Average Error: 1.8 → 1.8
Time: 1.6m
Precision: 64
\[\frac{\pi}{\sin \left(\pi \cdot z\right)} \cdot \left(\left(\left(\sqrt{\pi \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)\]
↓
\[\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\]
\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(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-06}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-07}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right)
↓
\left(\left(\left(\sqrt{2 \cdot \pi} \cdot {\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}^{\left(0.5 + \left(\left(1 - z\right) - 1\right)\right)}\right) \cdot e^{-\left(\left(7 + \left(\left(1 - z\right) - 1\right)\right) + 0.5\right)}\right) \cdot \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 + \left(\left(1 - z\right) - 1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-06}}{7 + \left(\left(1 - z\right) - 1\right)} + \left(\left(\left(\frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4} + \left(\frac{771.3234287776531}{3 + \left(\left(1 - z\right) - 1\right)} + \left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{6 + \left(\left(1 - z\right) - 1\right)}\right)\right)\right)\right) \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}
double f(double z) {
        double r1831514 = atan2(1.0, 0.0);
        double r1831515 = z;
        double r1831516 = r1831514 * r1831515;
        double r1831517 = sin(r1831516);
        double r1831518 = r1831514 / r1831517;
        double r1831519 = 2.0;
        double r1831520 = r1831514 * r1831519;
        double r1831521 = sqrt(r1831520);
        double r1831522 = 1.0;
        double r1831523 = r1831522 - r1831515;
        double r1831524 = r1831523 - r1831522;
        double r1831525 = 7.0;
        double r1831526 = r1831524 + r1831525;
        double r1831527 = 0.5;
        double r1831528 = r1831526 + r1831527;
        double r1831529 = r1831524 + r1831527;
        double r1831530 = pow(r1831528, r1831529);
        double r1831531 = r1831521 * r1831530;
        double r1831532 = -r1831528;
        double r1831533 = exp(r1831532);
        double r1831534 = r1831531 * r1831533;
        double r1831535 = 0.9999999999998099;
        double r1831536 = 676.5203681218851;
        double r1831537 = r1831524 + r1831522;
        double r1831538 = r1831536 / r1831537;
        double r1831539 = r1831535 + r1831538;
        double r1831540 = -1259.1392167224028;
        double r1831541 = r1831524 + r1831519;
        double r1831542 = r1831540 / r1831541;
        double r1831543 = r1831539 + r1831542;
        double r1831544 = 771.3234287776531;
        double r1831545 = 3.0;
        double r1831546 = r1831524 + r1831545;
        double r1831547 = r1831544 / r1831546;
        double r1831548 = r1831543 + r1831547;
        double r1831549 = -176.6150291621406;
        double r1831550 = 4.0;
        double r1831551 = r1831524 + r1831550;
        double r1831552 = r1831549 / r1831551;
        double r1831553 = r1831548 + r1831552;
        double r1831554 = 12.507343278686905;
        double r1831555 = 5.0;
        double r1831556 = r1831524 + r1831555;
        double r1831557 = r1831554 / r1831556;
        double r1831558 = r1831553 + r1831557;
        double r1831559 = -0.13857109526572012;
        double r1831560 = 6.0;
        double r1831561 = r1831524 + r1831560;
        double r1831562 = r1831559 / r1831561;
        double r1831563 = r1831558 + r1831562;
        double r1831564 = 9.984369578019572e-06;
        double r1831565 = r1831564 / r1831526;
        double r1831566 = r1831563 + r1831565;
        double r1831567 = 1.5056327351493116e-07;
        double r1831568 = 8.0;
        double r1831569 = r1831524 + r1831568;
        double r1831570 = r1831567 / r1831569;
        double r1831571 = r1831566 + r1831570;
        double r1831572 = r1831534 * r1831571;
        double r1831573 = r1831518 * r1831572;
        return r1831573;
}
↓
double f(double z) {
        double r1831574 = 2.0;
        double r1831575 = atan2(1.0, 0.0);
        double r1831576 = r1831574 * r1831575;
        double r1831577 = sqrt(r1831576);
        double r1831578 = 7.0;
        double r1831579 = 1.0;
        double r1831580 = z;
        double r1831581 = r1831579 - r1831580;
        double r1831582 = r1831581 - r1831579;
        double r1831583 = r1831578 + r1831582;
        double r1831584 = 0.5;
        double r1831585 = r1831583 + r1831584;
        double r1831586 = r1831584 + r1831582;
        double r1831587 = pow(r1831585, r1831586);
        double r1831588 = r1831577 * r1831587;
        double r1831589 = -r1831585;
        double r1831590 = exp(r1831589);
        double r1831591 = r1831588 * r1831590;
        double r1831592 = 1.5056327351493116e-07;
        double r1831593 = 8.0;
        double r1831594 = r1831593 + r1831582;
        double r1831595 = r1831592 / r1831594;
        double r1831596 = 9.984369578019572e-06;
        double r1831597 = r1831596 / r1831583;
        double r1831598 = -176.6150291621406;
        double r1831599 = 4.0;
        double r1831600 = r1831582 + r1831599;
        double r1831601 = r1831598 / r1831600;
        double r1831602 = 771.3234287776531;
        double r1831603 = 3.0;
        double r1831604 = r1831603 + r1831582;
        double r1831605 = r1831602 / r1831604;
        double r1831606 = 0.9999999999998099;
        double r1831607 = 676.5203681218851;
        double r1831608 = r1831582 + r1831579;
        double r1831609 = r1831607 / r1831608;
        double r1831610 = r1831606 + r1831609;
        double r1831611 = -1259.1392167224028;
        double r1831612 = r1831582 + r1831574;
        double r1831613 = r1831611 / r1831612;
        double r1831614 = r1831610 + r1831613;
        double r1831615 = r1831605 + r1831614;
        double r1831616 = r1831601 + r1831615;
        double r1831617 = 12.507343278686905;
        double r1831618 = 5.0;
        double r1831619 = r1831582 + r1831618;
        double r1831620 = r1831617 / r1831619;
        double r1831621 = r1831616 + r1831620;
        double r1831622 = -0.13857109526572012;
        double r1831623 = 6.0;
        double r1831624 = r1831623 + r1831582;
        double r1831625 = r1831622 / r1831624;
        double r1831626 = r1831621 + r1831625;
        double r1831627 = r1831597 + r1831626;
        double r1831628 = r1831595 + r1831627;
        double r1831629 = r1831591 * r1831628;
        double r1831630 = r1831575 * r1831580;
        double r1831631 = sin(r1831630);
        double r1831632 = r1831575 / r1831631;
        double r1831633 = r1831629 * r1831632;
        return r1831633;
}

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

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

  2. Final simplification1.8

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

Reproduce

herbie shell --seed 2019154 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  (* (/ 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))))))