Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D

Average Error: 29.3 → 1.2

Time: 28.6s

Precision: 64

Math

\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

↓

\[\begin{array}{l} \mathbf{if}\;z \le -44597120873684739159803907641071830455810000 \lor \neg \left(z \le 4.625141675276618178345036173136869149202 \cdot 10^{62}\right):\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + \frac{y}{z} \cdot \frac{t}{z}\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(b + \left(z \cdot a + \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot \left(z \cdot z\right)\right)\right) \cdot \frac{1}{z \cdot \left(11.94009057210000079862766142468899488449 + \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}\right) + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a, double b) {
        double r277767 = x;
        double r277768 = y;
        double r277769 = z;
        double r277770 = 3.13060547623;
        double r277771 = r277769 * r277770;
        double r277772 = 11.1667541262;
        double r277773 = r277771 + r277772;
        double r277774 = r277773 * r277769;
        double r277775 = t;
        double r277776 = r277774 + r277775;
        double r277777 = r277776 * r277769;
        double r277778 = a;
        double r277779 = r277777 + r277778;
        double r277780 = r277779 * r277769;
        double r277781 = b;
        double r277782 = r277780 + r277781;
        double r277783 = r277768 * r277782;
        double r277784 = 15.234687407;
        double r277785 = r277769 + r277784;
        double r277786 = r277785 * r277769;
        double r277787 = 31.4690115749;
        double r277788 = r277786 + r277787;
        double r277789 = r277788 * r277769;
        double r277790 = 11.9400905721;
        double r277791 = r277789 + r277790;
        double r277792 = r277791 * r277769;
        double r277793 = 0.607771387771;
        double r277794 = r277792 + r277793;
        double r277795 = r277783 / r277794;
        double r277796 = r277767 + r277795;
        return r277796;
}

↓

double f(double x, double y, double z, double t, double a, double b) {
        double r277797 = z;
        double r277798 = -4.459712087368474e+43;
        bool r277799 = r277797 <= r277798;
        double r277800 = 4.625141675276618e+62;
        bool r277801 = r277797 <= r277800;
        double r277802 = !r277801;
        bool r277803 = r277799 || r277802;
        double r277804 = y;
        double r277805 = 3.13060547623;
        double r277806 = r277804 * r277805;
        double r277807 = 36.527041698806414;
        double r277808 = r277797 / r277804;
        double r277809 = r277807 / r277808;
        double r277810 = r277806 - r277809;
        double r277811 = r277804 / r277797;
        double r277812 = t;
        double r277813 = r277812 / r277797;
        double r277814 = r277811 * r277813;
        double r277815 = r277810 + r277814;
        double r277816 = x;
        double r277817 = r277815 + r277816;
        double r277818 = b;
        double r277819 = a;
        double r277820 = r277797 * r277819;
        double r277821 = r277805 * r277797;
        double r277822 = 11.1667541262;
        double r277823 = r277821 + r277822;
        double r277824 = r277823 * r277797;
        double r277825 = r277824 + r277812;
        double r277826 = r277797 * r277797;
        double r277827 = r277825 * r277826;
        double r277828 = r277820 + r277827;
        double r277829 = r277818 + r277828;
        double r277830 = 1.0;
        double r277831 = 11.9400905721;
        double r277832 = 15.234687407;
        double r277833 = r277797 + r277832;
        double r277834 = r277797 * r277833;
        double r277835 = 31.4690115749;
        double r277836 = r277834 + r277835;
        double r277837 = r277836 * r277797;
        double r277838 = r277831 + r277837;
        double r277839 = r277797 * r277838;
        double r277840 = 0.607771387771;
        double r277841 = r277839 + r277840;
        double r277842 = r277830 / r277841;
        double r277843 = r277829 * r277842;
        double r277844 = r277804 * r277843;
        double r277845 = r277844 + r277816;
        double r277846 = r277803 ? r277817 : r277845;
        return r277846;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	29.3
Target	1.0
Herbie	1.2

\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -4.459712087368474e+43 or 4.625141675276618e+62 < z

   1. Initial program 61.8

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 59.7

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227} \cdot y}\]

   3. Taylor expanded around inf 9.1

      \[\leadsto x + \color{blue}{\left(\left(\frac{t \cdot y}{{z}^{2}} + 3.130605476229999961645944495103321969509 \cdot y\right) - 36.52704169880641416057187598198652267456 \cdot \frac{y}{z}\right)}\]

   4. Simplified 1.1

      \[\leadsto x + \color{blue}{\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + \frac{t}{z} \cdot \frac{y}{z}\right)}\]

   if -4.459712087368474e+43 < z < 4.625141675276618e+62

   1. Initial program 3.0

      \[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]

   2. Simplified 1.2

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227} \cdot y}\]
   3. Using strategy rm

   4. Applied *-un-lft-identity 1.2

      \[\leadsto x + \frac{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}{\color{blue}{1 \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227\right)}} \cdot y\]

   5. Applied add-cube-cbrt 1.7

      \[\leadsto x + \frac{\color{blue}{\left(\sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b} \cdot \sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}\right) \cdot \sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}}}{1 \cdot \left(z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227\right)} \cdot y\]

   6. Applied times-frac 1.7

      \[\leadsto x + \color{blue}{\left(\frac{\sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b} \cdot \sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}}{1} \cdot \frac{\sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227}\right)} \cdot y\]

   7. Simplified 1.7

      \[\leadsto x + \left(\color{blue}{\left(\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)} \cdot \sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}\right)} \cdot \frac{\sqrt[3]{z \cdot \left(a + z \cdot \left(t + \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right) \cdot z\right)\right) + b}}{z \cdot \left(z \cdot \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) + 11.94009057210000079862766142468899488449\right) + 0.6077713877710000378584709324059076607227}\right) \cdot y\]

   8. Simplified 1.7

      \[\leadsto x + \left(\left(\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)} \cdot \sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}\right) \cdot \color{blue}{\frac{\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}}\right) \cdot y\]
   9. Using strategy rm

   10. Applied div-inv 1.7

       \[\leadsto x + \left(\left(\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)} \cdot \sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}\right) \cdot \color{blue}{\left(\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)} \cdot \frac{1}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}\right)}\right) \cdot y\]

   11. Applied associate-*r* 1.7

       \[\leadsto x + \color{blue}{\left(\left(\left(\sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)} \cdot \sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}\right) \cdot \sqrt[3]{b + z \cdot \left(\left(z \cdot \left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) + t\right) \cdot z + a\right)}\right) \cdot \frac{1}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}\right)} \cdot y\]

   12. Simplified 1.2

       \[\leadsto x + \left(\color{blue}{\left(z \cdot \left(a + \left(t + z \cdot \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right)\right) \cdot z\right) + b\right)} \cdot \frac{1}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}\right) \cdot y\]
   13. Using strategy rm

   14. Applied distribute-lft-in 1.2

       \[\leadsto x + \left(\left(\color{blue}{\left(z \cdot a + z \cdot \left(\left(t + z \cdot \left(11.16675412620000074070958362426608800888 + z \cdot 3.130605476229999961645944495103321969509\right)\right) \cdot z\right)\right)} + b\right) \cdot \frac{1}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}\right) \cdot y\]

   15. Simplified 1.4

       \[\leadsto x + \left(\left(\left(z \cdot a + \color{blue}{\left(\left(11.16675412620000074070958362426608800888 + 3.130605476229999961645944495103321969509 \cdot z\right) \cdot z + t\right) \cdot \left(z \cdot z\right)}\right) + b\right) \cdot \frac{1}{0.6077713877710000378584709324059076607227 + \left(\left(\left(15.2346874069999991263557603815570473671 + z\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z}\right) \cdot y\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -44597120873684739159803907641071830455810000 \lor \neg \left(z \le 4.625141675276618178345036173136869149202 \cdot 10^{62}\right):\\ \;\;\;\;\left(\left(y \cdot 3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{\frac{z}{y}}\right) + \frac{y}{z} \cdot \frac{t}{z}\right) + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\left(b + \left(z \cdot a + \left(\left(3.130605476229999961645944495103321969509 \cdot z + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot \left(z \cdot z\right)\right)\right) \cdot \frac{1}{z \cdot \left(11.94009057210000079862766142468899488449 + \left(z \cdot \left(z + 15.2346874069999991263557603815570473671\right) + 31.46901157490000144889563671313226222992\right) \cdot z\right) + 0.6077713877710000378584709324059076607227}\right) + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"

  :herbie-target
  (if (< z -6.499344996252632e+53) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0))) (if (< z 7.066965436914287e+59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771) (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547623 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1.0)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547623) 11.1667541262) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687407) z) 31.4690115749) z) 11.9400905721) z) 0.607771387771))))