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Fix tgamma() on some special args:
(1) tgamma(-Inf) returned +Inf and failed to raise any exception, but should always have raised an exception, and should behave like tgamma(negative integer). (2) tgamma(negative integer) returned +Inf and raised divide-by-zero, but should return NaN and raise "invalid" on any IEEEish system. (3) About half of the 2**52 negative intgers between -2**53 and -2**52 were misclassified as non-integers by using floor(x + 0.5) to round to nearest, so tgamma(x) was wrong (+-0 instead of +Inf and now NaN) on these args. The floor() expression is hard to use since rounding of (x + 0.5) may give x or x + 1, depending on |x| and the current rounding mode. The fixed version uses ceil(x) to classify x before operating on x and ends up being more efficient since ceil(x) is needed anyway. (4) On at least the problematic args in (3), tgamma() raised a spurious inexact. (5) tgamma(large positive) raised divide-by-zero but should raise overflow. (6) tgamma(+Inf) raised divide-by-zero but should not raise any exception. (7) Raise inexact for tiny |x| in a way that has some chance of not being optimized away. The fix for (5) and (6), and probably for (2), also prevents -O optimizing away the exception. PR: 112180 (2) Standards: Annex F in C99 (IEC 60559 binding) requires (1), (2) and (6).
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svn2git
2020-12-20 02:59:44 +00:00
svn path=/head/; revision=169212
@ -49,7 +49,7 @@ __FBSDID("$FreeBSD$");
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/* METHOD:
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/* METHOD:
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* x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
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* x < 0: Use reflection formula, G(x) = pi/(sin(pi*x)*x*G(x))
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* At negative integers, return +Inf and raise divide-by-zero.
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* At negative integers, return NaN and raise invalid.
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*
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*
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* x < 6.5:
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* x < 6.5:
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* Use argument reduction G(x+1) = xG(x) to reach the
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* Use argument reduction G(x+1) = xG(x) to reach the
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@ -66,12 +66,12 @@ __FBSDID("$FreeBSD$");
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* avoid premature round-off.
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* avoid premature round-off.
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*
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*
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* Special values:
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* Special values:
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* -Inf: return +Inf (without raising any exception!);
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* -Inf: return NaN and raise invalid;
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* negative integer: return +Inf and raise divide-by-zero;
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* negative integer: return NaN and raise invalid;
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* other x ~< 177.79: return +-0 and raise underflow;
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* other x ~< 177.79: return +-0 and raise underflow;
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* +-0: return +-Inf and raise divide-by-zero;
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* +-0: return +-Inf and raise divide-by-zero;
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* finite x ~> 171.63: return +Inf and raise divide-by-zero(!);
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* finite x ~> 171.63: return +Inf and raise overflow;
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* +Inf: return +Inf and raise divide-by-zero(!);
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* +Inf: return +Inf;
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* NaN: return NaN.
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* NaN: return NaN.
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*
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*
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* Accuracy: tgamma(x) is accurate to within
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* Accuracy: tgamma(x) is accurate to within
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@ -135,7 +135,7 @@ tgamma(x)
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if (x >= 6) {
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if (x >= 6) {
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if(x > 171.63)
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if(x > 171.63)
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return(one/zero);
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return (x / zero);
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u = large_gam(x);
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u = large_gam(x);
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return(__exp__D(u.a, u.b));
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return(__exp__D(u.a, u.b));
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} else if (x >= 1.0 + LEFT + x0)
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} else if (x >= 1.0 + LEFT + x0)
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@ -143,12 +143,11 @@ tgamma(x)
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else if (x > 1.e-17)
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else if (x > 1.e-17)
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return (smaller_gam(x));
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return (smaller_gam(x));
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else if (x > -1.e-17) {
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else if (x > -1.e-17) {
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if (x == 0.0)
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if (x != 0.0)
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return (one/x);
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u.a = one - tiny; /* raise inexact */
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one+1e-20; /* Raise inexact flag. */
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return (one/x);
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return (one/x);
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} else if (!finite(x))
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} else if (!finite(x))
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return (x*x); /* x = NaN, -Inf */
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return (x - x); /* x is NaN or -Inf */
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else
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else
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return (neg_gam(x));
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return (neg_gam(x));
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}
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}
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@ -282,11 +281,13 @@ neg_gam(x)
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struct Double lg, lsine;
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struct Double lg, lsine;
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double y, z;
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double y, z;
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y = floor(x + .5);
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y = ceil(x);
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if (y == x) /* Negative integer. */
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if (y == x) /* Negative integer. */
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return (one/zero);
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return ((x - x) / zero);
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z = fabs(x - y);
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z = y - x;
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y = .5*ceil(x);
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if (z > 0.5)
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z = one - z;
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y = 0.5 * y;
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if (y == ceil(y))
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if (y == ceil(y))
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sgn = -1;
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sgn = -1;
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if (z < .25)
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if (z < .25)
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