Average Error: 1.8 → 1.8
Time: 10.3s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
double f(double z) {
        double r136957 = atan2(1.0, 0.0);
        double r136958 = z;
        double r136959 = r136957 * r136958;
        double r136960 = sin(r136959);
        double r136961 = r136957 / r136960;
        double r136962 = 2.0;
        double r136963 = r136957 * r136962;
        double r136964 = sqrt(r136963);
        double r136965 = 1.0;
        double r136966 = r136965 - r136958;
        double r136967 = r136966 - r136965;
        double r136968 = 7.0;
        double r136969 = r136967 + r136968;
        double r136970 = 0.5;
        double r136971 = r136969 + r136970;
        double r136972 = r136967 + r136970;
        double r136973 = pow(r136971, r136972);
        double r136974 = r136964 * r136973;
        double r136975 = -r136971;
        double r136976 = exp(r136975);
        double r136977 = r136974 * r136976;
        double r136978 = 0.9999999999998099;
        double r136979 = 676.5203681218851;
        double r136980 = r136967 + r136965;
        double r136981 = r136979 / r136980;
        double r136982 = r136978 + r136981;
        double r136983 = -1259.1392167224028;
        double r136984 = r136967 + r136962;
        double r136985 = r136983 / r136984;
        double r136986 = r136982 + r136985;
        double r136987 = 771.3234287776531;
        double r136988 = 3.0;
        double r136989 = r136967 + r136988;
        double r136990 = r136987 / r136989;
        double r136991 = r136986 + r136990;
        double r136992 = -176.6150291621406;
        double r136993 = 4.0;
        double r136994 = r136967 + r136993;
        double r136995 = r136992 / r136994;
        double r136996 = r136991 + r136995;
        double r136997 = 12.507343278686905;
        double r136998 = 5.0;
        double r136999 = r136967 + r136998;
        double r137000 = r136997 / r136999;
        double r137001 = r136996 + r137000;
        double r137002 = -0.13857109526572012;
        double r137003 = 6.0;
        double r137004 = r136967 + r137003;
        double r137005 = r137002 / r137004;
        double r137006 = r137001 + r137005;
        double r137007 = 9.984369578019572e-06;
        double r137008 = r137007 / r136969;
        double r137009 = r137006 + r137008;
        double r137010 = 1.5056327351493116e-07;
        double r137011 = 8.0;
        double r137012 = r136967 + r137011;
        double r137013 = r137010 / r137012;
        double r137014 = r137009 + r137013;
        double r137015 = r136977 * r137014;
        double r137016 = r136961 * r137015;
        return r137016;
}
↓
double f(double z) {
        double r137017 = atan2(1.0, 0.0);
        double r137018 = z;
        double r137019 = r137017 * r137018;
        double r137020 = sin(r137019);
        double r137021 = r137017 / r137020;
        double r137022 = 2.0;
        double r137023 = r137017 * r137022;
        double r137024 = sqrt(r137023);
        double r137025 = 1.0;
        double r137026 = r137025 - r137018;
        double r137027 = r137026 - r137025;
        double r137028 = 7.0;
        double r137029 = r137027 + r137028;
        double r137030 = 0.5;
        double r137031 = r137029 + r137030;
        double r137032 = r137027 + r137030;
        double r137033 = pow(r137031, r137032);
        double r137034 = r137024 * r137033;
        double r137035 = -r137031;
        double r137036 = exp(r137035);
        double r137037 = r137034 * r137036;
        double r137038 = 0.9999999999998099;
        double r137039 = 676.5203681218851;
        double r137040 = r137027 + r137025;
        double r137041 = r137039 / r137040;
        double r137042 = r137038 + r137041;
        double r137043 = -1259.1392167224028;
        double r137044 = r137027 + r137022;
        double r137045 = r137043 / r137044;
        double r137046 = r137042 + r137045;
        double r137047 = 771.3234287776531;
        double r137048 = 3.0;
        double r137049 = r137027 + r137048;
        double r137050 = r137047 / r137049;
        double r137051 = r137046 + r137050;
        double r137052 = -176.6150291621406;
        double r137053 = 4.0;
        double r137054 = r137027 + r137053;
        double r137055 = r137052 / r137054;
        double r137056 = r137051 + r137055;
        double r137057 = 12.507343278686905;
        double r137058 = 5.0;
        double r137059 = r137027 + r137058;
        double r137060 = r137057 / r137059;
        double r137061 = r137056 + r137060;
        double r137062 = -0.13857109526572012;
        double r137063 = 6.0;
        double r137064 = r137027 + r137063;
        double r137065 = r137062 / r137064;
        double r137066 = r137061 + r137065;
        double r137067 = 9.984369578019572e-06;
        double r137068 = r137067 / r137029;
        double r137069 = r137066 + r137068;
        double r137070 = 1.5056327351493116e-07;
        double r137071 = 8.0;
        double r137072 = r137027 + r137071;
        double r137073 = r137070 / r137072;
        double r137074 = r137069 + r137073;
        double r137075 = r137037 * r137074;
        double r137076 = r137021 * r137075;
        return r137076;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

    \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

  2. Final simplification1.8

    \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

Reproduce

herbie shell --seed 2020021 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  (* (/ PI (sin (* PI z))) (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- (- 1 z) 1) 7) 0.5) (+ (- (- 1 z) 1) 0.5))) (exp (- (+ (+ (- (- 1 z) 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1 z) 1) 1))) (/ -1259.1392167224028 (+ (- (- 1 z) 1) 2))) (/ 771.3234287776531 (+ (- (- 1 z) 1) 3))) (/ -176.6150291621406 (+ (- (- 1 z) 1) 4))) (/ 12.507343278686905 (+ (- (- 1 z) 1) 5))) (/ -0.13857109526572012 (+ (- (- 1 z) 1) 6))) (/ 9.984369578019572e-06 (+ (- (- 1 z) 1) 7))) (/ 1.5056327351493116e-07 (+ (- (- 1 z) 1) 8))))))