Average Error: 1.8 → 1.8
Time: 51.0s
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 r111208 = atan2(1.0, 0.0);
        double r111209 = z;
        double r111210 = r111208 * r111209;
        double r111211 = sin(r111210);
        double r111212 = r111208 / r111211;
        double r111213 = 2.0;
        double r111214 = r111208 * r111213;
        double r111215 = sqrt(r111214);
        double r111216 = 1.0;
        double r111217 = r111216 - r111209;
        double r111218 = r111217 - r111216;
        double r111219 = 7.0;
        double r111220 = r111218 + r111219;
        double r111221 = 0.5;
        double r111222 = r111220 + r111221;
        double r111223 = r111218 + r111221;
        double r111224 = pow(r111222, r111223);
        double r111225 = r111215 * r111224;
        double r111226 = -r111222;
        double r111227 = exp(r111226);
        double r111228 = r111225 * r111227;
        double r111229 = 0.9999999999998099;
        double r111230 = 676.5203681218851;
        double r111231 = r111218 + r111216;
        double r111232 = r111230 / r111231;
        double r111233 = r111229 + r111232;
        double r111234 = -1259.1392167224028;
        double r111235 = r111218 + r111213;
        double r111236 = r111234 / r111235;
        double r111237 = r111233 + r111236;
        double r111238 = 771.3234287776531;
        double r111239 = 3.0;
        double r111240 = r111218 + r111239;
        double r111241 = r111238 / r111240;
        double r111242 = r111237 + r111241;
        double r111243 = -176.6150291621406;
        double r111244 = 4.0;
        double r111245 = r111218 + r111244;
        double r111246 = r111243 / r111245;
        double r111247 = r111242 + r111246;
        double r111248 = 12.507343278686905;
        double r111249 = 5.0;
        double r111250 = r111218 + r111249;
        double r111251 = r111248 / r111250;
        double r111252 = r111247 + r111251;
        double r111253 = -0.13857109526572012;
        double r111254 = 6.0;
        double r111255 = r111218 + r111254;
        double r111256 = r111253 / r111255;
        double r111257 = r111252 + r111256;
        double r111258 = 9.984369578019572e-06;
        double r111259 = r111258 / r111220;
        double r111260 = r111257 + r111259;
        double r111261 = 1.5056327351493116e-07;
        double r111262 = 8.0;
        double r111263 = r111218 + r111262;
        double r111264 = r111261 / r111263;
        double r111265 = r111260 + r111264;
        double r111266 = r111228 * r111265;
        double r111267 = r111212 * r111266;
        return r111267;
}
↓
double f(double z) {
        double r111268 = atan2(1.0, 0.0);
        double r111269 = z;
        double r111270 = r111268 * r111269;
        double r111271 = sin(r111270);
        double r111272 = r111268 / r111271;
        double r111273 = 2.0;
        double r111274 = r111268 * r111273;
        double r111275 = sqrt(r111274);
        double r111276 = 1.0;
        double r111277 = r111276 - r111269;
        double r111278 = r111277 - r111276;
        double r111279 = 7.0;
        double r111280 = r111278 + r111279;
        double r111281 = 0.5;
        double r111282 = r111280 + r111281;
        double r111283 = r111278 + r111281;
        double r111284 = pow(r111282, r111283);
        double r111285 = r111275 * r111284;
        double r111286 = -r111282;
        double r111287 = exp(r111286);
        double r111288 = r111285 * r111287;
        double r111289 = 0.9999999999998099;
        double r111290 = 676.5203681218851;
        double r111291 = r111278 + r111276;
        double r111292 = r111290 / r111291;
        double r111293 = r111289 + r111292;
        double r111294 = -1259.1392167224028;
        double r111295 = r111278 + r111273;
        double r111296 = r111294 / r111295;
        double r111297 = r111293 + r111296;
        double r111298 = 771.3234287776531;
        double r111299 = 3.0;
        double r111300 = r111278 + r111299;
        double r111301 = r111298 / r111300;
        double r111302 = r111297 + r111301;
        double r111303 = -176.6150291621406;
        double r111304 = 4.0;
        double r111305 = r111278 + r111304;
        double r111306 = r111303 / r111305;
        double r111307 = r111302 + r111306;
        double r111308 = 12.507343278686905;
        double r111309 = 5.0;
        double r111310 = r111278 + r111309;
        double r111311 = r111308 / r111310;
        double r111312 = r111307 + r111311;
        double r111313 = -0.13857109526572012;
        double r111314 = 6.0;
        double r111315 = r111278 + r111314;
        double r111316 = r111313 / r111315;
        double r111317 = r111312 + r111316;
        double r111318 = 9.984369578019572e-06;
        double r111319 = r111318 / r111280;
        double r111320 = r111317 + r111319;
        double r111321 = 1.5056327351493116e-07;
        double r111322 = 8.0;
        double r111323 = r111278 + r111322;
        double r111324 = r111321 / r111323;
        double r111325 = r111320 + r111324;
        double r111326 = r111288 * r111325;
        double r111327 = r111272 * r111326;
        return r111327;
}

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 2020064 
(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))))))