Jmat.Real.gamma, branch z less than 0.5

Average Error: 1.8 → 1.8

Time: 1.3m

Precision: 64

• could not determine a ground truth for program body

  1. z = -2.637695780548612e+29

Math

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

↓

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

C

double f(double z) {
        double r4002700 = atan2(1.0, 0.0);
        double r4002701 = z;
        double r4002702 = r4002700 * r4002701;
        double r4002703 = sin(r4002702);
        double r4002704 = r4002700 / r4002703;
        double r4002705 = 2.0;
        double r4002706 = r4002700 * r4002705;
        double r4002707 = sqrt(r4002706);
        double r4002708 = 1.0;
        double r4002709 = r4002708 - r4002701;
        double r4002710 = r4002709 - r4002708;
        double r4002711 = 7.0;
        double r4002712 = r4002710 + r4002711;
        double r4002713 = 0.5;
        double r4002714 = r4002712 + r4002713;
        double r4002715 = r4002710 + r4002713;
        double r4002716 = pow(r4002714, r4002715);
        double r4002717 = r4002707 * r4002716;
        double r4002718 = -r4002714;
        double r4002719 = exp(r4002718);
        double r4002720 = r4002717 * r4002719;
        double r4002721 = 0.9999999999998099;
        double r4002722 = 676.5203681218851;
        double r4002723 = r4002710 + r4002708;
        double r4002724 = r4002722 / r4002723;
        double r4002725 = r4002721 + r4002724;
        double r4002726 = -1259.1392167224028;
        double r4002727 = r4002710 + r4002705;
        double r4002728 = r4002726 / r4002727;
        double r4002729 = r4002725 + r4002728;
        double r4002730 = 771.3234287776531;
        double r4002731 = 3.0;
        double r4002732 = r4002710 + r4002731;
        double r4002733 = r4002730 / r4002732;
        double r4002734 = r4002729 + r4002733;
        double r4002735 = -176.6150291621406;
        double r4002736 = 4.0;
        double r4002737 = r4002710 + r4002736;
        double r4002738 = r4002735 / r4002737;
        double r4002739 = r4002734 + r4002738;
        double r4002740 = 12.507343278686905;
        double r4002741 = 5.0;
        double r4002742 = r4002710 + r4002741;
        double r4002743 = r4002740 / r4002742;
        double r4002744 = r4002739 + r4002743;
        double r4002745 = -0.13857109526572012;
        double r4002746 = 6.0;
        double r4002747 = r4002710 + r4002746;
        double r4002748 = r4002745 / r4002747;
        double r4002749 = r4002744 + r4002748;
        double r4002750 = 9.984369578019572e-06;
        double r4002751 = r4002750 / r4002712;
        double r4002752 = r4002749 + r4002751;
        double r4002753 = 1.5056327351493116e-07;
        double r4002754 = 8.0;
        double r4002755 = r4002710 + r4002754;
        double r4002756 = r4002753 / r4002755;
        double r4002757 = r4002752 + r4002756;
        double r4002758 = r4002720 * r4002757;
        double r4002759 = r4002704 * r4002758;
        return r4002759;
}

↓

double f(double z) {
        double r4002760 = 1.5056327351493116e-07;
        double r4002761 = 8.0;
        double r4002762 = 1.0;
        double r4002763 = z;
        double r4002764 = r4002762 - r4002763;
        double r4002765 = r4002764 - r4002762;
        double r4002766 = r4002761 + r4002765;
        double r4002767 = r4002760 / r4002766;
        double r4002768 = 9.984369578019572e-06;
        double r4002769 = 7.0;
        double r4002770 = cbrt(r4002765);
        double r4002771 = r4002770 * r4002770;
        double r4002772 = r4002771 * r4002770;
        double r4002773 = cbrt(r4002772);
        double r4002774 = r4002773 * r4002771;
        double r4002775 = r4002769 + r4002774;
        double r4002776 = r4002768 / r4002775;
        double r4002777 = 12.507343278686905;
        double r4002778 = 5.0;
        double r4002779 = r4002778 + r4002774;
        double r4002780 = r4002777 / r4002779;
        double r4002781 = -176.6150291621406;
        double r4002782 = 4.0;
        double r4002783 = r4002765 + r4002782;
        double r4002784 = r4002781 / r4002783;
        double r4002785 = 771.3234287776531;
        double r4002786 = 3.0;
        double r4002787 = r4002786 + r4002765;
        double r4002788 = r4002785 / r4002787;
        double r4002789 = 0.9999999999998099;
        double r4002790 = 676.5203681218851;
        double r4002791 = r4002765 + r4002762;
        double r4002792 = r4002790 / r4002791;
        double r4002793 = r4002789 + r4002792;
        double r4002794 = -1259.1392167224028;
        double r4002795 = 2.0;
        double r4002796 = r4002765 + r4002795;
        double r4002797 = r4002794 / r4002796;
        double r4002798 = r4002793 + r4002797;
        double r4002799 = r4002788 + r4002798;
        double r4002800 = r4002784 + r4002799;
        double r4002801 = r4002780 + r4002800;
        double r4002802 = -0.13857109526572012;
        double r4002803 = 6.0;
        double r4002804 = r4002765 + r4002803;
        double r4002805 = r4002802 / r4002804;
        double r4002806 = r4002801 + r4002805;
        double r4002807 = r4002776 + r4002806;
        double r4002808 = r4002767 + r4002807;
        double r4002809 = atan2(1.0, 0.0);
        double r4002810 = r4002795 * r4002809;
        double r4002811 = sqrt(r4002810);
        double r4002812 = r4002769 + r4002765;
        double r4002813 = 0.5;
        double r4002814 = r4002812 + r4002813;
        double r4002815 = r4002813 + r4002765;
        double r4002816 = pow(r4002814, r4002815);
        double r4002817 = r4002811 * r4002816;
        double r4002818 = -r4002814;
        double r4002819 = exp(r4002818);
        double r4002820 = r4002817 * r4002819;
        double r4002821 = r4002808 * r4002820;
        double r4002822 = r4002809 * r4002763;
        double r4002823 = sin(r4002822);
        double r4002824 = r4002809 / r4002823;
        double r4002825 = r4002821 * r4002824;
        return r4002825;
}

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. Using strategy rm

3. Applied add-cube-cbrt 1.8

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

5. Applied add-cube-cbrt 1.8

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

7. Applied add-cube-cbrt 1.8

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

9. Applied add-cube-cbrt 1.8

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

10. Final simplification 1.8

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

Reproduce

herbie shell --seed 2019141 
(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))))))