Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.1

Time: 7.0m

Precision: 64

• could not determine a ground truth for program body

  1. z = -1.564289068419064e+191

Math

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

↓

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

C

double f(double z) {
        double r6157557 = atan2(1.0, 0.0);
        double r6157558 = z;
        double r6157559 = r6157557 * r6157558;
        double r6157560 = sin(r6157559);
        double r6157561 = r6157557 / r6157560;
        double r6157562 = 2.0;
        double r6157563 = r6157557 * r6157562;
        double r6157564 = sqrt(r6157563);
        double r6157565 = 1.0;
        double r6157566 = r6157565 - r6157558;
        double r6157567 = r6157566 - r6157565;
        double r6157568 = 7.0;
        double r6157569 = r6157567 + r6157568;
        double r6157570 = 0.5;
        double r6157571 = r6157569 + r6157570;
        double r6157572 = r6157567 + r6157570;
        double r6157573 = pow(r6157571, r6157572);
        double r6157574 = r6157564 * r6157573;
        double r6157575 = -r6157571;
        double r6157576 = exp(r6157575);
        double r6157577 = r6157574 * r6157576;
        double r6157578 = 0.9999999999998099;
        double r6157579 = 676.5203681218851;
        double r6157580 = r6157567 + r6157565;
        double r6157581 = r6157579 / r6157580;
        double r6157582 = r6157578 + r6157581;
        double r6157583 = -1259.1392167224028;
        double r6157584 = r6157567 + r6157562;
        double r6157585 = r6157583 / r6157584;
        double r6157586 = r6157582 + r6157585;
        double r6157587 = 771.3234287776531;
        double r6157588 = 3.0;
        double r6157589 = r6157567 + r6157588;
        double r6157590 = r6157587 / r6157589;
        double r6157591 = r6157586 + r6157590;
        double r6157592 = -176.6150291621406;
        double r6157593 = 4.0;
        double r6157594 = r6157567 + r6157593;
        double r6157595 = r6157592 / r6157594;
        double r6157596 = r6157591 + r6157595;
        double r6157597 = 12.507343278686905;
        double r6157598 = 5.0;
        double r6157599 = r6157567 + r6157598;
        double r6157600 = r6157597 / r6157599;
        double r6157601 = r6157596 + r6157600;
        double r6157602 = -0.13857109526572012;
        double r6157603 = 6.0;
        double r6157604 = r6157567 + r6157603;
        double r6157605 = r6157602 / r6157604;
        double r6157606 = r6157601 + r6157605;
        double r6157607 = 9.984369578019572e-06;
        double r6157608 = r6157607 / r6157569;
        double r6157609 = r6157606 + r6157608;
        double r6157610 = 1.5056327351493116e-07;
        double r6157611 = 8.0;
        double r6157612 = r6157567 + r6157611;
        double r6157613 = r6157610 / r6157612;
        double r6157614 = r6157609 + r6157613;
        double r6157615 = r6157577 * r6157614;
        double r6157616 = r6157561 * r6157615;
        return r6157616;
}

↓

double f(double z) {
        double r6157617 = -176.6150291621406;
        double r6157618 = 2.0;
        double r6157619 = z;
        double r6157620 = r6157618 - r6157619;
        double r6157621 = r6157618 + r6157620;
        double r6157622 = r6157617 / r6157621;
        double r6157623 = 1.5056327351493116e-07;
        double r6157624 = 8.0;
        double r6157625 = r6157624 - r6157619;
        double r6157626 = r6157623 / r6157625;
        double r6157627 = 9.984369578019572e-06;
        double r6157628 = 7.0;
        double r6157629 = r6157628 - r6157619;
        double r6157630 = r6157627 / r6157629;
        double r6157631 = r6157626 + r6157630;
        double r6157632 = -0.13857109526572012;
        double r6157633 = 6.0;
        double r6157634 = r6157633 - r6157619;
        double r6157635 = r6157632 / r6157634;
        double r6157636 = 0.9999999999998099;
        double r6157637 = -1259.1392167224028;
        double r6157638 = r6157637 / r6157620;
        double r6157639 = 676.5203681218851;
        double r6157640 = 1.0;
        double r6157641 = r6157640 - r6157619;
        double r6157642 = r6157639 / r6157641;
        double r6157643 = r6157638 + r6157642;
        double r6157644 = r6157636 + r6157643;
        double r6157645 = 12.507343278686905;
        double r6157646 = -1.0;
        double r6157647 = r6157634 + r6157646;
        double r6157648 = r6157645 / r6157647;
        double r6157649 = r6157644 + r6157648;
        double r6157650 = 771.3234287776531;
        double r6157651 = -2.0;
        double r6157652 = r6157641 - r6157651;
        double r6157653 = r6157650 / r6157652;
        double r6157654 = r6157649 + r6157653;
        double r6157655 = r6157635 + r6157654;
        double r6157656 = r6157631 + r6157655;
        double r6157657 = r6157622 + r6157656;
        double r6157658 = atan2(1.0, 0.0);
        double r6157659 = r6157658 * r6157619;
        double r6157660 = sin(r6157659);
        double r6157661 = r6157658 / r6157660;
        double r6157662 = r6157658 * r6157618;
        double r6157663 = sqrt(r6157662);
        double r6157664 = r6157661 * r6157663;
        double r6157665 = r6157657 * r6157664;
        double r6157666 = 0.5;
        double r6157667 = r6157629 + r6157666;
        double r6157668 = cbrt(r6157667);
        double r6157669 = r6157668 * r6157668;
        double r6157670 = -r6157619;
        double r6157671 = r6157670 + r6157666;
        double r6157672 = pow(r6157669, r6157671);
        double r6157673 = exp(r6157667);
        double r6157674 = pow(r6157668, r6157671);
        double r6157675 = r6157673 / r6157674;
        double r6157676 = r6157672 / r6157675;
        double r6157677 = r6157665 * r6157676;
        return r6157677;
}

Error

Bits error versus z

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. Simplified 1.1

   \[\leadsto \color{blue}{\left(\left(\frac{-176.6150291621406}{\left(2 - z\right) + 2} + \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{12.507343278686905}{-1 + \left(6 - z\right)}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \frac{{\left(\left(7 - z\right) + 0.5\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 1.1

   \[\leadsto \left(\left(\frac{-176.6150291621406}{\left(2 - z\right) + 2} + \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{12.507343278686905}{-1 + \left(6 - z\right)}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \frac{{\color{blue}{\left(\left(\sqrt[3]{\left(7 - z\right) + 0.5} \cdot \sqrt[3]{\left(7 - z\right) + 0.5}\right) \cdot \sqrt[3]{\left(7 - z\right) + 0.5}\right)}}^{\left(0.5 + \left(0 - z\right)\right)}}{e^{\left(7 - z\right) + 0.5}}\]

5. Applied unpow-prod-down 1.1

   \[\leadsto \left(\left(\frac{-176.6150291621406}{\left(2 - z\right) + 2} + \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{12.507343278686905}{-1 + \left(6 - z\right)}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \frac{\color{blue}{{\left(\sqrt[3]{\left(7 - z\right) + 0.5} \cdot \sqrt[3]{\left(7 - z\right) + 0.5}\right)}^{\left(0.5 + \left(0 - z\right)\right)} \cdot {\left(\sqrt[3]{\left(7 - z\right) + 0.5}\right)}^{\left(0.5 + \left(0 - z\right)\right)}}}{e^{\left(7 - z\right) + 0.5}}\]

6. Applied associate-/l* 1.1

   \[\leadsto \left(\left(\frac{-176.6150291621406}{\left(2 - z\right) + 2} + \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{2 - z}\right)\right) + \frac{12.507343278686905}{-1 + \left(6 - z\right)}\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-0.13857109526572012}{6 - z}\right) + \left(\frac{1.5056327351493116 \cdot 10^{-07}}{8 - z} + \frac{9.984369578019572 \cdot 10^{-06}}{7 - z}\right)\right)\right) \cdot \left(\sqrt{2 \cdot \pi} \cdot \frac{\pi}{\sin \left(\pi \cdot z\right)}\right)\right) \cdot \color{blue}{\frac{{\left(\sqrt[3]{\left(7 - z\right) + 0.5} \cdot \sqrt[3]{\left(7 - z\right) + 0.5}\right)}^{\left(0.5 + \left(0 - z\right)\right)}}{\frac{e^{\left(7 - z\right) + 0.5}}{{\left(\sqrt[3]{\left(7 - z\right) + 0.5}\right)}^{\left(0.5 + \left(0 - z\right)\right)}}}}\]

7. Final simplification 1.1

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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))))))