-27 */
if(huge+x>one) return x;/* return x with inexact if x!=0*/
} else {
t = x*x;
w = R(t);
return x+x*w;
}
}
/* 1> |x|>= 0.5 */
w = one-fabs(x);
t = w*0.5;
r = R(t);
s = js_sqrt(t);
if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
w = r;
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
} else {
w = zero_low(s);
c = (t-w*w)/(s+w);
p = 2.0*s*r-(pio2_lo-2.0*c);
q = pio4_hi-2.0*w;
t = pio4_hi-(p-q);
}
if(hx>0) return t; else return -t;
}
/* @(#)e_acos.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_acos(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 const double
pi = 3.14159265358979311600e+00; /* 0x400921FB, 0x54442D18 */
double js_acos(double x)
{
double z,r,w,s,c,df;
int hx,ix;
hx = get_high_word(x);
ix = hx&0x7fffffff;
if(ix>=0x3ff00000) { /* |x| >= 1 */
if(((ix-0x3ff00000)|get_low_word(x))==0) { /* |x|==1 */
if(hx>0) return 0.0; /* acos(1) = 0 */
else return pi+2.0*pio2_lo; /* acos(-1)= pi */
}
return (x-x)/(x-x); /* acos(|x|>1) is NaN */
}
if(ix<0x3fe00000) { /* |x| < 0.5 */
if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
z = x*x;
r = R(z);
return pio2_hi - (x - (pio2_lo-x*r));
} else {
z = (one-fabs(x))*0.5;
r = R(z);
s = js_sqrt(z);
if (hx<0) { /* x < -0.5 */
w = r*s-pio2_lo;
return pi - 2.0*(s+w);
} else { /* x > 0.5 */
df = zero_low(s);
c = (z-df*df)/(s+df);
w = r*s+c;
return 2.0*(df+w);
}
}
}
/* @(#)s_atan.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* atan(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 const double atanhi[] = {
4.63647609000806093515e-01, /* atan(0.5)hi 0x3FDDAC67, 0x0561BB4F */
7.85398163397448278999e-01, /* atan(1.0)hi 0x3FE921FB, 0x54442D18 */
9.82793723247329054082e-01, /* atan(1.5)hi 0x3FEF730B, 0xD281F69B */
1.57079632679489655800e+00, /* atan(inf)hi 0x3FF921FB, 0x54442D18 */
};
static const double atanlo[] = {
2.26987774529616870924e-17, /* atan(0.5)lo 0x3C7A2B7F, 0x222F65E2 */
3.06161699786838301793e-17, /* atan(1.0)lo 0x3C81A626, 0x33145C07 */
1.39033110312309984516e-17, /* atan(1.5)lo 0x3C700788, 0x7AF0CBBD */
6.12323399573676603587e-17, /* atan(inf)lo 0x3C91A626, 0x33145C07 */
};
static const double aT_even[] = {
3.33333333333329318027e-01, /* 0x3FD55555, 0x5555550D */
1.42857142725034663711e-01, /* 0x3FC24924, 0x920083FF */
9.09088713343650656196e-02, /* 0x3FB745CD, 0xC54C206E */
6.66107313738753120669e-02, /* 0x3FB10D66, 0xA0D03D51 */
4.97687799461593236017e-02, /* 0x3FA97B4B, 0x24760DEB */
1.62858201153657823623e-02, /* 0x3F90AD3A, 0xE322DA11 */
};
static const double aT_odd[] = {
-1.99999999998764832476e-01, /* 0xBFC99999, 0x9998EBC4 */
-1.11111104054623557880e-01, /* 0xBFBC71C6, 0xFE231671 */
-7.69187620504482999495e-02, /* 0xBFB3B0F2, 0xAF749A6D */
-5.83357013379057348645e-02, /* 0xBFADDE2D, 0x52DEFD9A */
-3.65315727442169155270e-02, /* 0xBFA2B444, 0x2C6A6C2F */
};
double js_atan(double x)
{
double w,s1,s2,z;
int ix,hx,id;
hx = get_high_word(x);
ix = hx&0x7fffffff;
if(ix>=0x44100000) { /* if |x| >= 2^66 */
if(ix>0x7ff00000||
(ix==0x7ff00000&&(get_low_word(x)!=0)))
return x+x; /* NaN */
if(hx>0) return atanhi[3]+atanlo[3];
else return -atanhi[3]-atanlo[3];
} if (ix < 0x3fdc0000) { /* |x| < 0.4375 */
if (ix < 0x3e200000) { /* |x| < 2^-29 */
if(huge+x>one) return x; /* raise inexact */
}
id = -1;
} else {
x = fabs(x);
if (ix < 0x3ff30000) { /* |x| < 1.1875 */
if (ix < 0x3fe60000) { /* 7/16 <=|x|<11/16 */
id = 0; x = (2.0*x-one)/(2.0+x);
} else { /* 11/16<=|x|< 19/16 */
id = 1; x = (x-one)/(x+one);
}
} else {
if (ix < 0x40038000) { /* |x| < 2.4375 */
id = 2; x = (x-1.5)/(one+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*eval_poly(w, aT_even, 6);
s2 = w*eval_poly(w, aT_odd, 5);
if (id<0) return x - x*(s1+s2);
else {
z = atanhi[id] - ((x*(s1+s2) - atanlo[id]) - x);
return (hx<0)? -z:z;
}
}
/* @(#)e_atan2.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
*/
/* __ieee754_atan2(y,x)
* 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 const double
pi_o_4 = 7.8539816339744827900E-01, /* 0x3FE921FB, 0x54442D18 */
pi_o_2 = 1.5707963267948965580E+00, /* 0x3FF921FB, 0x54442D18 */
pi_lo = 1.2246467991473531772E-16; /* 0x3CA1A626, 0x33145C07 */
double js_atan2(double y, double x)
{
double z;
int k,m,hx,hy,ix,iy;
unsigned lx,ly;
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = hx&0x7fffffff;
iy = hy&0x7fffffff;
if(((ix|((lx|-lx)>>31))>0x7ff00000)||
((iy|((ly|-ly)>>31))>0x7ff00000)) /* x or y is NaN */
return x+y;
if(((hx-0x3ff00000)|lx)==0)
return js_atan(y); /* x=1.0 */
m = ((hy>>31)&1)|((hx>>30)&2); /* 2*sign(x)+sign(y) */
/* when y = 0 */
if((iy|ly)==0) {
z = 0;
goto done;
}
/* when x = 0 */
if((ix|lx)==0) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
/* when x is INF */
if(ix==0x7ff00000) {
if(iy==0x7ff00000) {
z = pi_o_4;
} else {
z = 0;
}
goto done;
}
/* when y is INF */
if(iy==0x7ff00000) return (hy<0)? -pi_o_2-tiny: pi_o_2+tiny;
/* compute y/x */
k = (iy-ix)>>20;
if(k > 60) {
z=pi_o_2+0.5*pi_lo; /* |y/x| > 2**60 */
} else if(hx<0&&k<-60) {
z=0.0; /* |y|/x < -2**60 */
} else {
z=js_atan(fabs(y/x)); /* safe to do y/x */
}
done:
switch (m) {
case 0:
return z ; /* atan(+,+) */
case 1:
z = set_high_word(z, get_high_word(z) ^ 0x80000000);
return z ; /* atan(-,+) */
case 2:
return pi-(z-pi_lo);/* atan(+,-) */
default: /* case 3 */
return (z-pi_lo)-pi;/* atan(-,-) */
}
}
/* @(#)e_exp.c 1.6 04/04/22 */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_exp(x)
* 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 Remes 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 const double
two = 2.0,
halF[2] = {0.5,-0.5,},
twom1000= 9.33263618503218878990e-302, /* 2**-1000=0x01700000,0*/
o_threshold= 7.09782712893383973096e+02, /* 0x40862E42, 0xFEFA39EF */
u_threshold= -7.45133219101941108420e+02, /* 0xc0874910, 0xD52D3051 */
ln2HI[2] ={ 6.93147180369123816490e-01, /* 0x3fe62e42, 0xfee00000 */
-6.93147180369123816490e-01,},/* 0xbfe62e42, 0xfee00000 */
ln2LO[2] ={ 1.90821492927058770002e-10, /* 0x3dea39ef, 0x35793c76 */
-1.90821492927058770002e-10,},/* 0xbdea39ef, 0x35793c76 */
invln2 = 1.44269504088896338700e+00; /* 0x3ff71547, 0x652b82fe */
static const double P[] = {
/* P1 */ 1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
/* P2 */ -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
/* P3 */ 6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
/* P4 */ -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
/* P5 */ 4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
};
/* compute exp(z+w)*2^n */
static double kernel_exp(double z, double w, double lo, double hi, int n)
{
int j;
double t, t1, r;
t = z*z;
t1 = z - t*eval_poly(t, P, 5);
r = (z*t1)/(t1-two) - (w+z*w);
z = one-((lo + r)-hi);
j = get_high_word(z);
j += (n<<20);
if((j>>20)<=0)
z = js_scalbn(z,n); /* subnormal output */
else
z = set_high_word(z, get_high_word(z) + (n<<20));
return z;
}
double js_exp(double x)
{
double hi,lo,t;
int k,xsb;
unsigned hx;
hx = get_high_word(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)|get_low_word(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 */
k = 0; /* avoid warning */
}
else k = 0;
/* x is now in primary range */
if (k == 0) {
lo = 0;
hi = x;
}
return kernel_exp(x, 0, lo, hi, k);
}
/* @(#)e_pow.c 1.5 04/04/22 SMI */
/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/* __ieee754_pow(x,y) return 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.
*
* 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 const double
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
two53 = 9007199254740992.0, /* 0x43400000, 0x00000000 */
/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
lg2 = 6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
lg2_h = 6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
lg2_l = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
ovt = 8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
cp = 9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
cp_h = 9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
cp_l = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
ivln2 = 1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
ivln2_h = 1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
ivln2_l = 1.92596299112661746887e-08, /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
ivlg10b2 = 0.3010299956639812, /* 0x3fd34413509f79ff 1/log2(10) */
ivlg10b2_h = 0.30102992057800293, /* 0x3fd3441300000000 1/log2(10) high */
ivlg10b2_l = 7.508597826552624e-8; /* 0x3e7427de7fbcc47c 1/log2(10) low */
static const double L_tab[] = {
/* L1 */ 5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
/* L2 */ 4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
/* L3 */ 3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
/* L4 */ 2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
/* L5 */ 2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
/* L6 */ 2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
};
/* compute (t1, t2) = log2(ax). is_small_ax is true if abs(ax)<= 2**-20 */
static void kernel_log2(double *pt1, double *pt2, double ax)
{
double t, u, v, t1, t2, r;
int n, j, ix, k;
double ss, s2, s_h, s_l, t_h, t_l, p_l, p_h, z_h, z_l;
n = 0;
ix = get_high_word(ax);
/* take care subnormal number */
if(ix<0x00100000)
{ax *= two53; n -= 53; ix = get_high_word(ax); }
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;}
ax = set_high_word(ax, ix);
/* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
u = ax-bp[k]; /* bp[0]=1.0, bp[1]=1.5 */
v = one/(ax+bp[k]);
ss = u*v;
s_h = zero_low(ss);
/* t_h=ax+bp[k] High */
t_h = zero;
t_h = set_high_word(t_h, ((ix>>1)|0x20000000)+0x00080000+(k<<18));
t_l = ax - (t_h-bp[k]);
s_l = v*((u-s_h*t_h)-s_h*t_l);
/* compute log(ax) */
s2 = ss*ss;
r = s2*s2*eval_poly(s2, L_tab, 6);
r += s_l*(s_h+ss);
s2 = s_h*s_h;
t_h = zero_low(3.0+s2+r);
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 = zero_low(u+v);
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(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
t = (double)n;
t1 = zero_low(((z_h+z_l)+dp_h[k])+t);
t2 = z_l-(((t1-t)-dp_h[k])-z_h);
*pt1 = t1;
*pt2 = t2;
}
/* flag = 0: log2()
flag = 1: log()
flag = 2: log10()
*/
static double js_log_internal(double x, int flag)
{
double p_h, p_l, t, u, v;
int hx, lx;
EXTRACT_WORDS(hx, lx, x);
if (hx <= 0) {
if (((hx&0x7fffffff)|lx)==0)
return -INFINITY; /* log(+-0)=-inf */
if (hx<0)
return NAN; /* log(-#) = NaN */
} else if (hx >= 0x7ff00000) {
/* log(inf) = inf, log(nan) = nan */
return x+x;
}
kernel_log2(&p_h, &p_l, x);
t = p_h + p_l;
if (flag == 0) {
return t;
} else {
t = zero_low(t);
if (flag == 1) {
/* multiply (p_l+p_h) by lg2 */
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2+t*lg2_l;
} else {
/* mutiply (p_l+p_h) by 1/log2(10) */
u = t*ivlg10b2_h;
v = (p_l-(t-p_h))*ivlg10b2+t*ivlg10b2_l;
}
return u+v;
}
}
double js_log2(double x)
{
return js_log_internal(x, 0);
}
double js_log(double x)
{
return js_log_internal(x, 1);
}
double js_log10(double x)
{
return js_log_internal(x, 2);
}
double js_pow(double x, double y)
{
double z,ax,p_h,p_l;
double y1,t1,t2,s,t,u,v,w;
int i,j,k,yisint,n;
int hx,hy,ix,iy;
unsigned lx,ly;
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = hx&0x7fffffff; iy = hy&0x7fffffff;
/* y==zero: x**0 = 1 */
if((iy|ly)==0) return one;
/* +-NaN return x+y */
if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
return x+y;
/* determine if y is an odd int when x < 0
* yisint = 0 ... y is not an integer
* yisint = 1 ... y is an odd int
* yisint = 2 ... y is an even int
*/
yisint = 0;
if(hx<0) {
if(iy>=0x43400000) yisint = 2; /* even integer y */
else if(iy>=0x3ff00000) {
k = (iy>>20)-0x3ff; /* exponent */
if(k>20) {
j = ly>>(52-k);
if((j<<(52-k))==ly) yisint = 2-(j&1);
} else if(ly==0) {
j = iy>>(20-k);
if((j<<(20-k))==iy) yisint = 2-(j&1);
}
}
}
/* special value of y */
if(ly==0) {
if (iy==0x7ff00000) { /* y is +-inf */
if(((ix-0x3ff00000)|lx)==0)
return y - y; /* inf**+-1 is NaN */
else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
return (hy>=0)? y: zero;
else /* (|x|<1)**-,+inf = inf,0 */
return (hy<0)?-y: zero;
}
if(iy==0x3ff00000) { /* y is +-1 */
if(hy<0) return one/x; else return x;
}
if(hy==0x40000000) return x*x; /* y is 2 */
if(hy==0x3fe00000) { /* y is 0.5 */
if(hx>=0) /* x >= +0 */
return js_sqrt(x);
}
}
ax = fabs(x);
/* special value of x */
if(lx==0) {
if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
z = ax; /*x is +-0,+-inf,+-1*/
if(hy<0) z = one/z; /* z = (1/|x|) */
if(hx<0) {
if(((ix-0x3ff00000)|yisint)==0) {
z = (z-z)/(z-z); /* (-1)**non-int is NaN */
} else if(yisint==1)
z = -z; /* (x<0)**odd = -(|x|**odd) */
}
return z;
}
}
n = (hx>>31)+1;
/* (x<0)**(non-int) is NaN */
if((n|yisint)==0) return (x-x)/(x-x);
s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
/* |y| is huge */
if(iy>0x41e00000) { /* if |y| > 2**31 */
if(iy>0x43f00000){ /* if |y| > 2**64, must o/uflow */
if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
}
/* over/underflow if x is not close to one */
if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
t = ax-one; /* t has 20 trailing zeros */
w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
u = ivln2_h*t; /* ivln2_h has 21 sig. bits */
v = t*ivln2_l-w*ivln2;
t1 = zero_low(u+v);
t2 = v-(t1-u);
} else {
kernel_log2(&t1, &t2, ax);
}
/* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
y1 = zero_low(y);
p_l = (y-y1)*t1+y*t2;
p_h = y1*t1;
z = p_l+p_h;
EXTRACT_WORDS(j, i, z);
if (j>=0x40900000) { /* z >= 1024 */
if(((j-0x40900000)|i)!=0) /* if z > 1024 */
return s*huge*huge; /* overflow */
else {
if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
}
} else if((j&0x7fffffff)>=0x4090cc00 ) { /* z <= -1075 */
if(((j-0xc090cc00)|i)!=0) /* z < -1075 */
return s*tiny*tiny; /* underflow */
else {
if(p_l<=z-p_h) return s*tiny*tiny; /* underflow */
}
}
/*
* compute 2**(p_h+p_l)
*/
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 = zero;
t = set_high_word(t, n&~(0x000fffff>>k));
n = ((n&0x000fffff)|0x00100000)>>(20-k);
if(j<0) n = -n;
p_h -= t;
}
/* multiply (p_l+p_h) by lg2 */
t = zero_low(p_l+p_h);
u = t*lg2_h;
v = (p_l-(t-p_h))*lg2+t*lg2_l;
z = u+v;
w = v-(z-u);
return s * kernel_exp(z, w, 0, z, n);
}