Average Error: 26.8 → 0.8
Time: 52.3s
Precision: 64
\[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.3024317203567868 \cdot 10^{+29}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \le 1.2768733135552317 \cdot 10^{+36}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{x \cdot \left(y + x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right)\right) + z}{47.066876606 + x \cdot \left(313.399215894 + \sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}} \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\ \end{array}\]
\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.3024317203567868 \cdot 10^{+29}:\\
\;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\

\mathbf{elif}\;x \le 1.2768733135552317 \cdot 10^{+36}:\\
\;\;\;\;\left(x - 2.0\right) \cdot \frac{x \cdot \left(y + x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right)\right) + z}{47.066876606 + x \cdot \left(313.399215894 + \sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}} \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot x\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r18295816 = x;
        double r18295817 = 2.0;
        double r18295818 = r18295816 - r18295817;
        double r18295819 = 4.16438922228;
        double r18295820 = r18295816 * r18295819;
        double r18295821 = 78.6994924154;
        double r18295822 = r18295820 + r18295821;
        double r18295823 = r18295822 * r18295816;
        double r18295824 = 137.519416416;
        double r18295825 = r18295823 + r18295824;
        double r18295826 = r18295825 * r18295816;
        double r18295827 = y;
        double r18295828 = r18295826 + r18295827;
        double r18295829 = r18295828 * r18295816;
        double r18295830 = z;
        double r18295831 = r18295829 + r18295830;
        double r18295832 = r18295818 * r18295831;
        double r18295833 = 43.3400022514;
        double r18295834 = r18295816 + r18295833;
        double r18295835 = r18295834 * r18295816;
        double r18295836 = 263.505074721;
        double r18295837 = r18295835 + r18295836;
        double r18295838 = r18295837 * r18295816;
        double r18295839 = 313.399215894;
        double r18295840 = r18295838 + r18295839;
        double r18295841 = r18295840 * r18295816;
        double r18295842 = 47.066876606;
        double r18295843 = r18295841 + r18295842;
        double r18295844 = r18295832 / r18295843;
        return r18295844;
}
↓
double f(double x, double y, double z) {
        double r18295845 = x;
        double r18295846 = -2.3024317203567868e+29;
        bool r18295847 = r18295845 <= r18295846;
        double r18295848 = 2.0;
        double r18295849 = r18295845 - r18295848;
        double r18295850 = 4.16438922228;
        double r18295851 = 101.7851458539211;
        double r18295852 = r18295851 / r18295845;
        double r18295853 = r18295850 - r18295852;
        double r18295854 = y;
        double r18295855 = r18295845 * r18295845;
        double r18295856 = r18295854 / r18295855;
        double r18295857 = r18295856 / r18295845;
        double r18295858 = r18295853 + r18295857;
        double r18295859 = r18295849 * r18295858;
        double r18295860 = 1.2768733135552317e+36;
        bool r18295861 = r18295845 <= r18295860;
        double r18295862 = r18295850 * r18295845;
        double r18295863 = 78.6994924154;
        double r18295864 = r18295862 + r18295863;
        double r18295865 = r18295845 * r18295864;
        double r18295866 = 137.519416416;
        double r18295867 = r18295865 + r18295866;
        double r18295868 = r18295845 * r18295867;
        double r18295869 = r18295854 + r18295868;
        double r18295870 = r18295845 * r18295869;
        double r18295871 = z;
        double r18295872 = r18295870 + r18295871;
        double r18295873 = 47.066876606;
        double r18295874 = 313.399215894;
        double r18295875 = 43.3400022514;
        double r18295876 = r18295845 + r18295875;
        double r18295877 = r18295876 * r18295845;
        double r18295878 = 263.505074721;
        double r18295879 = r18295877 + r18295878;
        double r18295880 = sqrt(r18295879);
        double r18295881 = cbrt(r18295880);
        double r18295882 = r18295881 * r18295881;
        double r18295883 = r18295882 * r18295881;
        double r18295884 = r18295883 * r18295845;
        double r18295885 = r18295880 * r18295884;
        double r18295886 = r18295874 + r18295885;
        double r18295887 = r18295845 * r18295886;
        double r18295888 = r18295873 + r18295887;
        double r18295889 = r18295872 / r18295888;
        double r18295890 = r18295849 * r18295889;
        double r18295891 = r18295861 ? r18295890 : r18295859;
        double r18295892 = r18295847 ? r18295859 : r18295891;
        return r18295892;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original26.8
Target0.6
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;x \lt -3.326128725870005 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \mathbf{elif}\;x \lt 9.429991714554673 \cdot 10^{+55}:\\ \;\;\;\;\frac{x - 2.0}{1.0} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.505074721 \cdot x + \left(43.3400022514 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922228 \cdot x\right) - 110.1139242984811\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2.3024317203567868e+29 or 1.2768733135552317e+36 < x

    1. Initial program 58.7

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity58.7

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

    4. Applied times-frac54.7

      \[\leadsto \color{blue}{\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

    5. Simplified54.7

      \[\leadsto \color{blue}{\left(x - 2.0\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt54.7

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}\right)} \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    8. Applied associate-*l*54.7

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot x\right)} + 313.399215894\right) \cdot x + 47.066876606}\]

    9. Taylor expanded around inf 1.1

      \[\leadsto \left(x - 2.0\right) \cdot \color{blue}{\left(\left(\frac{y}{{x}^{3}} + 4.16438922228\right) - 101.7851458539211 \cdot \frac{1}{x}\right)}\]

    10. Simplified1.1

      \[\leadsto \left(x - 2.0\right) \cdot \color{blue}{\left(\frac{\frac{y}{x \cdot x}}{x} + \left(4.16438922228 - \frac{101.7851458539211}{x}\right)\right)}\]

    if -2.3024317203567868e+29 < x < 1.2768733135552317e+36

    1. Initial program 0.6

      \[\frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.6

      \[\leadsto \frac{\left(x - 2.0\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z\right)}{\color{blue}{1 \cdot \left(\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606\right)}}\]

    4. Applied times-frac0.4

      \[\leadsto \color{blue}{\frac{x - 2.0}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}}\]

    5. Simplified0.4

      \[\leadsto \color{blue}{\left(x - 2.0\right)} \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\left(\left(x + 43.3400022514\right) \cdot x + 263.505074721\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]
    6. Using strategy rm

    7. Applied add-sqr-sqrt0.4

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\left(\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}\right)} \cdot x + 313.399215894\right) \cdot x + 47.066876606}\]

    8. Applied associate-*l*0.4

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\color{blue}{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot x\right)} + 313.399215894\right) \cdot x + 47.066876606}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt0.5

      \[\leadsto \left(x - 2.0\right) \cdot \frac{\left(\left(\left(x \cdot 4.16438922228 + 78.6994924154\right) \cdot x + 137.519416416\right) \cdot x + y\right) \cdot x + z}{\left(\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}} \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right)} \cdot x\right) + 313.399215894\right) \cdot x + 47.066876606}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.3024317203567868 \cdot 10^{+29}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\ \mathbf{elif}\;x \le 1.2768733135552317 \cdot 10^{+36}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \frac{x \cdot \left(y + x \cdot \left(x \cdot \left(4.16438922228 \cdot x + 78.6994924154\right) + 137.519416416\right)\right) + z}{47.066876606 + x \cdot \left(313.399215894 + \sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721} \cdot \left(\left(\left(\sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}} \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot \sqrt[3]{\sqrt{\left(x + 43.3400022514\right) \cdot x + 263.505074721}}\right) \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x - 2.0\right) \cdot \left(\left(4.16438922228 - \frac{101.7851458539211}{x}\right) + \frac{\frac{y}{x \cdot x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2.0) 1.0) (/ (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z) (+ (* (+ (+ (* 263.505074721 x) (+ (* 43.3400022514 (* x x)) (* x (* x x)))) 313.399215894) x) 47.066876606))) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811)))

  (/ (* (- x 2.0) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))