Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 56.4s

Precision: 64

• could not determine a ground truth for program body

  1. z = -1.8061903386524708e+183

Math

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

↓

\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\mathsf{fma}\left(1, -1, {\left(\sqrt[3]{1 - z}\right)}^{3}\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

C

double f(double z) {
        double r148217 = atan2(1.0, 0.0);
        double r148218 = z;
        double r148219 = r148217 * r148218;
        double r148220 = sin(r148219);
        double r148221 = r148217 / r148220;
        double r148222 = 2.0;
        double r148223 = r148217 * r148222;
        double r148224 = sqrt(r148223);
        double r148225 = 1.0;
        double r148226 = r148225 - r148218;
        double r148227 = r148226 - r148225;
        double r148228 = 7.0;
        double r148229 = r148227 + r148228;
        double r148230 = 0.5;
        double r148231 = r148229 + r148230;
        double r148232 = r148227 + r148230;
        double r148233 = pow(r148231, r148232);
        double r148234 = r148224 * r148233;
        double r148235 = -r148231;
        double r148236 = exp(r148235);
        double r148237 = r148234 * r148236;
        double r148238 = 0.9999999999998099;
        double r148239 = 676.5203681218851;
        double r148240 = r148227 + r148225;
        double r148241 = r148239 / r148240;
        double r148242 = r148238 + r148241;
        double r148243 = -1259.1392167224028;
        double r148244 = r148227 + r148222;
        double r148245 = r148243 / r148244;
        double r148246 = r148242 + r148245;
        double r148247 = 771.3234287776531;
        double r148248 = 3.0;
        double r148249 = r148227 + r148248;
        double r148250 = r148247 / r148249;
        double r148251 = r148246 + r148250;
        double r148252 = -176.6150291621406;
        double r148253 = 4.0;
        double r148254 = r148227 + r148253;
        double r148255 = r148252 / r148254;
        double r148256 = r148251 + r148255;
        double r148257 = 12.507343278686905;
        double r148258 = 5.0;
        double r148259 = r148227 + r148258;
        double r148260 = r148257 / r148259;
        double r148261 = r148256 + r148260;
        double r148262 = -0.13857109526572012;
        double r148263 = 6.0;
        double r148264 = r148227 + r148263;
        double r148265 = r148262 / r148264;
        double r148266 = r148261 + r148265;
        double r148267 = 9.984369578019572e-06;
        double r148268 = r148267 / r148229;
        double r148269 = r148266 + r148268;
        double r148270 = 1.5056327351493116e-07;
        double r148271 = 8.0;
        double r148272 = r148227 + r148271;
        double r148273 = r148270 / r148272;
        double r148274 = r148269 + r148273;
        double r148275 = r148237 * r148274;
        double r148276 = r148221 * r148275;
        return r148276;
}

↓

double f(double z) {
        double r148277 = atan2(1.0, 0.0);
        double r148278 = z;
        double r148279 = r148277 * r148278;
        double r148280 = sin(r148279);
        double r148281 = r148277 / r148280;
        double r148282 = 2.0;
        double r148283 = r148277 * r148282;
        double r148284 = sqrt(r148283);
        double r148285 = 1.0;
        double r148286 = r148285 - r148278;
        double r148287 = r148286 - r148285;
        double r148288 = 7.0;
        double r148289 = r148287 + r148288;
        double r148290 = 0.5;
        double r148291 = r148289 + r148290;
        double r148292 = r148287 + r148290;
        double r148293 = pow(r148291, r148292);
        double r148294 = r148284 * r148293;
        double r148295 = -r148291;
        double r148296 = exp(r148295);
        double r148297 = r148294 * r148296;
        double r148298 = 0.9999999999998099;
        double r148299 = 676.5203681218851;
        double r148300 = r148287 + r148285;
        double r148301 = r148299 / r148300;
        double r148302 = r148298 + r148301;
        double r148303 = -1259.1392167224028;
        double r148304 = r148287 + r148282;
        double r148305 = r148303 / r148304;
        double r148306 = r148302 + r148305;
        double r148307 = 771.3234287776531;
        double r148308 = 3.0;
        double r148309 = r148287 + r148308;
        double r148310 = r148307 / r148309;
        double r148311 = r148306 + r148310;
        double r148312 = -176.6150291621406;
        double r148313 = 4.0;
        double r148314 = r148287 + r148313;
        double r148315 = r148312 / r148314;
        double r148316 = r148311 + r148315;
        double r148317 = 12.507343278686905;
        double r148318 = -1.0;
        double r148319 = cbrt(r148286);
        double r148320 = 3.0;
        double r148321 = pow(r148319, r148320);
        double r148322 = fma(r148285, r148318, r148321);
        double r148323 = 0.0;
        double r148324 = r148322 + r148323;
        double r148325 = 5.0;
        double r148326 = r148324 + r148325;
        double r148327 = r148317 / r148326;
        double r148328 = r148316 + r148327;
        double r148329 = -0.13857109526572012;
        double r148330 = 6.0;
        double r148331 = r148287 + r148330;
        double r148332 = r148329 / r148331;
        double r148333 = r148328 + r148332;
        double r148334 = 9.984369578019572e-06;
        double r148335 = r148334 / r148289;
        double r148336 = r148333 + r148335;
        double r148337 = 1.5056327351493116e-07;
        double r148338 = 8.0;
        double r148339 = r148287 + r148338;
        double r148340 = r148337 / r148339;
        double r148341 = r148336 + r148340;
        double r148342 = r148297 * r148341;
        double r148343 = r148281 * r148342;
        return r148343;
}

Error

Bits error versus z

Derivation

1. Initial program 1.8

   \[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
2. Using strategy rm

3. Applied add-cube-cbrt 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

4. Applied add-cube-cbrt 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}} - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

5. Applied prod-diff 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}, \sqrt[3]{1 - z}, -\sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right)} + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

6. Simplified 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\color{blue}{\mathsf{fma}\left(1, -1, {\left(\sqrt[3]{1 - z}\right)}^{3}\right)} + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

7. Simplified 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\mathsf{fma}\left(1, -1, {\left(\sqrt[3]{1 - z}\right)}^{3}\right) + \color{blue}{0}\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

8. Final simplification 1.8

   \[\leadsto \frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\mathsf{fma}\left(1, -1, {\left(\sqrt[3]{1 - z}\right)}^{3}\right) + 0\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]

Reproduce

herbie shell --seed 2020062 +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))))))