Annotation of src/lib/libm/src/s_cbrtl.c, Revision 1.1.8.2
1.1.8.2 ! tls 1: /*-
! 2: * ====================================================
! 3: * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
! 4: * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
! 5: *
! 6: * Developed at SunPro, a Sun Microsystems, Inc. business.
! 7: * Permission to use, copy, modify, and distribute this
! 8: * software is freely granted, provided that this notice
! 9: * is preserved.
! 10: * ====================================================
! 11: *
! 12: * The argument reduction and testing for exceptional cases was
! 13: * written by Steven G. Kargl with input from Bruce D. Evans
! 14: * and David A. Schultz.
! 15: */
! 16:
! 17: #include <sys/cdefs.h>
! 18: __RCSID("$NetBSD$");
! 19: #if 0
! 20: __FBSDID("$FreeBSD: head/lib/msun/src/s_cbrtl.c 238924 2012-07-30 21:58:28Z kargl $");
! 21: #endif
! 22:
! 23: #include "namespace.h"
! 24: #include <machine/ieee.h>
! 25: #include <float.h>
! 26:
! 27: #include "math.h"
! 28: #include "math_private.h"
! 29:
! 30: #ifdef __HAVE_LONG_DOUBLE
! 31: __weak_alias(cbrtl, _cbrtl)
! 32:
! 33: static const unsigned
! 34: B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
! 35:
! 36: long double
! 37: cbrtl(long double x)
! 38: {
! 39: union ieee_ext_u ux, vx;
! 40: long double r, s, t, w;
! 41: double dr, dt, dx;
! 42: float ft, fx;
! 43: uint32_t hx;
! 44: int k;
! 45:
! 46: ux.extu_ld = x;
! 47:
! 48:
! 49: /*
! 50: * If x = +-Inf, then cbrt(x) = +-Inf.
! 51: * If x = NaN, then cbrt(x) = NaN.
! 52: */
! 53: if (ux.extu_exp == EXT_EXP_INFNAN)
! 54: return (x + x);
! 55: if ((ux.extu_frach | ux.extu_fracl | ux.extu_exp) == 0)
! 56: return (x);
! 57:
! 58: vx.extu_ld = 1;
! 59: vx.extu_ext.ext_sign = ux.extu_ext.ext_sign;
! 60: ux.extu_ext.ext_sign = 0;
! 61: if (ux.extu_exp == 0) {
! 62: /* Adjust subnormal numbers. */
! 63: ux.extu_ld *= 0x1.0p514;
! 64: k = ux.extu_exp - EXT_EXP_BIAS - 514;
! 65: } else {
! 66: k = ux.extu_exp - EXT_EXP_BIAS;
! 67: }
! 68:
! 69: ux.extu_exp = EXT_EXP_BIAS;
! 70: x = ux.extu_ld;
! 71: switch (k % 3) {
! 72: case 1:
! 73: case -2:
! 74: x = 2*x;
! 75: k--;
! 76: break;
! 77: case 2:
! 78: case -1:
! 79: x = 4*x;
! 80: k -= 2;
! 81: break;
! 82: }
! 83: vx.extu_exp = EXT_EXP_BIAS + k / 3;
! 84:
! 85: /*
! 86: * The following is the guts of s_cbrtf, with the handling of
! 87: * special values removed and extra care for accuracy not taken,
! 88: * but with most of the extra accuracy not discarded.
! 89: */
! 90:
! 91: /* ~5-bit estimate: */
! 92: fx = x;
! 93: GET_FLOAT_WORD(hx, fx);
! 94: SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
! 95:
! 96: /* ~16-bit estimate: */
! 97: dx = x;
! 98: dt = ft;
! 99: dr = dt * dt * dt;
! 100: dt = dt * (dx + dx + dr) / (dx + dr + dr);
! 101:
! 102: /* ~47-bit estimate: */
! 103: dr = dt * dt * dt;
! 104: dt = dt * (dx + dx + dr) / (dx + dr + dr);
! 105:
! 106: #if LDBL_MANT_DIG == 64
! 107: /*
! 108: * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
! 109: * Round it away from zero to 32 bits (32 so that t*t is exact, and
! 110: * away from zero for technical reasons).
! 111: */
! 112: volatile double vd2 = 0x1.0p32;
! 113: volatile double vd1 = 0x1.0p-31;
! 114: #define vd ((long double)vd2 + vd1)
! 115:
! 116: t = dt + vd - 0x1.0p32;
! 117: #elif LDBL_MANT_DIG == 113
! 118: /*
! 119: * Round dt away from zero to 47 bits. Since we don't trust the 47,
! 120: * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
! 121: * might be avoidable in this case, since on most machines dt will
! 122: * have been evaluated in 53-bit precision and the technical reasons
! 123: * for rounding up might not apply to either case in cbrtl() since
! 124: * dt is much more accurate than needed.
! 125: */
! 126: t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
! 127: #else
! 128: #error "Unsupported long double format"
! 129: #endif
! 130:
! 131: /*
! 132: * Final step Newton iteration to 64 or 113 bits with
! 133: * error < 0.667 ulps
! 134: */
! 135: s=t*t; /* t*t is exact */
! 136: r=x/s; /* error <= 0.5 ulps; |r| < |t| */
! 137: w=t+t; /* t+t is exact */
! 138: r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
! 139: t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
! 140:
! 141: t *= vx.extu_ld;
! 142: return t;
! 143: }
! 144:
! 145: #endif /* __HAVE_LONG_DOUBLE */
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