Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 1.2m

Precision: 64

• could not determine a ground truth for program body

  1. z = -4.662649594744133e+254

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{-0.13857109526572012}{6 + \frac{{\left(1 - z\right)}^{3} - 1}{\left(1 + \left(1 - z\right)\right) + \left(1 - z\right) \cdot \left(1 - z\right)}} + \left(\frac{12.507343278686905}{5 + \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}} + \left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]

C

double f(double z) {
        double r6516063 = atan2(1.0, 0.0);
        double r6516064 = z;
        double r6516065 = r6516063 * r6516064;
        double r6516066 = sin(r6516065);
        double r6516067 = r6516063 / r6516066;
        double r6516068 = 2.0;
        double r6516069 = r6516063 * r6516068;
        double r6516070 = sqrt(r6516069);
        double r6516071 = 1.0;
        double r6516072 = r6516071 - r6516064;
        double r6516073 = r6516072 - r6516071;
        double r6516074 = 7.0;
        double r6516075 = r6516073 + r6516074;
        double r6516076 = 0.5;
        double r6516077 = r6516075 + r6516076;
        double r6516078 = r6516073 + r6516076;
        double r6516079 = pow(r6516077, r6516078);
        double r6516080 = r6516070 * r6516079;
        double r6516081 = -r6516077;
        double r6516082 = exp(r6516081);
        double r6516083 = r6516080 * r6516082;
        double r6516084 = 0.9999999999998099;
        double r6516085 = 676.5203681218851;
        double r6516086 = r6516073 + r6516071;
        double r6516087 = r6516085 / r6516086;
        double r6516088 = r6516084 + r6516087;
        double r6516089 = -1259.1392167224028;
        double r6516090 = r6516073 + r6516068;
        double r6516091 = r6516089 / r6516090;
        double r6516092 = r6516088 + r6516091;
        double r6516093 = 771.3234287776531;
        double r6516094 = 3.0;
        double r6516095 = r6516073 + r6516094;
        double r6516096 = r6516093 / r6516095;
        double r6516097 = r6516092 + r6516096;
        double r6516098 = -176.6150291621406;
        double r6516099 = 4.0;
        double r6516100 = r6516073 + r6516099;
        double r6516101 = r6516098 / r6516100;
        double r6516102 = r6516097 + r6516101;
        double r6516103 = 12.507343278686905;
        double r6516104 = 5.0;
        double r6516105 = r6516073 + r6516104;
        double r6516106 = r6516103 / r6516105;
        double r6516107 = r6516102 + r6516106;
        double r6516108 = -0.13857109526572012;
        double r6516109 = 6.0;
        double r6516110 = r6516073 + r6516109;
        double r6516111 = r6516108 / r6516110;
        double r6516112 = r6516107 + r6516111;
        double r6516113 = 9.984369578019572e-06;
        double r6516114 = r6516113 / r6516075;
        double r6516115 = r6516112 + r6516114;
        double r6516116 = 1.5056327351493116e-07;
        double r6516117 = 8.0;
        double r6516118 = r6516073 + r6516117;
        double r6516119 = r6516116 / r6516118;
        double r6516120 = r6516115 + r6516119;
        double r6516121 = r6516083 * r6516120;
        double r6516122 = r6516067 * r6516121;
        return r6516122;
}

↓

double f(double z) {
        double r6516123 = 2.0;
        double r6516124 = atan2(1.0, 0.0);
        double r6516125 = r6516123 * r6516124;
        double r6516126 = sqrt(r6516125);
        double r6516127 = 7.0;
        double r6516128 = 1.0;
        double r6516129 = z;
        double r6516130 = r6516128 - r6516129;
        double r6516131 = r6516130 - r6516128;
        double r6516132 = r6516127 + r6516131;
        double r6516133 = 0.5;
        double r6516134 = r6516132 + r6516133;
        double r6516135 = r6516133 + r6516131;
        double r6516136 = pow(r6516134, r6516135);
        double r6516137 = r6516126 * r6516136;
        double r6516138 = -r6516134;
        double r6516139 = exp(r6516138);
        double r6516140 = r6516137 * r6516139;
        double r6516141 = 1.5056327351493116e-07;
        double r6516142 = 8.0;
        double r6516143 = r6516131 + r6516142;
        double r6516144 = r6516141 / r6516143;
        double r6516145 = 9.984369578019572e-06;
        double r6516146 = r6516145 / r6516132;
        double r6516147 = -0.13857109526572012;
        double r6516148 = 6.0;
        double r6516149 = 3.0;
        double r6516150 = pow(r6516130, r6516149);
        double r6516151 = r6516150 - r6516128;
        double r6516152 = r6516128 + r6516130;
        double r6516153 = r6516130 * r6516130;
        double r6516154 = r6516152 + r6516153;
        double r6516155 = r6516151 / r6516154;
        double r6516156 = r6516148 + r6516155;
        double r6516157 = r6516147 / r6516156;
        double r6516158 = 12.507343278686905;
        double r6516159 = 5.0;
        double r6516160 = cbrt(r6516131);
        double r6516161 = r6516160 * r6516160;
        double r6516162 = r6516161 * r6516160;
        double r6516163 = r6516159 + r6516162;
        double r6516164 = r6516158 / r6516163;
        double r6516165 = -176.6150291621406;
        double r6516166 = 4.0;
        double r6516167 = r6516131 + r6516166;
        double r6516168 = r6516165 / r6516167;
        double r6516169 = 771.3234287776531;
        double r6516170 = r6516149 + r6516131;
        double r6516171 = r6516169 / r6516170;
        double r6516172 = 0.9999999999998099;
        double r6516173 = 676.5203681218851;
        double r6516174 = r6516131 + r6516128;
        double r6516175 = r6516173 / r6516174;
        double r6516176 = r6516172 + r6516175;
        double r6516177 = -1259.1392167224028;
        double r6516178 = r6516131 + r6516123;
        double r6516179 = r6516177 / r6516178;
        double r6516180 = r6516176 + r6516179;
        double r6516181 = r6516171 + r6516180;
        double r6516182 = r6516168 + r6516181;
        double r6516183 = r6516164 + r6516182;
        double r6516184 = r6516157 + r6516183;
        double r6516185 = r6516146 + r6516184;
        double r6516186 = r6516144 + r6516185;
        double r6516187 = r6516140 * r6516186;
        double r6516188 = r6516124 * r6516129;
        double r6516189 = sin(r6516188);
        double r6516190 = r6516124 / r6516189;
        double r6516191 = r6516187 * r6516190;
        return r6516191;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\color{blue}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}} + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
4. Using strategy rm

5. Applied flip3-- 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1} + 5}\right) + \frac{-0.13857109526572012}{\color{blue}{\frac{{\left(1 - z\right)}^{3} - {1}^{3}}{\left(1 - z\right) \cdot \left(1 - z\right) + \left(1 \cdot 1 + \left(1 - z\right) \cdot 1\right)}} + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

6. Final simplification 1.8

   \[\leadsto \left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{-0.13857109526572012}{6 + \frac{{\left(1 - z\right)}^{3} - 1}{\left(1 + \left(1 - z\right)\right) + \left(1 - z\right) \cdot \left(1 - z\right)}} + \left(\frac{12.507343278686905}{5 + \left(\sqrt[3]{\left(1 - z\right) - 1} \cdot \sqrt[3]{\left(1 - z\right) - 1}\right) \cdot \sqrt[3]{\left(1 - z\right) - 1}} + \left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right)\right)\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ 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))))))