Average Error: 1.8 → 1.8
Time: 51.2s
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 r99045 = atan2(1.0, 0.0);
        double r99046 = z;
        double r99047 = r99045 * r99046;
        double r99048 = sin(r99047);
        double r99049 = r99045 / r99048;
        double r99050 = 2.0;
        double r99051 = r99045 * r99050;
        double r99052 = sqrt(r99051);
        double r99053 = 1.0;
        double r99054 = r99053 - r99046;
        double r99055 = r99054 - r99053;
        double r99056 = 7.0;
        double r99057 = r99055 + r99056;
        double r99058 = 0.5;
        double r99059 = r99057 + r99058;
        double r99060 = r99055 + r99058;
        double r99061 = pow(r99059, r99060);
        double r99062 = r99052 * r99061;
        double r99063 = -r99059;
        double r99064 = exp(r99063);
        double r99065 = r99062 * r99064;
        double r99066 = 0.9999999999998099;
        double r99067 = 676.5203681218851;
        double r99068 = r99055 + r99053;
        double r99069 = r99067 / r99068;
        double r99070 = r99066 + r99069;
        double r99071 = -1259.1392167224028;
        double r99072 = r99055 + r99050;
        double r99073 = r99071 / r99072;
        double r99074 = r99070 + r99073;
        double r99075 = 771.3234287776531;
        double r99076 = 3.0;
        double r99077 = r99055 + r99076;
        double r99078 = r99075 / r99077;
        double r99079 = r99074 + r99078;
        double r99080 = -176.6150291621406;
        double r99081 = 4.0;
        double r99082 = r99055 + r99081;
        double r99083 = r99080 / r99082;
        double r99084 = r99079 + r99083;
        double r99085 = 12.507343278686905;
        double r99086 = 5.0;
        double r99087 = r99055 + r99086;
        double r99088 = r99085 / r99087;
        double r99089 = r99084 + r99088;
        double r99090 = -0.13857109526572012;
        double r99091 = 6.0;
        double r99092 = r99055 + r99091;
        double r99093 = r99090 / r99092;
        double r99094 = r99089 + r99093;
        double r99095 = 9.984369578019572e-06;
        double r99096 = r99095 / r99057;
        double r99097 = r99094 + r99096;
        double r99098 = 1.5056327351493116e-07;
        double r99099 = 8.0;
        double r99100 = r99055 + r99099;
        double r99101 = r99098 / r99100;
        double r99102 = r99097 + r99101;
        double r99103 = r99065 * r99102;
        double r99104 = r99049 * r99103;
        return r99104;
}
↓
double f(double z) {
        double r99105 = atan2(1.0, 0.0);
        double r99106 = z;
        double r99107 = r99105 * r99106;
        double r99108 = sin(r99107);
        double r99109 = r99105 / r99108;
        double r99110 = 2.0;
        double r99111 = r99105 * r99110;
        double r99112 = sqrt(r99111);
        double r99113 = 1.0;
        double r99114 = r99113 - r99106;
        double r99115 = r99114 - r99113;
        double r99116 = 7.0;
        double r99117 = r99115 + r99116;
        double r99118 = 0.5;
        double r99119 = r99117 + r99118;
        double r99120 = r99115 + r99118;
        double r99121 = pow(r99119, r99120);
        double r99122 = r99112 * r99121;
        double r99123 = -r99119;
        double r99124 = exp(r99123);
        double r99125 = r99122 * r99124;
        double r99126 = 0.9999999999998099;
        double r99127 = 676.5203681218851;
        double r99128 = r99115 + r99113;
        double r99129 = r99127 / r99128;
        double r99130 = r99126 + r99129;
        double r99131 = -1259.1392167224028;
        double r99132 = r99115 + r99110;
        double r99133 = r99131 / r99132;
        double r99134 = r99130 + r99133;
        double r99135 = 771.3234287776531;
        double r99136 = 3.0;
        double r99137 = r99115 + r99136;
        double r99138 = r99135 / r99137;
        double r99139 = r99134 + r99138;
        double r99140 = -176.6150291621406;
        double r99141 = 4.0;
        double r99142 = r99115 + r99141;
        double r99143 = r99140 / r99142;
        double r99144 = r99139 + r99143;
        double r99145 = 12.507343278686905;
        double r99146 = 5.0;
        double r99147 = r99115 + r99146;
        double r99148 = r99145 / r99147;
        double r99149 = r99144 + r99148;
        double r99150 = -0.13857109526572012;
        double r99151 = 6.0;
        double r99152 = r99115 + r99151;
        double r99153 = r99150 / r99152;
        double r99154 = r99149 + r99153;
        double r99155 = 9.984369578019572e-06;
        double r99156 = r99155 / r99117;
        double r99157 = r99154 + r99156;
        double r99158 = 1.5056327351493116e-07;
        double r99159 = 8.0;
        double r99160 = r99115 + r99159;
        double r99161 = r99158 / r99160;
        double r99162 = r99157 + r99161;
        double r99163 = r99125 * r99162;
        double r99164 = r99109 * r99163;
        return r99164;
}

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 2020018 
(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))))))