Average Error: 1.8 → 1.8
Time: 51.6s
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 r122492 = atan2(1.0, 0.0);
        double r122493 = z;
        double r122494 = r122492 * r122493;
        double r122495 = sin(r122494);
        double r122496 = r122492 / r122495;
        double r122497 = 2.0;
        double r122498 = r122492 * r122497;
        double r122499 = sqrt(r122498);
        double r122500 = 1.0;
        double r122501 = r122500 - r122493;
        double r122502 = r122501 - r122500;
        double r122503 = 7.0;
        double r122504 = r122502 + r122503;
        double r122505 = 0.5;
        double r122506 = r122504 + r122505;
        double r122507 = r122502 + r122505;
        double r122508 = pow(r122506, r122507);
        double r122509 = r122499 * r122508;
        double r122510 = -r122506;
        double r122511 = exp(r122510);
        double r122512 = r122509 * r122511;
        double r122513 = 0.9999999999998099;
        double r122514 = 676.5203681218851;
        double r122515 = r122502 + r122500;
        double r122516 = r122514 / r122515;
        double r122517 = r122513 + r122516;
        double r122518 = -1259.1392167224028;
        double r122519 = r122502 + r122497;
        double r122520 = r122518 / r122519;
        double r122521 = r122517 + r122520;
        double r122522 = 771.3234287776531;
        double r122523 = 3.0;
        double r122524 = r122502 + r122523;
        double r122525 = r122522 / r122524;
        double r122526 = r122521 + r122525;
        double r122527 = -176.6150291621406;
        double r122528 = 4.0;
        double r122529 = r122502 + r122528;
        double r122530 = r122527 / r122529;
        double r122531 = r122526 + r122530;
        double r122532 = 12.507343278686905;
        double r122533 = 5.0;
        double r122534 = r122502 + r122533;
        double r122535 = r122532 / r122534;
        double r122536 = r122531 + r122535;
        double r122537 = -0.13857109526572012;
        double r122538 = 6.0;
        double r122539 = r122502 + r122538;
        double r122540 = r122537 / r122539;
        double r122541 = r122536 + r122540;
        double r122542 = 9.984369578019572e-06;
        double r122543 = r122542 / r122504;
        double r122544 = r122541 + r122543;
        double r122545 = 1.5056327351493116e-07;
        double r122546 = 8.0;
        double r122547 = r122502 + r122546;
        double r122548 = r122545 / r122547;
        double r122549 = r122544 + r122548;
        double r122550 = r122512 * r122549;
        double r122551 = r122496 * r122550;
        return r122551;
}
↓
double f(double z) {
        double r122552 = atan2(1.0, 0.0);
        double r122553 = z;
        double r122554 = r122552 * r122553;
        double r122555 = sin(r122554);
        double r122556 = r122552 / r122555;
        double r122557 = 2.0;
        double r122558 = r122552 * r122557;
        double r122559 = sqrt(r122558);
        double r122560 = 1.0;
        double r122561 = r122560 - r122553;
        double r122562 = r122561 - r122560;
        double r122563 = 7.0;
        double r122564 = r122562 + r122563;
        double r122565 = 0.5;
        double r122566 = r122564 + r122565;
        double r122567 = r122562 + r122565;
        double r122568 = pow(r122566, r122567);
        double r122569 = r122559 * r122568;
        double r122570 = -r122566;
        double r122571 = exp(r122570);
        double r122572 = r122569 * r122571;
        double r122573 = 0.9999999999998099;
        double r122574 = 676.5203681218851;
        double r122575 = r122562 + r122560;
        double r122576 = r122574 / r122575;
        double r122577 = r122573 + r122576;
        double r122578 = -1259.1392167224028;
        double r122579 = r122562 + r122557;
        double r122580 = r122578 / r122579;
        double r122581 = r122577 + r122580;
        double r122582 = 771.3234287776531;
        double r122583 = 3.0;
        double r122584 = r122562 + r122583;
        double r122585 = r122582 / r122584;
        double r122586 = r122581 + r122585;
        double r122587 = -176.6150291621406;
        double r122588 = 4.0;
        double r122589 = r122562 + r122588;
        double r122590 = r122587 / r122589;
        double r122591 = r122586 + r122590;
        double r122592 = 12.507343278686905;
        double r122593 = 5.0;
        double r122594 = r122562 + r122593;
        double r122595 = r122592 / r122594;
        double r122596 = r122591 + r122595;
        double r122597 = -0.13857109526572012;
        double r122598 = 6.0;
        double r122599 = r122562 + r122598;
        double r122600 = r122597 / r122599;
        double r122601 = r122596 + r122600;
        double r122602 = 9.984369578019572e-06;
        double r122603 = r122602 / r122564;
        double r122604 = r122601 + r122603;
        double r122605 = 1.5056327351493116e-07;
        double r122606 = 8.0;
        double r122607 = r122562 + r122606;
        double r122608 = r122605 / r122607;
        double r122609 = r122604 + r122608;
        double r122610 = r122572 * r122609;
        double r122611 = r122556 * r122610;
        return r122611;
}

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 2020049 
(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))))))