Index: fusl/src/math/cbrt.c |
diff --git a/fusl/src/math/cbrt.c b/fusl/src/math/cbrt.c |
new file mode 100644 |
index 0000000000000000000000000000000000000000..7599d3e37d2f6f81f21321b62f1e97aae5e34167 |
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+++ b/fusl/src/math/cbrt.c |
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+/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */ |
+/* |
+ * ==================================================== |
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
+ * |
+ * Developed at SunPro, a Sun Microsystems, Inc. business. |
+ * Permission to use, copy, modify, and distribute this |
+ * software is freely granted, provided that this notice |
+ * is preserved. |
+ * ==================================================== |
+ * |
+ * Optimized by Bruce D. Evans. |
+ */ |
+/* cbrt(x) |
+ * Return cube root of x |
+ */ |
+ |
+#include <math.h> |
+#include <stdint.h> |
+ |
+static const uint32_t |
+B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */ |
+B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */ |
+ |
+/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */ |
+static const double |
+P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */ |
+P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */ |
+P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */ |
+P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */ |
+P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */ |
+ |
+double cbrt(double x) |
+{ |
+ union {double f; uint64_t i;} u = {x}; |
+ double_t r,s,t,w; |
+ uint32_t hx = u.i>>32 & 0x7fffffff; |
+ |
+ if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */ |
+ return x+x; |
+ |
+ /* |
+ * Rough cbrt to 5 bits: |
+ * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3) |
+ * where e is integral and >= 0, m is real and in [0, 1), and "/" and |
+ * "%" are integer division and modulus with rounding towards minus |
+ * infinity. The RHS is always >= the LHS and has a maximum relative |
+ * error of about 1 in 16. Adding a bias of -0.03306235651 to the |
+ * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE |
+ * floating point representation, for finite positive normal values, |
+ * ordinary integer divison of the value in bits magically gives |
+ * almost exactly the RHS of the above provided we first subtract the |
+ * exponent bias (1023 for doubles) and later add it back. We do the |
+ * subtraction virtually to keep e >= 0 so that ordinary integer |
+ * division rounds towards minus infinity; this is also efficient. |
+ */ |
+ if (hx < 0x00100000) { /* zero or subnormal? */ |
+ u.f = x*0x1p54; |
+ hx = u.i>>32 & 0x7fffffff; |
+ if (hx == 0) |
+ return x; /* cbrt(0) is itself */ |
+ hx = hx/3 + B2; |
+ } else |
+ hx = hx/3 + B1; |
+ u.i &= 1ULL<<63; |
+ u.i |= (uint64_t)hx << 32; |
+ t = u.f; |
+ |
+ /* |
+ * New cbrt to 23 bits: |
+ * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x) |
+ * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r) |
+ * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation |
+ * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this |
+ * gives us bounds for r = t**3/x. |
+ * |
+ * Try to optimize for parallel evaluation as in __tanf.c. |
+ */ |
+ r = (t*t)*(t/x); |
+ t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4)); |
+ |
+ /* |
+ * Round t away from zero to 23 bits (sloppily except for ensuring that |
+ * the result is larger in magnitude than cbrt(x) but not much more than |
+ * 2 23-bit ulps larger). With rounding towards zero, the error bound |
+ * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps |
+ * in the rounded t, the infinite-precision error in the Newton |
+ * approximation barely affects third digit in the final error |
+ * 0.667; the error in the rounded t can be up to about 3 23-bit ulps |
+ * before the final error is larger than 0.667 ulps. |
+ */ |
+ u.f = t; |
+ u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL; |
+ t = u.f; |
+ |
+ /* one step Newton iteration to 53 bits with error < 0.667 ulps */ |
+ s = t*t; /* t*t is exact */ |
+ r = x/s; /* error <= 0.5 ulps; |r| < |t| */ |
+ w = t+t; /* t+t is exact */ |
+ r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */ |
+ t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */ |
+ return t; |
+} |