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Diffstat (limited to 'sysdeps/ieee754/ldbl-128/e_powl.c')
| -rw-r--r-- | sysdeps/ieee754/ldbl-128/e_powl.c | 422 |
1 files changed, 422 insertions, 0 deletions
diff --git a/sysdeps/ieee754/ldbl-128/e_powl.c b/sysdeps/ieee754/ldbl-128/e_powl.c new file mode 100644 index 0000000000..6d5b9d52d2 --- /dev/null +++ b/sysdeps/ieee754/ldbl-128/e_powl.c @@ -0,0 +1,422 @@ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* Expansions and modifications for 128-bit long double contributed by + Stephen L. Moshier <moshier@na-net.ornl.gov> */ + +/* __ieee754_powl(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 113-53 = 60 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: + * 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 + * + */ + +#include "math.h" +#include "math_private.h" + +static const long double bp[] = { + 1.0L, + 1.5L, +}; + +/* log_2(1.5) */ +static const long double dp_h[] = { + 0.0, + 5.8496250072115607565592654282227158546448E-1L +}; + +/* Low part of log_2(1.5) */ +static const long double dp_l[] = { + 0.0, + 1.0579781240112554492329533686862998106046E-16L +}; + +static const long double zero = 0.0L, + one = 1.0L, + two = 2.0L, + two113 = 1.0384593717069655257060992658440192E34L, + huge = 1.0e3000L, + tiny = 1.0e-3000L; + +/* 3/2 log x = 3 z + z^3 + z^3 (z^2 R(z^2)) + z = (x-1)/(x+1) + 1 <= x <= 1.25 + Peak relative error 2.3e-37 */ +static const long double LN[] = +{ + -3.0779177200290054398792536829702930623200E1L, + 6.5135778082209159921251824580292116201640E1L, + -4.6312921812152436921591152809994014413540E1L, + 1.2510208195629420304615674658258363295208E1L, + -9.9266909031921425609179910128531667336670E-1L +}; +static const long double LD[] = +{ + -5.129862866715009066465422805058933131960E1L, + 1.452015077564081884387441590064272782044E2L, + -1.524043275549860505277434040464085593165E2L, + 7.236063513651544224319663428634139768808E1L, + -1.494198912340228235853027849917095580053E1L + /* 1.0E0 */ +}; + +/* exp(x) = 1 + x - x / (1 - 2 / (x - x^2 R(x^2))) + 0 <= x <= 0.5 + Peak relative error 5.7e-38 */ +static const long double PN[] = +{ + 5.081801691915377692446852383385968225675E8L, + 9.360895299872484512023336636427675327355E6L, + 4.213701282274196030811629773097579432957E4L, + 5.201006511142748908655720086041570288182E1L, + 9.088368420359444263703202925095675982530E-3L, +}; +static const long double PD[] = +{ + 3.049081015149226615468111430031590411682E9L, + 1.069833887183886839966085436512368982758E8L, + 8.259257717868875207333991924545445705394E5L, + 1.872583833284143212651746812884298360922E3L, + /* 1.0E0 */ +}; + +static const long double + /* ln 2 */ + lg2 = 6.9314718055994530941723212145817656807550E-1L, + lg2_h = 6.9314718055994528622676398299518041312695E-1L, + lg2_l = 2.3190468138462996154948554638754786504121E-17L, + ovt = 8.0085662595372944372e-0017L, + /* 2/(3*log(2)) */ + cp = 9.6179669392597560490661645400126142495110E-1L, + cp_h = 9.6179669392597555432899980587535537779331E-1L, + cp_l = 5.0577616648125906047157785230014751039424E-17L; + +#ifdef __STDC__ +long double +__ieee754_powl (long double x, long double y) +#else +long double +__ieee754_powl (x, y) + long double x, y; +#endif +{ + long double z, ax, z_h, z_l, p_h, p_l; + long double y1, t1, t2, r, s, t, u, v, w; + long double s2, s_h, s_l, t_h, t_l; + int32_t i, j, k, yisint, n; + u_int32_t ix, iy; + int32_t hx, hy; + ieee854_long_double_shape_type o, p, q; + + p.value = x; + hx = p.parts32.w0; + ix = hx & 0x7fffffff; + + q.value = y; + hy = q.parts32.w0; + iy = hy & 0x7fffffff; + + + /* y==zero: x**0 = 1 */ + if ((iy | q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) + return one; + + /* +-NaN return x+y */ + if ((ix > 0x7fff0000) + || ((ix == 0x7fff0000) + && ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) != 0)) + || (iy > 0x7fff0000) + || ((iy == 0x7fff0000) + && ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) != 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 >= 0x40700000) /* 2^113 */ + yisint = 2; /* even integer y */ + else if (iy >= 0x3fff0000) /* 1.0 */ + { + if (__floorl (y) == y) + { + z = 0.5 * y; + if (__floorl (z) == z) + yisint = 2; + else + yisint = 1; + } + } + } + + /* special value of y */ + if ((q.parts32.w1 | q.parts32.w2 | q.parts32.w3) == 0) + { + if (iy == 0x7fff0000) /* y is +-inf */ + { + if (((ix - 0x3fff0000) | p.parts32.w1 | p.parts32.w2 | p.parts32.w3) + == 0) + return y - y; /* inf**+-1 is NaN */ + else if (ix >= 0x3fff0000) /* (|x|>1)**+-inf = inf,0 */ + return (hy >= 0) ? y : zero; + else /* (|x|<1)**-,+inf = inf,0 */ + return (hy < 0) ? -y : zero; + } + if (iy == 0x3fff0000) + { /* y is +-1 */ + if (hy < 0) + return one / x; + else + return x; + } + if (hy == 0x40000000) + return x * x; /* y is 2 */ + if (hy == 0x3ffe0000) + { /* y is 0.5 */ + if (hx >= 0) /* x >= +0 */ + return __ieee754_sqrtl (x); + } + } + + ax = fabsl (x); + /* special value of x */ + if ((p.parts32.w1 | p.parts32.w2 | p.parts32.w3) == 0) + { + if (ix == 0x7fff0000 || ix == 0 || ix == 0x3fff0000) + { + z = ax; /*x is +-0,+-inf,+-1 */ + if (hy < 0) + z = one / z; /* z = (1/|x|) */ + if (hx < 0) + { + if (((ix - 0x3fff0000) | 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; + } + } + + /* (x<0)**(non-int) is NaN */ + if (((((u_int32_t) hx >> 31) - 1) | yisint) == 0) + return (x - x) / (x - x); + + /* |y| is huge. + 2^-16495 = 1/2 of smallest representable value. + If (1 - 1/131072)^y underflows, y > 1.4986e9 */ + if (iy > 0x401d654b) + { + /* if (1 - 2^-113)^y underflows, y > 1.1873e38 */ + if (iy > 0x407d654b) + { + if (ix <= 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix >= 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + /* over/underflow if x is not close to one */ + if (ix < 0x3ffeffff) + return (hy < 0) ? huge * huge : tiny * tiny; + if (ix > 0x3fff0000) + return (hy > 0) ? huge * huge : tiny * tiny; + } + + n = 0; + /* take care subnormal number */ + if (ix < 0x00010000) + { + ax *= two113; + n -= 113; + o.value = ax; + ix = o.parts32.w0; + } + n += ((ix) >> 16) - 0x3fff; + j = ix & 0x0000ffff; + /* determine interval */ + ix = j | 0x3fff0000; /* normalize ix */ + if (j <= 0x3988) + k = 0; /* |x|<sqrt(3/2) */ + else if (j < 0xbb67) + k = 1; /* |x|<sqrt(3) */ + else + { + k = 0; + n += 1; + ix -= 0x00010000; + } + + o.value = ax; + o.parts32.w0 = ix; + ax = o.value; + + /* compute s = 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]); + s = u * v; + s_h = s; + + o.value = s_h; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + s_h = o.value; + /* t_h=ax+bp[k] High */ + t_h = ax + bp[k]; + o.value = t_h; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + t_h = o.value; + t_l = ax - (t_h - bp[k]); + s_l = v * ((u - s_h * t_h) - s_h * t_l); + /* compute log(ax) */ + s2 = s * s; + u = LN[0] + s2 * (LN[1] + s2 * (LN[2] + s2 * (LN[3] + s2 * LN[4]))); + v = LD[0] + s2 * (LD[1] + s2 * (LD[2] + s2 * (LD[3] + s2 * (LD[4] + s2)))); + r = s2 * s2 * u / v; + r += s_l * (s_h + s); + s2 = s_h * s_h; + t_h = 3.0 + s2 + r; + o.value = t_h; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + t_h = o.value; + t_l = r - ((t_h - 3.0) - s2); + /* u+v = s*(1+...) */ + u = s_h * t_h; + v = s_l * t_h + t_l * s; + /* 2/(3log2)*(s+...) */ + p_h = u + v; + o.value = p_h; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + p_h = o.value; + 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) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */ + t = (long double) n; + t1 = (((z_h + z_l) + dp_h[k]) + t); + o.value = t1; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + t1 = o.value; + t2 = z_l - (((t1 - t) - dp_h[k]) - z_h); + + /* s (sign of result -ve**odd) = -1 else = 1 */ + s = one; + if (((((u_int32_t) hx >> 31) - 1) | (yisint - 1)) == 0) + s = -one; /* (-ve)**(odd int) */ + + /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */ + y1 = y; + o.value = y1; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + y1 = o.value; + p_l = (y - y1) * t1 + y * t2; + p_h = y1 * t1; + z = p_l + p_h; + o.value = z; + j = o.parts32.w0; + if (j >= 0x400d0000) /* z >= 16384 */ + { + /* if z > 16384 */ + if (((j - 0x400d0000) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) != 0) + return s * huge * huge; /* overflow */ + else + { + if (p_l + ovt > z - p_h) + return s * huge * huge; /* overflow */ + } + } + else if ((j & 0x7fffffff) >= 0x400d01b9) /* z <= -16495 */ + { + /* z < -16495 */ + if (((j - 0xc00d01bc) | o.parts32.w1 | o.parts32.w2 | o.parts32.w3) + != 0) + 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 >> 16) - 0x3fff; + n = 0; + if (i > 0x3ffe0000) + { /* if |z| > 0.5, set n = [z+0.5] */ + n = __floorl (z + 0.5L); + t = n; + p_h -= t; + } + t = p_l + p_h; + o.value = t; + o.parts32.w3 = 0; + o.parts32.w2 &= 0xf8000000; + t = o.value; + u = t * lg2_h; + v = (p_l - (t - p_h)) * lg2 + t * lg2_l; + z = u + v; + w = v - (z - u); + /* exp(z) */ + t = z * z; + u = PN[0] + t * (PN[1] + t * (PN[2] + t * (PN[3] + t * PN[4]))); + v = PD[0] + t * (PD[1] + t * (PD[2] + t * (PD[3] + t))); + t1 = z - t * u / v; + r = (z * t1) / (t1 - two) - (w + z * w); + z = one - (r - z); + o.value = z; + j = o.parts32.w0; + j += (n << 16); + if ((j >> 16) <= 0) + z = __scalbnl (z, n); /* subnormal output */ + else + { + o.parts32.w0 = j; + z = o.value; + } + return s * z; +} |