1/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
  2/*-
  3 * ====================================================
  4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  5 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
  6 *
  7 * Developed at SunPro, a Sun Microsystems, Inc. business.
  8 * Permission to use, copy, modify, and distribute this
  9 * software is freely granted, provided that this notice
 10 * is preserved.
 11 * ====================================================
 12 *
 13 * The argument reduction and testing for exceptional cases was
 14 * written by Steven G. Kargl with input from Bruce D. Evans
 15 * and David A. Schultz.
 16 */
 17
 18#include "libm.h"
 19
 20#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
 21long double cbrtl(long double x)
 22{
 23	return cbrt(x);
 24}
 25#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
 26static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
 27
 28long double cbrtl(long double x)
 29{
 30	union ldshape u = {x}, v;
 31	union {float f; uint32_t i;} uft;
 32	long double r, s, t, w;
 33	double_t dr, dt, dx;
 34	float_t ft;
 35	int e = u.i.se & 0x7fff;
 36	int sign = u.i.se & 0x8000;
 37
 38	/*
 39	 * If x = +-Inf, then cbrt(x) = +-Inf.
 40	 * If x = NaN, then cbrt(x) = NaN.
 41	 */
 42	if (e == 0x7fff)
 43		return x + x;
 44	if (e == 0) {
 45		/* Adjust subnormal numbers. */
 46		u.f *= 0x1p120;
 47		e = u.i.se & 0x7fff;
 48		/* If x = +-0, then cbrt(x) = +-0. */
 49		if (e == 0)
 50			return x;
 51		e -= 120;
 52	}
 53	e -= 0x3fff;
 54	u.i.se = 0x3fff;
 55	x = u.f;
 56	switch (e % 3) {
 57	case 1:
 58	case -2:
 59		x *= 2;
 60		e--;
 61		break;
 62	case 2:
 63	case -1:
 64		x *= 4;
 65		e -= 2;
 66		break;
 67	}
 68	v.f = 1.0;
 69	v.i.se = sign | (0x3fff + e/3);
 70
 71	/*
 72	 * The following is the guts of s_cbrtf, with the handling of
 73	 * special values removed and extra care for accuracy not taken,
 74	 * but with most of the extra accuracy not discarded.
 75	 */
 76
 77	/* ~5-bit estimate: */
 78	uft.f = x;
 79	uft.i = (uft.i & 0x7fffffff)/3 + B1;
 80	ft = uft.f;
 81
 82	/* ~16-bit estimate: */
 83	dx = x;
 84	dt = ft;
 85	dr = dt * dt * dt;
 86	dt = dt * (dx + dx + dr) / (dx + dr + dr);
 87
 88	/* ~47-bit estimate: */
 89	dr = dt * dt * dt;
 90	dt = dt * (dx + dx + dr) / (dx + dr + dr);
 91
 92#if LDBL_MANT_DIG == 64
 93	/*
 94	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
 95	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
 96	 * away from zero for technical reasons).
 97	 */
 98	t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
 99#elif LDBL_MANT_DIG == 113
100	/*
101	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
102	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
103	 * might be avoidable in this case, since on most machines dt will
104	 * have been evaluated in 53-bit precision and the technical reasons
105	 * for rounding up might not apply to either case in cbrtl() since
106	 * dt is much more accurate than needed.
107	 */
108	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
109#endif
110
111	/*
112	 * Final step Newton iteration to 64 or 113 bits with
113	 * error < 0.667 ulps
114	 */
115	s = t*t;         /* t*t is exact */
116	r = x/s;         /* error <= 0.5 ulps; |r| < |t| */
117	w = t+t;         /* t+t is exact */
118	r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
119	t = t+t*r;       /* error <= 0.5 + 0.5/3 + epsilon */
120
121	t *= v.f;
122	return t;
123}
124#endif