Average Error: 1.8 → 1.8
Time: 1.0m
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 r142381 = atan2(1.0, 0.0);
        double r142382 = z;
        double r142383 = r142381 * r142382;
        double r142384 = sin(r142383);
        double r142385 = r142381 / r142384;
        double r142386 = 2.0;
        double r142387 = r142381 * r142386;
        double r142388 = sqrt(r142387);
        double r142389 = 1.0;
        double r142390 = r142389 - r142382;
        double r142391 = r142390 - r142389;
        double r142392 = 7.0;
        double r142393 = r142391 + r142392;
        double r142394 = 0.5;
        double r142395 = r142393 + r142394;
        double r142396 = r142391 + r142394;
        double r142397 = pow(r142395, r142396);
        double r142398 = r142388 * r142397;
        double r142399 = -r142395;
        double r142400 = exp(r142399);
        double r142401 = r142398 * r142400;
        double r142402 = 0.9999999999998099;
        double r142403 = 676.5203681218851;
        double r142404 = r142391 + r142389;
        double r142405 = r142403 / r142404;
        double r142406 = r142402 + r142405;
        double r142407 = -1259.1392167224028;
        double r142408 = r142391 + r142386;
        double r142409 = r142407 / r142408;
        double r142410 = r142406 + r142409;
        double r142411 = 771.3234287776531;
        double r142412 = 3.0;
        double r142413 = r142391 + r142412;
        double r142414 = r142411 / r142413;
        double r142415 = r142410 + r142414;
        double r142416 = -176.6150291621406;
        double r142417 = 4.0;
        double r142418 = r142391 + r142417;
        double r142419 = r142416 / r142418;
        double r142420 = r142415 + r142419;
        double r142421 = 12.507343278686905;
        double r142422 = 5.0;
        double r142423 = r142391 + r142422;
        double r142424 = r142421 / r142423;
        double r142425 = r142420 + r142424;
        double r142426 = -0.13857109526572012;
        double r142427 = 6.0;
        double r142428 = r142391 + r142427;
        double r142429 = r142426 / r142428;
        double r142430 = r142425 + r142429;
        double r142431 = 9.984369578019572e-06;
        double r142432 = r142431 / r142393;
        double r142433 = r142430 + r142432;
        double r142434 = 1.5056327351493116e-07;
        double r142435 = 8.0;
        double r142436 = r142391 + r142435;
        double r142437 = r142434 / r142436;
        double r142438 = r142433 + r142437;
        double r142439 = r142401 * r142438;
        double r142440 = r142385 * r142439;
        return r142440;
}
↓
double f(double z) {
        double r142441 = atan2(1.0, 0.0);
        double r142442 = z;
        double r142443 = r142441 * r142442;
        double r142444 = sin(r142443);
        double r142445 = r142441 / r142444;
        double r142446 = 2.0;
        double r142447 = r142441 * r142446;
        double r142448 = sqrt(r142447);
        double r142449 = 1.0;
        double r142450 = r142449 - r142442;
        double r142451 = r142450 - r142449;
        double r142452 = 7.0;
        double r142453 = r142451 + r142452;
        double r142454 = 0.5;
        double r142455 = r142453 + r142454;
        double r142456 = r142451 + r142454;
        double r142457 = pow(r142455, r142456);
        double r142458 = r142448 * r142457;
        double r142459 = -r142455;
        double r142460 = exp(r142459);
        double r142461 = r142458 * r142460;
        double r142462 = 0.9999999999998099;
        double r142463 = 676.5203681218851;
        double r142464 = r142451 + r142449;
        double r142465 = r142463 / r142464;
        double r142466 = r142462 + r142465;
        double r142467 = -1259.1392167224028;
        double r142468 = r142451 + r142446;
        double r142469 = r142467 / r142468;
        double r142470 = r142466 + r142469;
        double r142471 = 771.3234287776531;
        double r142472 = 3.0;
        double r142473 = r142451 + r142472;
        double r142474 = r142471 / r142473;
        double r142475 = r142470 + r142474;
        double r142476 = -176.6150291621406;
        double r142477 = 4.0;
        double r142478 = r142451 + r142477;
        double r142479 = r142476 / r142478;
        double r142480 = r142475 + r142479;
        double r142481 = 12.507343278686905;
        double r142482 = 5.0;
        double r142483 = r142451 + r142482;
        double r142484 = r142481 / r142483;
        double r142485 = r142480 + r142484;
        double r142486 = -0.13857109526572012;
        double r142487 = 6.0;
        double r142488 = r142451 + r142487;
        double r142489 = r142486 / r142488;
        double r142490 = r142485 + r142489;
        double r142491 = 9.984369578019572e-06;
        double r142492 = r142491 / r142453;
        double r142493 = r142490 + r142492;
        double r142494 = 1.5056327351493116e-07;
        double r142495 = 8.0;
        double r142496 = r142451 + r142495;
        double r142497 = r142494 / r142496;
        double r142498 = r142493 + r142497;
        double r142499 = r142461 * r142498;
        double r142500 = r142445 * r142499;
        return r142500;
}

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 +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))))))