119 lines
4.1 KiB
C
119 lines
4.1 KiB
C
/* @(#)s_cbrt.c 5.1 93/09/24 */
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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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*
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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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* Optimized by Bruce D. Evans.
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*/
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#include "cdefs-compat.h"
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//__FBSDID("$FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.17 2011/03/12 16:50:39 kargl Exp $");
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#include <openlibm_math.h>
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#include "math_private.h"
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/* cbrt(x)
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* Return cube root of x
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*/
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static const u_int32_t
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B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
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B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
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/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
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static const double
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P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
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P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */
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P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
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P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
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P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
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OLM_DLLEXPORT double
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cbrt(double x)
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{
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int32_t hx;
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union {
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double value;
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u_int64_t bits;
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} u;
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double r,s,t=0.0,w;
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u_int32_t sign;
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u_int32_t high,low;
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EXTRACT_WORDS(hx,low,x);
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sign=hx&0x80000000; /* sign= sign(x) */
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hx ^=sign;
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if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
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/*
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* Rough cbrt to 5 bits:
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* cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
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* where e is integral and >= 0, m is real and in [0, 1), and "/" and
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* "%" are integer division and modulus with rounding towards minus
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* infinity. The RHS is always >= the LHS and has a maximum relative
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* error of about 1 in 16. Adding a bias of -0.03306235651 to the
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* (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
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* floating point representation, for finite positive normal values,
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* ordinary integer divison of the value in bits magically gives
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* almost exactly the RHS of the above provided we first subtract the
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* exponent bias (1023 for doubles) and later add it back. We do the
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* subtraction virtually to keep e >= 0 so that ordinary integer
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* division rounds towards minus infinity; this is also efficient.
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*/
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if(hx<0x00100000) { /* zero or subnormal? */
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if((hx|low)==0)
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return(x); /* cbrt(0) is itself */
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SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
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t*=x;
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GET_HIGH_WORD(high,t);
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INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
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} else
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INSERT_WORDS(t,sign|(hx/3+B1),0);
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/*
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* New cbrt to 23 bits:
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* cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
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* where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
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* to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
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* has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
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* gives us bounds for r = t**3/x.
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*
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* Try to optimize for parallel evaluation as in k_tanf.c.
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*/
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r=(t*t)*(t/x);
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t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
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/*
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* Round t away from zero to 23 bits (sloppily except for ensuring that
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* the result is larger in magnitude than cbrt(x) but not much more than
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* 2 23-bit ulps larger). With rounding towards zero, the error bound
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* would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
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* in the rounded t, the infinite-precision error in the Newton
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* approximation barely affects third digit in the final error
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* 0.667; the error in the rounded t can be up to about 3 23-bit ulps
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* before the final error is larger than 0.667 ulps.
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*/
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u.value=t;
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u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
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t=u.value;
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/* one step Newton iteration to 53 bits with error < 0.667 ulps */
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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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return(t);
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}
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#if (LDBL_MANT_DIG == 53)
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openlibm_weak_reference(cbrt, cbrtl);
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#endif
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