Average Error: 1.8 → 1.8
Time: 58.0s
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 r190971 = atan2(1.0, 0.0);
        double r190972 = z;
        double r190973 = r190971 * r190972;
        double r190974 = sin(r190973);
        double r190975 = r190971 / r190974;
        double r190976 = 2.0;
        double r190977 = r190971 * r190976;
        double r190978 = sqrt(r190977);
        double r190979 = 1.0;
        double r190980 = r190979 - r190972;
        double r190981 = r190980 - r190979;
        double r190982 = 7.0;
        double r190983 = r190981 + r190982;
        double r190984 = 0.5;
        double r190985 = r190983 + r190984;
        double r190986 = r190981 + r190984;
        double r190987 = pow(r190985, r190986);
        double r190988 = r190978 * r190987;
        double r190989 = -r190985;
        double r190990 = exp(r190989);
        double r190991 = r190988 * r190990;
        double r190992 = 0.9999999999998099;
        double r190993 = 676.5203681218851;
        double r190994 = r190981 + r190979;
        double r190995 = r190993 / r190994;
        double r190996 = r190992 + r190995;
        double r190997 = -1259.1392167224028;
        double r190998 = r190981 + r190976;
        double r190999 = r190997 / r190998;
        double r191000 = r190996 + r190999;
        double r191001 = 771.3234287776531;
        double r191002 = 3.0;
        double r191003 = r190981 + r191002;
        double r191004 = r191001 / r191003;
        double r191005 = r191000 + r191004;
        double r191006 = -176.6150291621406;
        double r191007 = 4.0;
        double r191008 = r190981 + r191007;
        double r191009 = r191006 / r191008;
        double r191010 = r191005 + r191009;
        double r191011 = 12.507343278686905;
        double r191012 = 5.0;
        double r191013 = r190981 + r191012;
        double r191014 = r191011 / r191013;
        double r191015 = r191010 + r191014;
        double r191016 = -0.13857109526572012;
        double r191017 = 6.0;
        double r191018 = r190981 + r191017;
        double r191019 = r191016 / r191018;
        double r191020 = r191015 + r191019;
        double r191021 = 9.984369578019572e-06;
        double r191022 = r191021 / r190983;
        double r191023 = r191020 + r191022;
        double r191024 = 1.5056327351493116e-07;
        double r191025 = 8.0;
        double r191026 = r190981 + r191025;
        double r191027 = r191024 / r191026;
        double r191028 = r191023 + r191027;
        double r191029 = r190991 * r191028;
        double r191030 = r190975 * r191029;
        return r191030;
}
↓
double f(double z) {
        double r191031 = atan2(1.0, 0.0);
        double r191032 = z;
        double r191033 = r191031 * r191032;
        double r191034 = sin(r191033);
        double r191035 = r191031 / r191034;
        double r191036 = 2.0;
        double r191037 = r191031 * r191036;
        double r191038 = sqrt(r191037);
        double r191039 = 1.0;
        double r191040 = r191039 - r191032;
        double r191041 = r191040 - r191039;
        double r191042 = 7.0;
        double r191043 = r191041 + r191042;
        double r191044 = 0.5;
        double r191045 = r191043 + r191044;
        double r191046 = r191041 + r191044;
        double r191047 = pow(r191045, r191046);
        double r191048 = r191038 * r191047;
        double r191049 = -r191045;
        double r191050 = exp(r191049);
        double r191051 = r191048 * r191050;
        double r191052 = 0.9999999999998099;
        double r191053 = 676.5203681218851;
        double r191054 = r191041 + r191039;
        double r191055 = r191053 / r191054;
        double r191056 = r191052 + r191055;
        double r191057 = -1259.1392167224028;
        double r191058 = r191041 + r191036;
        double r191059 = r191057 / r191058;
        double r191060 = r191056 + r191059;
        double r191061 = 771.3234287776531;
        double r191062 = 3.0;
        double r191063 = r191041 + r191062;
        double r191064 = r191061 / r191063;
        double r191065 = r191060 + r191064;
        double r191066 = -176.6150291621406;
        double r191067 = 4.0;
        double r191068 = r191041 + r191067;
        double r191069 = r191066 / r191068;
        double r191070 = r191065 + r191069;
        double r191071 = 12.507343278686905;
        double r191072 = 5.0;
        double r191073 = r191041 + r191072;
        double r191074 = r191071 / r191073;
        double r191075 = r191070 + r191074;
        double r191076 = -0.13857109526572012;
        double r191077 = 6.0;
        double r191078 = r191041 + r191077;
        double r191079 = r191076 / r191078;
        double r191080 = r191075 + r191079;
        double r191081 = 9.984369578019572e-06;
        double r191082 = r191081 / r191043;
        double r191083 = r191080 + r191082;
        double r191084 = 1.5056327351493116e-07;
        double r191085 = 8.0;
        double r191086 = r191041 + r191085;
        double r191087 = r191084 / r191086;
        double r191088 = r191083 + r191087;
        double r191089 = r191051 * r191088;
        double r191090 = r191035 * r191089;
        return r191090;
}

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 2020060 +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))))))