Average Error: 5.7 → 3.8
Time: 13.5s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 4.93214915957164702 \cdot 10^{38}:\\ \;\;\;\;\left(\left(\log \left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left({\left(\frac{1}{x}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]
\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 4.93214915957164702 \cdot 10^{38}:\\
\;\;\;\;\left(\left(\log \left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left({\left(\frac{1}{x}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right) - x\right)\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(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\

\end{array}
double f(double x, double y, double z) {
        double r448873 = x;
        double r448874 = 0.5;
        double r448875 = r448873 - r448874;
        double r448876 = log(r448873);
        double r448877 = r448875 * r448876;
        double r448878 = r448877 - r448873;
        double r448879 = 0.91893853320467;
        double r448880 = r448878 + r448879;
        double r448881 = y;
        double r448882 = 0.0007936500793651;
        double r448883 = r448881 + r448882;
        double r448884 = z;
        double r448885 = r448883 * r448884;
        double r448886 = 0.0027777777777778;
        double r448887 = r448885 - r448886;
        double r448888 = r448887 * r448884;
        double r448889 = 0.083333333333333;
        double r448890 = r448888 + r448889;
        double r448891 = r448890 / r448873;
        double r448892 = r448880 + r448891;
        return r448892;
}
↓
double f(double x, double y, double z) {
        double r448893 = x;
        double r448894 = 4.932149159571647e+38;
        bool r448895 = r448893 <= r448894;
        double r448896 = sqrt(r448893);
        double r448897 = cbrt(r448896);
        double r448898 = r448897 * r448897;
        double r448899 = cbrt(r448893);
        double r448900 = r448898 * r448899;
        double r448901 = log(r448900);
        double r448902 = 0.5;
        double r448903 = r448893 - r448902;
        double r448904 = r448901 * r448903;
        double r448905 = 1.0;
        double r448906 = r448905 / r448893;
        double r448907 = -0.3333333333333333;
        double r448908 = cbrt(r448907);
        double r448909 = r448908 * r448908;
        double r448910 = pow(r448906, r448909);
        double r448911 = pow(r448910, r448908);
        double r448912 = log(r448911);
        double r448913 = r448903 * r448912;
        double r448914 = r448913 - r448893;
        double r448915 = r448904 + r448914;
        double r448916 = 0.91893853320467;
        double r448917 = r448915 + r448916;
        double r448918 = y;
        double r448919 = 0.0007936500793651;
        double r448920 = r448918 + r448919;
        double r448921 = z;
        double r448922 = r448920 * r448921;
        double r448923 = 0.0027777777777778;
        double r448924 = r448922 - r448923;
        double r448925 = r448924 * r448921;
        double r448926 = 0.083333333333333;
        double r448927 = r448925 + r448926;
        double r448928 = r448927 / r448893;
        double r448929 = r448917 + r448928;
        double r448930 = log(r448893);
        double r448931 = r448903 * r448930;
        double r448932 = r448931 - r448893;
        double r448933 = r448932 + r448916;
        double r448934 = 2.0;
        double r448935 = pow(r448921, r448934);
        double r448936 = r448935 / r448893;
        double r448937 = r448936 * r448920;
        double r448938 = r448921 / r448893;
        double r448939 = r448923 * r448938;
        double r448940 = r448937 - r448939;
        double r448941 = r448933 + r448940;
        double r448942 = r448895 ? r448929 : r448941;
        return r448942;
}

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

Original5.7
Target1.2
Herbie3.8
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < 4.932149159571647e+38

    1. Initial program 0.3

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

    3. Applied add-cube-cbrt0.3

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

    4. Applied log-prod0.3

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

    5. Applied distribute-rgt-in0.3

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

    6. Applied associate--l+0.3

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

    7. Simplified0.3

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

    8. Taylor expanded around inf 0.3

      \[\leadsto \left(\left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right)} - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
    9. Using strategy rm

    10. Applied add-sqr-sqrt0.3

      \[\leadsto \left(\left(\log \left(\sqrt[3]{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    11. Applied cbrt-prod0.3

      \[\leadsto \left(\left(\log \left(\color{blue}{\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right)} \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{x}\right)}^{\frac{-1}{3}}\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
    12. Using strategy rm

    13. Applied add-cube-cbrt0.3

      \[\leadsto \left(\left(\log \left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left(\frac{1}{x}\right)}^{\color{blue}{\left(\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right) \cdot \sqrt[3]{\frac{-1}{3}}\right)}}\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    14. Applied pow-unpow0.3

      \[\leadsto \left(\left(\log \left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \color{blue}{\left({\left({\left(\frac{1}{x}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right)} - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]

    if 4.932149159571647e+38 < x

    1. Initial program 10.7

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

    2. Taylor expanded around inf 10.8

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

    3. Simplified7.0

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \color{blue}{\left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 4.93214915957164702 \cdot 10^{38}:\\ \;\;\;\;\left(\left(\log \left(\left(\sqrt[3]{\sqrt{x}} \cdot \sqrt[3]{\sqrt{x}}\right) \cdot \sqrt[3]{x}\right) \cdot \left(x - 0.5\right) + \left(\left(x - 0.5\right) \cdot \log \left({\left({\left(\frac{1}{x}\right)}^{\left(\sqrt[3]{\frac{-1}{3}} \cdot \sqrt[3]{\frac{-1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{-1}{3}}\right)}\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \left(\frac{{z}^{2}}{x} \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778 \cdot \frac{z}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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)))