| Index: fusl/src/math/cbrt.c
|
| diff --git a/fusl/src/math/cbrt.c b/fusl/src/math/cbrt.c
|
| index 7599d3e37d2f6f81f21321b62f1e97aae5e34167..d6879df9b40856430f24a6aa5d33c8116d433e66 100644
|
| --- a/fusl/src/math/cbrt.c
|
| +++ b/fusl/src/math/cbrt.c
|
| @@ -19,85 +19,86 @@
|
| #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 */
|
| + 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 */
|
| +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;
|
| +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;
|
| + 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;
|
| + /*
|
| + * 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));
|
| + /*
|
| + * 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;
|
| + /*
|
| + * 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;
|
| + /* 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;
|
| }
|
|
|