Average Error: 1.8 → 0.5
Time: 58.7s
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.99999999999980993 + \frac{676.520368121885099}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.13921672240281}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.32342877765313}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.615029162140587}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.5073432786869052}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.138571095265720118}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.98436957801957158 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.50563273514931162 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\frac{{\left(0.5 + \left(7 - z\right)\right)}^{\left(0.5 - z\right)} \cdot \left(\left(\sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}} \cdot \sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}}\right) \cdot \sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}}\right)}{\frac{\sin \left(\pi \cdot z\right)}{\left(\left(\frac{771.32342877765313}{3 - z} + \left(\left(0.99999999999980993 + \frac{676.520368121885099}{1 - z}\right) + \frac{-1259.13921672240281}{2 - z}\right)\right) + \frac{-176.615029162140587}{4 - z}\right) + \left(\left(\frac{1.50563273514931162 \cdot 10^{-7}}{8 - z} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{7 - z} + \frac{-0.138571095265720118}{6 - z}\right)\right) + \frac{12.5073432786869052}{5 - z}\right)} \cdot e^{0.5 + \left(7 - z\right)}}\]
\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{{\left(0.5 + \left(7 - z\right)\right)}^{\left(0.5 - z\right)} \cdot \left(\left(\sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}} \cdot \sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}}\right) \cdot \sqrt[3]{\pi \cdot \sqrt{\pi \cdot 2}}\right)}{\frac{\sin \left(\pi \cdot z\right)}{\left(\left(\frac{771.32342877765313}{3 - z} + \left(\left(0.99999999999980993 + \frac{676.520368121885099}{1 - z}\right) + \frac{-1259.13921672240281}{2 - z}\right)\right) + \frac{-176.615029162140587}{4 - z}\right) + \left(\left(\frac{1.50563273514931162 \cdot 10^{-7}}{8 - z} + \left(\frac{9.98436957801957158 \cdot 10^{-6}}{7 - z} + \frac{-0.138571095265720118}{6 - z}\right)\right) + \frac{12.5073432786869052}{5 - z}\right)} \cdot e^{0.5 + \left(7 - z\right)}}
double f(double z) {
        double r129181 = atan2(1.0, 0.0);
        double r129182 = z;
        double r129183 = r129181 * r129182;
        double r129184 = sin(r129183);
        double r129185 = r129181 / r129184;
        double r129186 = 2.0;
        double r129187 = r129181 * r129186;
        double r129188 = sqrt(r129187);
        double r129189 = 1.0;
        double r129190 = r129189 - r129182;
        double r129191 = r129190 - r129189;
        double r129192 = 7.0;
        double r129193 = r129191 + r129192;
        double r129194 = 0.5;
        double r129195 = r129193 + r129194;
        double r129196 = r129191 + r129194;
        double r129197 = pow(r129195, r129196);
        double r129198 = r129188 * r129197;
        double r129199 = -r129195;
        double r129200 = exp(r129199);
        double r129201 = r129198 * r129200;
        double r129202 = 0.9999999999998099;
        double r129203 = 676.5203681218851;
        double r129204 = r129191 + r129189;
        double r129205 = r129203 / r129204;
        double r129206 = r129202 + r129205;
        double r129207 = -1259.1392167224028;
        double r129208 = r129191 + r129186;
        double r129209 = r129207 / r129208;
        double r129210 = r129206 + r129209;
        double r129211 = 771.3234287776531;
        double r129212 = 3.0;
        double r129213 = r129191 + r129212;
        double r129214 = r129211 / r129213;
        double r129215 = r129210 + r129214;
        double r129216 = -176.6150291621406;
        double r129217 = 4.0;
        double r129218 = r129191 + r129217;
        double r129219 = r129216 / r129218;
        double r129220 = r129215 + r129219;
        double r129221 = 12.507343278686905;
        double r129222 = 5.0;
        double r129223 = r129191 + r129222;
        double r129224 = r129221 / r129223;
        double r129225 = r129220 + r129224;
        double r129226 = -0.13857109526572012;
        double r129227 = 6.0;
        double r129228 = r129191 + r129227;
        double r129229 = r129226 / r129228;
        double r129230 = r129225 + r129229;
        double r129231 = 9.984369578019572e-06;
        double r129232 = r129231 / r129193;
        double r129233 = r129230 + r129232;
        double r129234 = 1.5056327351493116e-07;
        double r129235 = 8.0;
        double r129236 = r129191 + r129235;
        double r129237 = r129234 / r129236;
        double r129238 = r129233 + r129237;
        double r129239 = r129201 * r129238;
        double r129240 = r129185 * r129239;
        return r129240;
}
↓
double f(double z) {
        double r129241 = 0.5;
        double r129242 = 7.0;
        double r129243 = z;
        double r129244 = r129242 - r129243;
        double r129245 = r129241 + r129244;
        double r129246 = r129241 - r129243;
        double r129247 = pow(r129245, r129246);
        double r129248 = atan2(1.0, 0.0);
        double r129249 = 2.0;
        double r129250 = r129248 * r129249;
        double r129251 = sqrt(r129250);
        double r129252 = r129248 * r129251;
        double r129253 = cbrt(r129252);
        double r129254 = r129253 * r129253;
        double r129255 = r129254 * r129253;
        double r129256 = r129247 * r129255;
        double r129257 = r129248 * r129243;
        double r129258 = sin(r129257);
        double r129259 = 771.3234287776531;
        double r129260 = 3.0;
        double r129261 = r129260 - r129243;
        double r129262 = r129259 / r129261;
        double r129263 = 0.9999999999998099;
        double r129264 = 676.5203681218851;
        double r129265 = 1.0;
        double r129266 = r129265 - r129243;
        double r129267 = r129264 / r129266;
        double r129268 = r129263 + r129267;
        double r129269 = -1259.1392167224028;
        double r129270 = r129249 - r129243;
        double r129271 = r129269 / r129270;
        double r129272 = r129268 + r129271;
        double r129273 = r129262 + r129272;
        double r129274 = -176.6150291621406;
        double r129275 = 4.0;
        double r129276 = r129275 - r129243;
        double r129277 = r129274 / r129276;
        double r129278 = r129273 + r129277;
        double r129279 = 1.5056327351493116e-07;
        double r129280 = 8.0;
        double r129281 = r129280 - r129243;
        double r129282 = r129279 / r129281;
        double r129283 = 9.984369578019572e-06;
        double r129284 = r129283 / r129244;
        double r129285 = -0.13857109526572012;
        double r129286 = 6.0;
        double r129287 = r129286 - r129243;
        double r129288 = r129285 / r129287;
        double r129289 = r129284 + r129288;
        double r129290 = r129282 + r129289;
        double r129291 = 12.507343278686905;
        double r129292 = 5.0;
        double r129293 = r129292 - r129243;
        double r129294 = r129291 / r129293;
        double r129295 = r129290 + r129294;
        double r129296 = r129278 + r129295;
        double r129297 = r129258 / r129296;
        double r129298 = exp(r129245);
        double r129299 = r129297 * r129298;
        double r129300 = r129256 / r129299;
        return r129300;
}

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. Simplified1.5

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

  4. Applied add-cube-cbrt0.5

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

  5. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 
(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))))))