Average Error: 59.7 → 0.8
Time: 3.5m
Precision: 64
\[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]
↓
\[\left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)} \cdot \left(\frac{\sqrt{2}}{e^{\log \left(6.5 + z\right) \cdot \left(0.5 - z\right)}} \cdot \sqrt{\pi}\right)\right) \cdot \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z - -5} + \left(\left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\right)\]
\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)
↓
\left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)} \cdot \left(\frac{\sqrt{2}}{e^{\log \left(6.5 + z\right) \cdot \left(0.5 - z\right)}} \cdot \sqrt{\pi}\right)\right) \cdot \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z - -5} + \left(\left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\right)
double f(double z) {
        double r24404232 = atan2(1.0, 0.0);
        double r24404233 = 2.0;
        double r24404234 = r24404232 * r24404233;
        double r24404235 = sqrt(r24404234);
        double r24404236 = z;
        double r24404237 = 1.0;
        double r24404238 = r24404236 - r24404237;
        double r24404239 = 7.0;
        double r24404240 = r24404238 + r24404239;
        double r24404241 = 0.5;
        double r24404242 = r24404240 + r24404241;
        double r24404243 = r24404238 + r24404241;
        double r24404244 = pow(r24404242, r24404243);
        double r24404245 = r24404235 * r24404244;
        double r24404246 = -r24404242;
        double r24404247 = exp(r24404246);
        double r24404248 = r24404245 * r24404247;
        double r24404249 = 0.9999999999998099;
        double r24404250 = 676.5203681218851;
        double r24404251 = r24404238 + r24404237;
        double r24404252 = r24404250 / r24404251;
        double r24404253 = r24404249 + r24404252;
        double r24404254 = -1259.1392167224028;
        double r24404255 = r24404238 + r24404233;
        double r24404256 = r24404254 / r24404255;
        double r24404257 = r24404253 + r24404256;
        double r24404258 = 771.3234287776531;
        double r24404259 = 3.0;
        double r24404260 = r24404238 + r24404259;
        double r24404261 = r24404258 / r24404260;
        double r24404262 = r24404257 + r24404261;
        double r24404263 = -176.6150291621406;
        double r24404264 = 4.0;
        double r24404265 = r24404238 + r24404264;
        double r24404266 = r24404263 / r24404265;
        double r24404267 = r24404262 + r24404266;
        double r24404268 = 12.507343278686905;
        double r24404269 = 5.0;
        double r24404270 = r24404238 + r24404269;
        double r24404271 = r24404268 / r24404270;
        double r24404272 = r24404267 + r24404271;
        double r24404273 = -0.13857109526572012;
        double r24404274 = 6.0;
        double r24404275 = r24404238 + r24404274;
        double r24404276 = r24404273 / r24404275;
        double r24404277 = r24404272 + r24404276;
        double r24404278 = 9.984369578019572e-06;
        double r24404279 = r24404278 / r24404240;
        double r24404280 = r24404277 + r24404279;
        double r24404281 = 1.5056327351493116e-07;
        double r24404282 = 8.0;
        double r24404283 = r24404238 + r24404282;
        double r24404284 = r24404281 / r24404283;
        double r24404285 = r24404280 + r24404284;
        double r24404286 = r24404248 * r24404285;
        return r24404286;
}
↓
double f(double z) {
        double r24404287 = 1.5056327351493116e-07;
        double r24404288 = z;
        double r24404289 = 7.0;
        double r24404290 = r24404288 + r24404289;
        double r24404291 = r24404287 / r24404290;
        double r24404292 = 9.984369578019572e-06;
        double r24404293 = 1.0;
        double r24404294 = r24404288 - r24404293;
        double r24404295 = r24404289 + r24404294;
        double r24404296 = r24404292 / r24404295;
        double r24404297 = r24404291 + r24404296;
        double r24404298 = 0.5;
        double r24404299 = r24404295 + r24404298;
        double r24404300 = r24404298 + r24404294;
        double r24404301 = pow(r24404299, r24404300);
        double r24404302 = atan2(1.0, 0.0);
        double r24404303 = 2.0;
        double r24404304 = r24404302 * r24404303;
        double r24404305 = sqrt(r24404304);
        double r24404306 = r24404301 * r24404305;
        double r24404307 = -r24404299;
        double r24404308 = exp(r24404307);
        double r24404309 = r24404306 * r24404308;
        double r24404310 = r24404297 * r24404309;
        double r24404311 = sqrt(r24404303);
        double r24404312 = 6.5;
        double r24404313 = r24404312 + r24404288;
        double r24404314 = log(r24404313);
        double r24404315 = r24404298 - r24404288;
        double r24404316 = r24404314 * r24404315;
        double r24404317 = exp(r24404316);
        double r24404318 = r24404311 / r24404317;
        double r24404319 = sqrt(r24404302);
        double r24404320 = r24404318 * r24404319;
        double r24404321 = r24404308 * r24404320;
        double r24404322 = 12.507343278686905;
        double r24404323 = 4.0;
        double r24404324 = r24404288 + r24404323;
        double r24404325 = r24404322 / r24404324;
        double r24404326 = -0.13857109526572012;
        double r24404327 = -5.0;
        double r24404328 = r24404288 - r24404327;
        double r24404329 = r24404326 / r24404328;
        double r24404330 = 676.5203681218851;
        double r24404331 = r24404330 / r24404288;
        double r24404332 = 0.9999999999998099;
        double r24404333 = -1259.1392167224028;
        double r24404334 = -1.0;
        double r24404335 = r24404288 - r24404334;
        double r24404336 = r24404333 / r24404335;
        double r24404337 = r24404332 + r24404336;
        double r24404338 = r24404331 + r24404337;
        double r24404339 = 771.3234287776531;
        double r24404340 = r24404288 + r24404303;
        double r24404341 = r24404339 / r24404340;
        double r24404342 = r24404338 + r24404341;
        double r24404343 = -176.6150291621406;
        double r24404344 = 3.0;
        double r24404345 = r24404344 + r24404288;
        double r24404346 = r24404343 / r24404345;
        double r24404347 = r24404342 + r24404346;
        double r24404348 = r24404329 + r24404347;
        double r24404349 = r24404325 + r24404348;
        double r24404350 = r24404321 * r24404349;
        double r24404351 = r24404310 + r24404350;
        return r24404351;
}

Error

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 59.7

    \[\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(z - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(z - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(z - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(z - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(z - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(z - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(z - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(z - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(z - 1\right) + 8}\right)\]

  2. Simplified0.7

    \[\leadsto \color{blue}{\left(\frac{12.507343278686905}{z + 4} + \left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \frac{-0.13857109526572012}{z - -5}\right)\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right)}\]

  3. Taylor expanded around -inf 0.8

    \[\leadsto \left(\frac{12.507343278686905}{z + 4} + \left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \frac{-0.13857109526572012}{z - -5}\right)\right) \cdot \left(\color{blue}{\left(\left(\sqrt{2} \cdot e^{-1 \cdot \left(\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)\right)}\right) \cdot \sqrt{\pi}\right)} \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right)\]

  4. Simplified0.8

    \[\leadsto \left(\frac{12.507343278686905}{z + 4} + \left(\left(\left(\frac{771.3234287776531}{z + 2} + \left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right)\right) + \frac{-176.6150291621406}{z + 3}\right) + \frac{-0.13857109526572012}{z - -5}\right)\right) \cdot \left(\color{blue}{\left(\frac{\sqrt{2}}{e^{\log \left(z + 6.5\right) \cdot \left(0.5 - z\right)}} \cdot \sqrt{\pi}\right)} \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)} + \frac{1.5056327351493116 \cdot 10^{-07}}{z + 7}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(\left(z - 1\right) + 0.5\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right)\]

  5. Final simplification0.8

    \[\leadsto \left(\frac{1.5056327351493116 \cdot 10^{-07}}{z + 7} + \frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(z - 1\right)}\right) \cdot \left(\left({\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(z - 1\right)\right)} \cdot \sqrt{\pi \cdot 2}\right) \cdot e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)}\right) + \left(e^{-\left(\left(7 + \left(z - 1\right)\right) + 0.5\right)} \cdot \left(\frac{\sqrt{2}}{e^{\log \left(6.5 + z\right) \cdot \left(0.5 - z\right)}} \cdot \sqrt{\pi}\right)\right) \cdot \left(\frac{12.507343278686905}{z + 4} + \left(\frac{-0.13857109526572012}{z - -5} + \left(\left(\left(\frac{676.5203681218851}{z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{z - -1}\right)\right) + \frac{771.3234287776531}{z + 2}\right) + \frac{-176.6150291621406}{3 + z}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (z)
  :name "Jmat.Real.gamma, branch z greater than 0.5"
  (* (* (* (sqrt (* PI 2)) (pow (+ (+ (- z 1) 7) 0.5) (+ (- z 1) 0.5))) (exp (- (+ (+ (- z 1) 7) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- z 1) 1))) (/ -1259.1392167224028 (+ (- z 1) 2))) (/ 771.3234287776531 (+ (- z 1) 3))) (/ -176.6150291621406 (+ (- z 1) 4))) (/ 12.507343278686905 (+ (- z 1) 5))) (/ -0.13857109526572012 (+ (- z 1) 6))) (/ 9.984369578019572e-06 (+ (- z 1) 7))) (/ 1.5056327351493116e-07 (+ (- z 1) 8)))))