Average Error: 6.1 → 5.2
Time: 10.0s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 9.4172685562653542 \cdot 10^{128}:\\ \;\;\;\;\left(\left(\left(3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right) - 1.5 \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 9.4172685562653542 \cdot 10^{128}:\\
\;\;\;\;\left(\left(\left(3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right) - 1.5 \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r507837 = x;
        double r507838 = 0.5;
        double r507839 = r507837 - r507838;
        double r507840 = log(r507837);
        double r507841 = r507839 * r507840;
        double r507842 = r507841 - r507837;
        double r507843 = 0.91893853320467;
        double r507844 = r507842 + r507843;
        double r507845 = y;
        double r507846 = 0.0007936500793651;
        double r507847 = r507845 + r507846;
        double r507848 = z;
        double r507849 = r507847 * r507848;
        double r507850 = 0.0027777777777778;
        double r507851 = r507849 - r507850;
        double r507852 = r507851 * r507848;
        double r507853 = 0.083333333333333;
        double r507854 = r507852 + r507853;
        double r507855 = r507854 / r507837;
        double r507856 = r507844 + r507855;
        return r507856;
}
↓
double f(double x, double y, double z) {
        double r507857 = x;
        double r507858 = 9.417268556265354e+128;
        bool r507859 = r507857 <= r507858;
        double r507860 = 3.0;
        double r507861 = 1.0;
        double r507862 = r507861 / r507857;
        double r507863 = -0.3333333333333333;
        double r507864 = pow(r507862, r507863);
        double r507865 = log(r507864);
        double r507866 = r507865 * r507857;
        double r507867 = r507860 * r507866;
        double r507868 = 1.5;
        double r507869 = r507868 * r507865;
        double r507870 = r507867 - r507869;
        double r507871 = r507870 - r507857;
        double r507872 = 0.91893853320467;
        double r507873 = r507871 + r507872;
        double r507874 = y;
        double r507875 = 0.0007936500793651;
        double r507876 = r507874 + r507875;
        double r507877 = z;
        double r507878 = r507876 * r507877;
        double r507879 = 0.0027777777777778;
        double r507880 = r507878 - r507879;
        double r507881 = r507880 * r507877;
        double r507882 = 0.083333333333333;
        double r507883 = r507881 + r507882;
        double r507884 = r507883 / r507857;
        double r507885 = r507873 + r507884;
        double r507886 = 0.3333333333333333;
        double r507887 = pow(r507857, r507886);
        double r507888 = log(r507887);
        double r507889 = r507860 * r507857;
        double r507890 = r507889 - r507868;
        double r507891 = r507888 * r507890;
        double r507892 = r507891 - r507857;
        double r507893 = r507892 + r507872;
        double r507894 = 2.0;
        double r507895 = pow(r507877, r507894);
        double r507896 = r507895 / r507857;
        double r507897 = r507875 * r507896;
        double r507898 = r507882 * r507862;
        double r507899 = r507897 + r507898;
        double r507900 = r507877 / r507857;
        double r507901 = r507879 * r507900;
        double r507902 = r507899 - r507901;
        double r507903 = r507893 + r507902;
        double r507904 = r507859 ? r507885 : r507903;
        return r507904;
}

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

Original6.1
Target1.3
Herbie5.2
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 9.417268556265354e+128

    1. Initial program 1.6

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt1.6

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    4. Applied log-prod1.7

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    5. Applied distribute-lft-in1.7

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    6. Simplified1.7

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    7. Taylor expanded around inf 1.7

      \[\leadsto \left(\left(\color{blue}{\left(3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right) - 1.5 \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    if 9.417268556265354e+128 < x

    1. Initial program 14.0

      \[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
    2. Using strategy rm

    3. Applied add-cube-cbrt14.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    4. Applied log-prod14.1

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    5. Applied distribute-lft-in14.1

      \[\leadsto \left(\left(\color{blue}{\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    6. Simplified14.1

      \[\leadsto \left(\left(\left(\color{blue}{\left(x - 0.5\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{x}\right)\right)} + \left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    7. Taylor expanded around 0 14.1

      \[\leadsto \left(\left(\color{blue}{\left(3 \cdot \left(x \cdot \log \left({x}^{\frac{1}{3}}\right)\right) - 1.5 \cdot \log \left({x}^{\frac{1}{3}}\right)\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    8. Simplified14.1

      \[\leadsto \left(\left(\color{blue}{\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right)} - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    9. Taylor expanded around 0 11.4

      \[\leadsto \left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification5.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 9.4172685562653542 \cdot 10^{128}:\\ \;\;\;\;\left(\left(\left(3 \cdot \left(\log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) \cdot x\right) - 1.5 \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)\right) - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\log \left({x}^{\frac{1}{3}}\right) \cdot \left(3 \cdot x - 1.5\right) - x\right) + 0.91893853320467001\right) + \left(\left(7.93650079365100015 \cdot 10^{-4} \cdot \frac{{z}^{2}}{x} + 0.0833333333333329956 \cdot \frac{1}{x}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:$slogFactorial from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (+ (+ (+ (* (- x 0.5) (log x)) (- 0.91893853320467 x)) (/ 0.083333333333333 x)) (* (/ z x) (- (* z (+ y 0.0007936500793651)) 0.0027777777777778)))

  (+ (+ (- (* (- x 0.5) (log x)) x) 0.91893853320467) (/ (+ (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) 0.083333333333333) x)))