Average Error: 1.8 → 1.8
Time: 59.7s
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 r137148 = atan2(1.0, 0.0);
        double r137149 = z;
        double r137150 = r137148 * r137149;
        double r137151 = sin(r137150);
        double r137152 = r137148 / r137151;
        double r137153 = 2.0;
        double r137154 = r137148 * r137153;
        double r137155 = sqrt(r137154);
        double r137156 = 1.0;
        double r137157 = r137156 - r137149;
        double r137158 = r137157 - r137156;
        double r137159 = 7.0;
        double r137160 = r137158 + r137159;
        double r137161 = 0.5;
        double r137162 = r137160 + r137161;
        double r137163 = r137158 + r137161;
        double r137164 = pow(r137162, r137163);
        double r137165 = r137155 * r137164;
        double r137166 = -r137162;
        double r137167 = exp(r137166);
        double r137168 = r137165 * r137167;
        double r137169 = 0.9999999999998099;
        double r137170 = 676.5203681218851;
        double r137171 = r137158 + r137156;
        double r137172 = r137170 / r137171;
        double r137173 = r137169 + r137172;
        double r137174 = -1259.1392167224028;
        double r137175 = r137158 + r137153;
        double r137176 = r137174 / r137175;
        double r137177 = r137173 + r137176;
        double r137178 = 771.3234287776531;
        double r137179 = 3.0;
        double r137180 = r137158 + r137179;
        double r137181 = r137178 / r137180;
        double r137182 = r137177 + r137181;
        double r137183 = -176.6150291621406;
        double r137184 = 4.0;
        double r137185 = r137158 + r137184;
        double r137186 = r137183 / r137185;
        double r137187 = r137182 + r137186;
        double r137188 = 12.507343278686905;
        double r137189 = 5.0;
        double r137190 = r137158 + r137189;
        double r137191 = r137188 / r137190;
        double r137192 = r137187 + r137191;
        double r137193 = -0.13857109526572012;
        double r137194 = 6.0;
        double r137195 = r137158 + r137194;
        double r137196 = r137193 / r137195;
        double r137197 = r137192 + r137196;
        double r137198 = 9.984369578019572e-06;
        double r137199 = r137198 / r137160;
        double r137200 = r137197 + r137199;
        double r137201 = 1.5056327351493116e-07;
        double r137202 = 8.0;
        double r137203 = r137158 + r137202;
        double r137204 = r137201 / r137203;
        double r137205 = r137200 + r137204;
        double r137206 = r137168 * r137205;
        double r137207 = r137152 * r137206;
        return r137207;
}
↓
double f(double z) {
        double r137208 = atan2(1.0, 0.0);
        double r137209 = z;
        double r137210 = r137208 * r137209;
        double r137211 = sin(r137210);
        double r137212 = r137208 / r137211;
        double r137213 = 2.0;
        double r137214 = r137208 * r137213;
        double r137215 = sqrt(r137214);
        double r137216 = 1.0;
        double r137217 = r137216 - r137209;
        double r137218 = r137217 - r137216;
        double r137219 = 7.0;
        double r137220 = r137218 + r137219;
        double r137221 = 0.5;
        double r137222 = r137220 + r137221;
        double r137223 = r137218 + r137221;
        double r137224 = pow(r137222, r137223);
        double r137225 = r137215 * r137224;
        double r137226 = -r137222;
        double r137227 = exp(r137226);
        double r137228 = r137225 * r137227;
        double r137229 = 0.9999999999998099;
        double r137230 = 676.5203681218851;
        double r137231 = r137218 + r137216;
        double r137232 = r137230 / r137231;
        double r137233 = r137229 + r137232;
        double r137234 = -1259.1392167224028;
        double r137235 = r137218 + r137213;
        double r137236 = r137234 / r137235;
        double r137237 = r137233 + r137236;
        double r137238 = 771.3234287776531;
        double r137239 = 3.0;
        double r137240 = r137218 + r137239;
        double r137241 = r137238 / r137240;
        double r137242 = r137237 + r137241;
        double r137243 = -176.6150291621406;
        double r137244 = 4.0;
        double r137245 = r137218 + r137244;
        double r137246 = r137243 / r137245;
        double r137247 = r137242 + r137246;
        double r137248 = 12.507343278686905;
        double r137249 = 5.0;
        double r137250 = r137218 + r137249;
        double r137251 = r137248 / r137250;
        double r137252 = r137247 + r137251;
        double r137253 = -0.13857109526572012;
        double r137254 = 6.0;
        double r137255 = r137218 + r137254;
        double r137256 = r137253 / r137255;
        double r137257 = r137252 + r137256;
        double r137258 = 9.984369578019572e-06;
        double r137259 = r137258 / r137220;
        double r137260 = r137257 + r137259;
        double r137261 = 1.5056327351493116e-07;
        double r137262 = 8.0;
        double r137263 = r137218 + r137262;
        double r137264 = r137261 / r137263;
        double r137265 = r137260 + r137264;
        double r137266 = r137228 * r137265;
        double r137267 = r137212 * r137266;
        return r137267;
}

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 2020034 +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))))))