Average Error: 5.9 → 6.2
Time: 10.3s
Precision: 64
\[\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 8.37321655534011972 \cdot 10^{233}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}\\ \end{array}\]
\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}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 8.37321655534011972 \cdot 10^{233}:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\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}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r560843 = x;
        double r560844 = 0.5;
        double r560845 = r560843 - r560844;
        double r560846 = log(r560843);
        double r560847 = r560845 * r560846;
        double r560848 = r560847 - r560843;
        double r560849 = 0.91893853320467;
        double r560850 = r560848 + r560849;
        double r560851 = y;
        double r560852 = 0.0007936500793651;
        double r560853 = r560851 + r560852;
        double r560854 = z;
        double r560855 = r560853 * r560854;
        double r560856 = 0.0027777777777778;
        double r560857 = r560855 - r560856;
        double r560858 = r560857 * r560854;
        double r560859 = 0.083333333333333;
        double r560860 = r560858 + r560859;
        double r560861 = r560860 / r560843;
        double r560862 = r560850 + r560861;
        return r560862;
}
↓
double f(double x, double y, double z) {
        double r560863 = y;
        double r560864 = 0.0007936500793651;
        double r560865 = r560863 + r560864;
        double r560866 = z;
        double r560867 = r560865 * r560866;
        double r560868 = 0.0027777777777778;
        double r560869 = r560867 - r560868;
        double r560870 = r560869 * r560866;
        double r560871 = 8.37321655534012e+233;
        bool r560872 = r560870 <= r560871;
        double r560873 = x;
        double r560874 = 0.5;
        double r560875 = r560873 - r560874;
        double r560876 = cbrt(r560873);
        double r560877 = r560876 * r560876;
        double r560878 = log(r560877);
        double r560879 = r560875 * r560878;
        double r560880 = log(r560876);
        double r560881 = r560880 * r560875;
        double r560882 = r560881 - r560873;
        double r560883 = r560879 + r560882;
        double r560884 = 0.91893853320467;
        double r560885 = r560883 + r560884;
        double r560886 = 0.083333333333333;
        double r560887 = r560870 + r560886;
        double r560888 = r560887 / r560873;
        double r560889 = r560885 + r560888;
        double r560890 = log(r560873);
        double r560891 = r560875 * r560890;
        double r560892 = r560891 - r560873;
        double r560893 = r560892 + r560884;
        double r560894 = 1.0;
        double r560895 = 0.4000000000000064;
        double r560896 = r560873 * r560866;
        double r560897 = r560895 * r560896;
        double r560898 = 12.000000000000048;
        double r560899 = r560898 * r560873;
        double r560900 = r560897 + r560899;
        double r560901 = 0.10095227809524161;
        double r560902 = 2.0;
        double r560903 = pow(r560866, r560902);
        double r560904 = r560873 * r560903;
        double r560905 = r560901 * r560904;
        double r560906 = r560900 - r560905;
        double r560907 = r560894 / r560906;
        double r560908 = r560893 + r560907;
        double r560909 = r560872 ? r560889 : r560908;
        return r560909;
}

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

Original5.9
Target1.2
Herbie6.2
\[\left(\left(\left(x - 0.5\right) \cdot \log x + \left(0.91893853320467001 - x\right)\right) + \frac{0.0833333333333329956}{x}\right) + \frac{z}{x} \cdot \left(z \cdot \left(y + 7.93650079365100015 \cdot 10^{-4}\right) - 0.0027777777777778\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z) < 8.37321655534012e+233

    1. Initial program 1.7

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

    3. Applied add-cube-cbrt1.8

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

    4. Applied log-prod1.8

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

    5. Applied distribute-lft-in1.8

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

    6. Applied associate--l+1.8

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

    7. Simplified1.8

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

    if 8.37321655534012e+233 < (* (- (* (+ y 0.0007936500793651) z) 0.0027777777777778) z)

    1. Initial program 40.2

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

    3. Applied clear-num40.2

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

    4. Taylor expanded around 0 43.6

      \[\leadsto \left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\color{blue}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z \le 8.37321655534011972 \cdot 10^{233}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) + \left(\log \left(\sqrt[3]{x}\right) \cdot \left(x - 0.5\right) - x\right)\right) + 0.91893853320467001\right) + \frac{\left(\left(y + 7.93650079365100015 \cdot 10^{-4}\right) \cdot z - 0.0027777777777778\right) \cdot z + 0.0833333333333329956}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(x - 0.5\right) \cdot \log x - x\right) + 0.91893853320467001\right) + \frac{1}{\left(0.400000000000006406 \cdot \left(x \cdot z\right) + 12.000000000000048 \cdot x\right) - 0.100952278095241613 \cdot \left(x \cdot {z}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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)))