[352] | 1 | /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | *
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| 12 | * Optimized by Bruce D. Evans.
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| 13 | */
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| 14 | /* cbrt(x)
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| 15 | * Return cube root of x
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| 16 | */
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| 17 |
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| 18 | #include <math.h>
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| 19 | #include <stdint.h>
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| 20 |
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| 21 | static const uint32_t
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| 22 | B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
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| 23 | B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
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| 24 |
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| 25 | /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
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| 26 | static const double
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| 27 | P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
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| 28 | P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */
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| 29 | P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
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| 30 | P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
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| 31 | P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
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| 32 |
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| 33 | double cbrt(double x)
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| 34 | {
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| 35 | union {double f; uint64_t i;} u = {x};
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| 36 | double_t r,s,t,w;
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| 37 | uint32_t hx = u.i>>32 & 0x7fffffff;
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| 38 |
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| 39 | if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */
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| 40 | return x+x;
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| 41 |
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| 42 | /*
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| 43 | * Rough cbrt to 5 bits:
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| 44 | * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
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| 45 | * where e is integral and >= 0, m is real and in [0, 1), and "/" and
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| 46 | * "%" are integer division and modulus with rounding towards minus
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| 47 | * infinity. The RHS is always >= the LHS and has a maximum relative
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| 48 | * error of about 1 in 16. Adding a bias of -0.03306235651 to the
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| 49 | * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
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| 50 | * floating point representation, for finite positive normal values,
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| 51 | * ordinary integer divison of the value in bits magically gives
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| 52 | * almost exactly the RHS of the above provided we first subtract the
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| 53 | * exponent bias (1023 for doubles) and later add it back. We do the
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| 54 | * subtraction virtually to keep e >= 0 so that ordinary integer
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| 55 | * division rounds towards minus infinity; this is also efficient.
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| 56 | */
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| 57 | if (hx < 0x00100000) { /* zero or subnormal? */
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| 58 | u.f = x*0x1p54;
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| 59 | hx = u.i>>32 & 0x7fffffff;
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| 60 | if (hx == 0)
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| 61 | return x; /* cbrt(0) is itself */
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| 62 | hx = hx/3 + B2;
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| 63 | } else
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| 64 | hx = hx/3 + B1;
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| 65 | u.i &= 1ULL<<63;
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| 66 | u.i |= (uint64_t)hx << 32;
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| 67 | t = u.f;
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| 68 |
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| 69 | /*
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| 70 | * New cbrt to 23 bits:
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| 71 | * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
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| 72 | * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
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| 73 | * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
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| 74 | * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
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| 75 | * gives us bounds for r = t**3/x.
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| 76 | *
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| 77 | * Try to optimize for parallel evaluation as in __tanf.c.
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| 78 | */
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| 79 | r = (t*t)*(t/x);
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| 80 | t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
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| 81 |
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| 82 | /*
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| 83 | * Round t away from zero to 23 bits (sloppily except for ensuring that
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| 84 | * the result is larger in magnitude than cbrt(x) but not much more than
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| 85 | * 2 23-bit ulps larger). With rounding towards zero, the error bound
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| 86 | * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
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| 87 | * in the rounded t, the infinite-precision error in the Newton
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| 88 | * approximation barely affects third digit in the final error
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| 89 | * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
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| 90 | * before the final error is larger than 0.667 ulps.
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| 91 | */
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| 92 | u.f = t;
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| 93 | u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
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| 94 | t = u.f;
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| 95 |
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| 96 | /* one step Newton iteration to 53 bits with error < 0.667 ulps */
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| 97 | s = t*t; /* t*t is exact */
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| 98 | r = x/s; /* error <= 0.5 ulps; |r| < |t| */
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| 99 | w = t+t; /* t+t is exact */
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| 100 | r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
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| 101 | t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
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| 102 | return t;
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| 103 | }
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