| Elliott Hughes | a0ee078 | 2013-01-30 19:06:37 -0800 | [diff] [blame] | 1 | /* @(#)s_cbrt.c 5.1 93/09/24 */ |
| 2 | /* |
| 3 | * ==================================================== |
| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| 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 | * Optimized by Bruce D. Evans. |
| 13 | */ |
| 14 | |
| 15 | #include <sys/cdefs.h> |
| 16 | __FBSDID("$FreeBSD$"); |
| 17 | |
| 18 | #include "math.h" |
| 19 | #include "math_private.h" |
| 20 | |
| 21 | /* cbrt(x) |
| 22 | * Return cube root of x |
| 23 | */ |
| 24 | static const u_int32_t |
| 25 | B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ |
| 26 | B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ |
| 27 | |
| 28 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ |
| 29 | static const double |
| 30 | P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ |
| 31 | P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ |
| 32 | P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ |
| 33 | P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ |
| 34 | P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ |
| 35 | |
| 36 | double |
| 37 | cbrt(double x) |
| 38 | { |
| 39 | int32_t hx; |
| 40 | union { |
| 41 | double value; |
| 42 | uint64_t bits; |
| 43 | } u; |
| 44 | double r,s,t=0.0,w; |
| 45 | u_int32_t sign; |
| 46 | u_int32_t high,low; |
| 47 | |
| 48 | EXTRACT_WORDS(hx,low,x); |
| 49 | sign=hx&0x80000000; /* sign= sign(x) */ |
| 50 | hx ^=sign; |
| 51 | if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */ |
| 52 | |
| 53 | /* |
| 54 | * Rough cbrt to 5 bits: |
| 55 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) |
| 56 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and |
| 57 | * "%" are integer division and modulus with rounding towards minus |
| 58 | * infinity. The RHS is always >= the LHS and has a maximum relative |
| 59 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the |
| 60 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE |
| 61 | * floating point representation, for finite positive normal values, |
| 62 | * ordinary integer divison of the value in bits magically gives |
| 63 | * almost exactly the RHS of the above provided we first subtract the |
| 64 | * exponent bias (1023 for doubles) and later add it back. We do the |
| 65 | * subtraction virtually to keep e >= 0 so that ordinary integer |
| 66 | * division rounds towards minus infinity; this is also efficient. |
| 67 | */ |
| 68 | if(hx<0x00100000) { /* zero or subnormal? */ |
| 69 | if((hx|low)==0) |
| 70 | return(x); /* cbrt(0) is itself */ |
| 71 | SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */ |
| 72 | t*=x; |
| 73 | GET_HIGH_WORD(high,t); |
| 74 | INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0); |
| 75 | } else |
| 76 | INSERT_WORDS(t,sign|(hx/3+B1),0); |
| 77 | |
| 78 | /* |
| 79 | * New cbrt to 23 bits: |
| 80 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) |
| 81 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) |
| 82 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation |
| 83 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this |
| 84 | * gives us bounds for r = t**3/x. |
| 85 | * |
| 86 | * Try to optimize for parallel evaluation as in k_tanf.c. |
| 87 | */ |
| 88 | r=(t*t)*(t/x); |
| 89 | t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); |
| 90 | |
| 91 | /* |
| 92 | * Round t away from zero to 23 bits (sloppily except for ensuring that |
| 93 | * the result is larger in magnitude than cbrt(x) but not much more than |
| 94 | * 2 23-bit ulps larger). With rounding towards zero, the error bound |
| 95 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps |
| 96 | * in the rounded t, the infinite-precision error in the Newton |
| 97 | * approximation barely affects third digit in the final error |
| 98 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps |
| 99 | * before the final error is larger than 0.667 ulps. |
| 100 | */ |
| 101 | u.value=t; |
| 102 | u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL; |
| 103 | t=u.value; |
| 104 | |
| 105 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */ |
| 106 | s=t*t; /* t*t is exact */ |
| 107 | r=x/s; /* error <= 0.5 ulps; |r| < |t| */ |
| 108 | w=t+t; /* t+t is exact */ |
| 109 | r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ |
| 110 | t=t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ |
| 111 | |
| 112 | return(t); |
| 113 | } |
| 114 | |
| 115 | #if (LDBL_MANT_DIG == 53) |
| 116 | __weak_reference(cbrt, cbrtl); |
| 117 | #endif |