Average Error: 29.7 → 4.3
Time: 22.9s
Precision: 64
\[x + \frac{y \cdot \left(\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b\right)}{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le -2.607950243078541092364020610547064160145 \cdot 10^{73} \lor \neg \left(z \le 300501737272537054707712\right):\\ \;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]
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 -2.607950243078541092364020610547064160145 \cdot 10^{73} \lor \neg \left(z \le 300501737272537054707712\right):\\
\;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\

\mathbf{else}:\\
\;\;\;\;x + \frac{y}{\frac{\left(\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r249746 = x;
        double r249747 = y;
        double r249748 = z;
        double r249749 = 3.13060547623;
        double r249750 = r249748 * r249749;
        double r249751 = 11.1667541262;
        double r249752 = r249750 + r249751;
        double r249753 = r249752 * r249748;
        double r249754 = t;
        double r249755 = r249753 + r249754;
        double r249756 = r249755 * r249748;
        double r249757 = a;
        double r249758 = r249756 + r249757;
        double r249759 = r249758 * r249748;
        double r249760 = b;
        double r249761 = r249759 + r249760;
        double r249762 = r249747 * r249761;
        double r249763 = 15.234687407;
        double r249764 = r249748 + r249763;
        double r249765 = r249764 * r249748;
        double r249766 = 31.4690115749;
        double r249767 = r249765 + r249766;
        double r249768 = r249767 * r249748;
        double r249769 = 11.9400905721;
        double r249770 = r249768 + r249769;
        double r249771 = r249770 * r249748;
        double r249772 = 0.607771387771;
        double r249773 = r249771 + r249772;
        double r249774 = r249762 / r249773;
        double r249775 = r249746 + r249774;
        return r249775;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r249776 = z;
        double r249777 = -2.607950243078541e+73;
        bool r249778 = r249776 <= r249777;
        double r249779 = 3.0050173727253705e+23;
        bool r249780 = r249776 <= r249779;
        double r249781 = !r249780;
        bool r249782 = r249778 || r249781;
        double r249783 = x;
        double r249784 = 7049496828096731.0;
        double r249785 = 2251799813685248.0;
        double r249786 = r249784 / r249785;
        double r249787 = y;
        double r249788 = r249786 * r249787;
        double r249789 = t;
        double r249790 = r249789 * r249787;
        double r249791 = 2.0;
        double r249792 = pow(r249776, r249791);
        double r249793 = r249790 / r249792;
        double r249794 = r249788 + r249793;
        double r249795 = 1285181026435087.0;
        double r249796 = 35184372088832.0;
        double r249797 = r249795 / r249796;
        double r249798 = r249787 / r249776;
        double r249799 = r249797 * r249798;
        double r249800 = r249794 - r249799;
        double r249801 = r249783 + r249800;
        double r249802 = 4288183283079449.0;
        double r249803 = 281474976710656.0;
        double r249804 = r249802 / r249803;
        double r249805 = r249776 + r249804;
        double r249806 = r249805 * r249776;
        double r249807 = 4428869650076171.0;
        double r249808 = 140737488355328.0;
        double r249809 = r249807 / r249808;
        double r249810 = r249806 + r249809;
        double r249811 = cbrt(r249776);
        double r249812 = r249811 * r249811;
        double r249813 = r249810 * r249812;
        double r249814 = r249813 * r249811;
        double r249815 = 3360836715704971.0;
        double r249816 = r249815 / r249803;
        double r249817 = r249814 + r249816;
        double r249818 = r249817 * r249776;
        double r249819 = 2737158995491925.0;
        double r249820 = 4503599627370496.0;
        double r249821 = r249819 / r249820;
        double r249822 = r249818 + r249821;
        double r249823 = r249776 * r249786;
        double r249824 = 3143161857605767.0;
        double r249825 = r249824 / r249803;
        double r249826 = r249823 + r249825;
        double r249827 = r249826 * r249776;
        double r249828 = r249827 + r249789;
        double r249829 = r249828 * r249776;
        double r249830 = a;
        double r249831 = r249829 + r249830;
        double r249832 = r249831 * r249776;
        double r249833 = b;
        double r249834 = r249832 + r249833;
        double r249835 = r249822 / r249834;
        double r249836 = r249787 / r249835;
        double r249837 = r249783 + r249836;
        double r249838 = r249782 ? r249801 : r249837;
        return r249838;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original29.7
Target0.9
Herbie4.3
\[\begin{array}{l} \mathbf{if}\;z \lt -6.499344996252631754123144978817242590467 \cdot 10^{53}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \mathbf{elif}\;z \lt 7.066965436914286795694558389038333165002 \cdot 10^{59}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(z + 15.2346874069999991263557603815570473671\right) \cdot z + 31.46901157490000144889563671313226222992\right) \cdot z + 11.94009057210000079862766142468899488449\right) \cdot z + 0.6077713877710000378584709324059076607227}{\left(\left(\left(z \cdot 3.130605476229999961645944495103321969509 + 11.16675412620000074070958362426608800888\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(3.130605476229999961645944495103321969509 - \frac{36.52704169880641416057187598198652267456}{z}\right) + \frac{t}{z \cdot z}\right) \cdot \frac{y}{1}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if z < -2.607950243078541e+73 or 3.0050173727253705e+23 < z

    1. Initial program 60.6

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

    2. Taylor expanded around inf 8.0

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

    3. Simplified8.0

      \[\leadsto x + \color{blue}{\left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)}\]

    if -2.607950243078541e+73 < z < 3.0050173727253705e+23

    1. Initial program 2.5

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

    3. Applied associate-/l*0.9

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

    4. Simplified0.9

      \[\leadsto x + \frac{y}{\color{blue}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot z + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt1.0

      \[\leadsto x + \frac{y}{\frac{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\]

    7. Applied associate-*r*1.0

      \[\leadsto x + \frac{y}{\frac{\left(\color{blue}{\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z}} + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification4.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -2.607950243078541092364020610547064160145 \cdot 10^{73} \lor \neg \left(z \le 300501737272537054707712\right):\\ \;\;\;\;x + \left(\left(\frac{7049496828096731}{2251799813685248} \cdot y + \frac{t \cdot y}{{z}^{2}}\right) - \frac{1285181026435087}{35184372088832} \cdot \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{\left(\left(\left(\left(z + \frac{4288183283079449}{281474976710656}\right) \cdot z + \frac{4428869650076171}{140737488355328}\right) \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \cdot \sqrt[3]{z} + \frac{3360836715704971}{281474976710656}\right) \cdot z + \frac{2737158995491925}{4503599627370496}}{\left(\left(\left(z \cdot \frac{7049496828096731}{2251799813685248} + \frac{3143161857605767}{281474976710656}\right) \cdot z + t\right) \cdot z + a\right) \cdot z + b}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, D"
  :precision binary64

  :herbie-target
  (if (< z -6.4993449962526318e53) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1))) (if (< z 7.0669654369142868e59) (+ x (/ y (/ (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004) (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)))) (+ x (* (+ (- 3.13060547622999996 (/ 36.527041698806414 z)) (/ t (* z z))) (/ y 1)))))

  (+ x (/ (* y (+ (* (+ (* (+ (* (+ (* z 3.13060547622999996) 11.166754126200001) z) t) z) a) z) b)) (+ (* (+ (* (+ (* (+ z 15.234687406999999) z) 31.469011574900001) z) 11.940090572100001) z) 0.60777138777100004))))