144 lines
3.3 KiB
C
144 lines
3.3 KiB
C
/*-
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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 <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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#include <float.h>
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#ifdef __i386__
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#include <ieeefp.h>
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#endif
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#include "fpmath.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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union IEEEl2bits u, v;
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long double 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;
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uint16_t expsign;
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int k;
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u.e = x;
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expsign = u.xbits.expsign;
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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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ENTERI();
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if (k == 0) {
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/* If x = +-0, then cbrt(x) = +-0. */
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if ((u.bits.manh | u.bits.manl) == 0)
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RETURNI(x);
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/* Adjust subnormal numbers. */
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u.e *= 0x1.0p514;
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k = u.bits.exp;
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k -= BIAS + 514;
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} else
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k -= BIAS;
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u.xbits.expsign = BIAS;
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v.e = 1;
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x = u.e;
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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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v.xbits.expsign = (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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#if LDBL_MANT_DIG == 64
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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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volatile double vd2 = 0x1.0p32;
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volatile double 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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#elif LDBL_MANT_DIG == 113
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/*
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* Round dt away from zero to 47 bits. Since we don't trust the 47,
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* add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
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* might be avoidable in this case, since on most machines dt will
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* have been evaluated in 53-bit precision and the technical reasons
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* for rounding up might not apply to either case in cbrtl() since
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* dt is much more accurate than needed.
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*/
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t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
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#else
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#error "Unsupported long double format"
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#endif
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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.e;
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RETURNI(t);
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}
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