Average Error: 1.8 → 1.8
Time: 57.2s
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 r120179 = atan2(1.0, 0.0);
        double r120180 = z;
        double r120181 = r120179 * r120180;
        double r120182 = sin(r120181);
        double r120183 = r120179 / r120182;
        double r120184 = 2.0;
        double r120185 = r120179 * r120184;
        double r120186 = sqrt(r120185);
        double r120187 = 1.0;
        double r120188 = r120187 - r120180;
        double r120189 = r120188 - r120187;
        double r120190 = 7.0;
        double r120191 = r120189 + r120190;
        double r120192 = 0.5;
        double r120193 = r120191 + r120192;
        double r120194 = r120189 + r120192;
        double r120195 = pow(r120193, r120194);
        double r120196 = r120186 * r120195;
        double r120197 = -r120193;
        double r120198 = exp(r120197);
        double r120199 = r120196 * r120198;
        double r120200 = 0.9999999999998099;
        double r120201 = 676.5203681218851;
        double r120202 = r120189 + r120187;
        double r120203 = r120201 / r120202;
        double r120204 = r120200 + r120203;
        double r120205 = -1259.1392167224028;
        double r120206 = r120189 + r120184;
        double r120207 = r120205 / r120206;
        double r120208 = r120204 + r120207;
        double r120209 = 771.3234287776531;
        double r120210 = 3.0;
        double r120211 = r120189 + r120210;
        double r120212 = r120209 / r120211;
        double r120213 = r120208 + r120212;
        double r120214 = -176.6150291621406;
        double r120215 = 4.0;
        double r120216 = r120189 + r120215;
        double r120217 = r120214 / r120216;
        double r120218 = r120213 + r120217;
        double r120219 = 12.507343278686905;
        double r120220 = 5.0;
        double r120221 = r120189 + r120220;
        double r120222 = r120219 / r120221;
        double r120223 = r120218 + r120222;
        double r120224 = -0.13857109526572012;
        double r120225 = 6.0;
        double r120226 = r120189 + r120225;
        double r120227 = r120224 / r120226;
        double r120228 = r120223 + r120227;
        double r120229 = 9.984369578019572e-06;
        double r120230 = r120229 / r120191;
        double r120231 = r120228 + r120230;
        double r120232 = 1.5056327351493116e-07;
        double r120233 = 8.0;
        double r120234 = r120189 + r120233;
        double r120235 = r120232 / r120234;
        double r120236 = r120231 + r120235;
        double r120237 = r120199 * r120236;
        double r120238 = r120183 * r120237;
        return r120238;
}
↓
double f(double z) {
        double r120239 = atan2(1.0, 0.0);
        double r120240 = z;
        double r120241 = r120239 * r120240;
        double r120242 = sin(r120241);
        double r120243 = r120239 / r120242;
        double r120244 = 2.0;
        double r120245 = r120239 * r120244;
        double r120246 = sqrt(r120245);
        double r120247 = 1.0;
        double r120248 = r120247 - r120240;
        double r120249 = r120248 - r120247;
        double r120250 = 7.0;
        double r120251 = r120249 + r120250;
        double r120252 = 0.5;
        double r120253 = r120251 + r120252;
        double r120254 = r120249 + r120252;
        double r120255 = pow(r120253, r120254);
        double r120256 = r120246 * r120255;
        double r120257 = -r120253;
        double r120258 = exp(r120257);
        double r120259 = r120256 * r120258;
        double r120260 = 0.9999999999998099;
        double r120261 = 676.5203681218851;
        double r120262 = r120249 + r120247;
        double r120263 = r120261 / r120262;
        double r120264 = r120260 + r120263;
        double r120265 = -1259.1392167224028;
        double r120266 = r120249 + r120244;
        double r120267 = r120265 / r120266;
        double r120268 = r120264 + r120267;
        double r120269 = 771.3234287776531;
        double r120270 = 3.0;
        double r120271 = r120249 + r120270;
        double r120272 = r120269 / r120271;
        double r120273 = r120268 + r120272;
        double r120274 = -176.6150291621406;
        double r120275 = 4.0;
        double r120276 = r120249 + r120275;
        double r120277 = r120274 / r120276;
        double r120278 = r120273 + r120277;
        double r120279 = 12.507343278686905;
        double r120280 = 5.0;
        double r120281 = r120249 + r120280;
        double r120282 = r120279 / r120281;
        double r120283 = r120278 + r120282;
        double r120284 = -0.13857109526572012;
        double r120285 = 6.0;
        double r120286 = r120249 + r120285;
        double r120287 = r120284 / r120286;
        double r120288 = r120283 + r120287;
        double r120289 = 9.984369578019572e-06;
        double r120290 = r120289 / r120251;
        double r120291 = r120288 + r120290;
        double r120292 = 1.5056327351493116e-07;
        double r120293 = 8.0;
        double r120294 = r120249 + r120293;
        double r120295 = r120292 / r120294;
        double r120296 = r120291 + r120295;
        double r120297 = r120259 * r120296;
        double r120298 = r120243 * r120297;
        return r120298;
}

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 2020089 
(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))))))