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/*- |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz. |
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* |
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* Developed at SunPro, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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* |
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* The argument reduction and testing for exceptional cases was |
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* written by Steven G. Kargl with input from Bruce D. Evans |
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* and David A. Schultz. |
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*/ |
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#include <float.h> |
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#include <ieeefp.h> |
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#include <math.h> |
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#include "math_private.h" |
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#define BIAS (LDBL_MAX_EXP - 1) |
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static const unsigned |
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B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */ |
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long double |
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cbrtl(long double x) |
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{ |
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long double v, r, s, t, w; |
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double dr, dt, dx; |
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float ft, fx; |
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uint32_t hx, lx; |
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uint16_t expsign, es; |
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int k; |
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volatile double vd1, vd2; |
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GET_LDOUBLE_EXP(expsign,x); |
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k = expsign & 0x7fff; |
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/* |
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* If x = +-Inf, then cbrt(x) = +-Inf. |
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* If x = NaN, then cbrt(x) = NaN. |
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*/ |
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if (k == BIAS + LDBL_MAX_EXP) |
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return (x + x); |
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if (k == 0) { |
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/* If x = +-0, then cbrt(x) = +-0. */ |
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GET_LDOUBLE_WORDS(es,hx,lx,x); |
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if ((hx|lx) == 0) { |
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return (x); |
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} |
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/* Adjust subnormal numbers. */ |
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x *= 0x1.0p514; |
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GET_LDOUBLE_EXP(k,x); |
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k &= 0x7fff; |
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k -= BIAS + 514; |
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} else |
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k -= BIAS; |
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SET_LDOUBLE_EXP(x,BIAS); |
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v = 1; |
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switch (k % 3) { |
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case 1: |
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case -2: |
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x = 2*x; |
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k--; |
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break; |
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case 2: |
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case -1: |
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x = 4*x; |
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k -= 2; |
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break; |
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} |
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SET_LDOUBLE_EXP(v, (expsign & 0x8000) | (BIAS + k / 3)); |
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/* |
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* The following is the guts of s_cbrtf, with the handling of |
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* special values removed and extra care for accuracy not taken, |
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* but with most of the extra accuracy not discarded. |
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*/ |
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/* ~5-bit estimate: */ |
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fx = x; |
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GET_FLOAT_WORD(hx, fx); |
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SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1)); |
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/* ~16-bit estimate: */ |
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dx = x; |
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dt = ft; |
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dr = dt * dt * dt; |
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dt = dt * (dx + dx + dr) / (dx + dr + dr); |
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/* ~47-bit estimate: */ |
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dr = dt * dt * dt; |
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dt = dt * (dx + dx + dr) / (dx + dr + dr); |
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/* |
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* dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8). |
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* Round it away from zero to 32 bits (32 so that t*t is exact, and |
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* away from zero for technical reasons). |
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*/ |
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vd2 = 0x1.0p32; |
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vd1 = 0x1.0p-31; |
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#define vd ((long double)vd2 + vd1) |
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t = dt + vd - 0x1.0p32; |
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/* |
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* Final step Newton iteration to 64 or 113 bits with |
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* error < 0.667 ulps |
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*/ |
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s=t*t; /* t*t is exact */ |
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r=x/s; /* error <= 0.5 ulps; |r| < |t| */ |
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w=t+t; /* t+t is exact */ |
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r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ |
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t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ |
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t *= v; |
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return (t); |
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} |