/* Copyright 2016 Samsung Electronics Co., Ltd.
* Copyright 2016 University of Szeged
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*
* This file is based on work under the following copyright and permission
* notice:
*
* 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.
*
* @(#)e_pow.c 1.5 04/04/22
*/
#include "jerry-libm-internal.h"
/* 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 muti-precision
* arithmetic, where |y'|<=0.5.
* 3. Return x**y = 2**n*exp(y'*log2)
*
* Special cases:
* 0. +1 ** (anything) is 1
* 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 1
* 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,
};
static const double dp_h[] =
{
0.0,
5.84962487220764160156e-01, /* 0x3FE2B803, 0x40000000 */
};
static const double dp_l[] =
{
0.0,
1.35003920212974897128e-08, /* 0x3E4CFDEB, 0x43CFD006 */
};
#define zero 0.0
#define one 1.0
#define two 2.0
#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */
#define huge 1.0e300
#define tiny 1.0e-300
/* poly coefs for (3/2) * (log(x) - 2s - 2/3 * s**3 */
#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */
#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */
#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */
#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */
#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */
#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */
#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */
#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */
#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */
#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */
#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */
#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */
#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */
#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */
#define ovt 8.0085662595372944372e-0017 /* -(1024-log2(ovfl+.5ulp)) */
#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD = 2 / (3 ln2) */
#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 = (float) cp */
#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 = tail of cp_h */
#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE = 1 / ln2 */
#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 = 24b 1 / ln2 */
#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 = 1 / ln2 tail */
double
pow (double x, double y)
{
double z, ax, z_h, z_l, p_h, p_l;
double y1, t1, t2, r, s, t, u, v, w;
int i, j, k, yisint, n;
int hx, hy, ix, iy;
unsigned lx, ly;
hx = __HI (x);
lx = __LO (x);
hy = __HI (y);
ly = __LO (y);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
/* x == one: 1**y = 1 */
if (((hx - 0x3ff00000) | lx) == 0)
{
return one;
}
/* 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) /* even integer y */
{
yisint = 2;
}
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) /* +-1**+-inf is 1 */
{
return one;
}
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) /* y is 2 */
{
return x * x;
}
if (hy == 0x3fe00000) /* y is 0.5 */
{
if (hx >= 0) /* x >= +0 */
{
return 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;
}
/* now |1 - x| is tiny <= 2**-20, suffice to compute
log(x) by x - x^2 / 2 + x^3 / 3 - x^4 / 4 */
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 = u + v;
__LO (t1) = 0;
t2 = v - (t1 - u);
}
else
{
double ss, s2, s_h, s_l, t_h, t_l;
n = 0;
/* take care subnormal number */
if (ix < 0x00100000)
{
ax *= two53;
n -= 53;
ix = __HI (ax);
}
n += ((ix) >> 20) - 0x3ff;
j = ix & 0x000fffff;
/* determine interval */
ix = j | 0x3ff00000; /* normalize ix */
if (j <= 0x3988E) /* |x| < sqrt(3/2) */
{
k = 0;
}
else if (j < 0xBB67A) /* |x| < sqrt(3) */
{
k = 1;
}
else
{
k = 0;
n += 1;
ix -= 0x00100000;
}
__HI (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 = ss;
__LO (s_h) = 0;
/* t_h = ax + bp[k] High */
t_h = zero;
__HI (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 * (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;
__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 / (3 * log2) * (ss + ...) */
p_h = u + v;
__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(ax) = (ss + ...) * 2 / (3 * log2) = n + dp_h + z_h + z_l */
t = (double) n;
t1 = (((z_h + z_l) + dp_h[k]) + t);
__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) */
y1 = y;
__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 * 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;
__HI (t) = (n & ~(0x000fffff >> k));
n = ((n & 0x000fffff) | 0x00100000) >> (20 - k);
if (j < 0)
{
n = -n;
}
p_h -= t;
}
t = p_l + p_h;
__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 - two) - (w + z * w);
z = one - (r - z);
j = __HI (z);
j += (n << 20);
if ((j >> 20) <= 0) /* subnormal output */
{
z = scalbn (z, n);
}
else
{
__HI (z) += (n << 20);
}
return s * z;
} /* pow */
#undef zero
#undef one
#undef two
#undef two53
#undef huge
#undef tiny
#undef L1
#undef L2
#undef L3
#undef L4
#undef L5
#undef L6
#undef P1
#undef P2
#undef P3
#undef P4
#undef P5
#undef lg2
#undef lg2_h
#undef lg2_l
#undef ovt
#undef cp
#undef cp_h
#undef cp_l
#undef ivln2
#undef ivln2_h
#undef ivln2_l