Average Error: 26.6 → 1.1
Time: 8.5s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -136760216645358404000 \lor \neg \left(x \le 5449213609141281790\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}
↓
\begin{array}{l}
\mathbf{if}\;x \le -136760216645358404000 \lor \neg \left(x \le 5449213609141281790\right):\\
\;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\

\end{array}
double f(double x, double y, double z) {
        double r367855 = x;
        double r367856 = 2.0;
        double r367857 = r367855 - r367856;
        double r367858 = 4.16438922228;
        double r367859 = r367855 * r367858;
        double r367860 = 78.6994924154;
        double r367861 = r367859 + r367860;
        double r367862 = r367861 * r367855;
        double r367863 = 137.519416416;
        double r367864 = r367862 + r367863;
        double r367865 = r367864 * r367855;
        double r367866 = y;
        double r367867 = r367865 + r367866;
        double r367868 = r367867 * r367855;
        double r367869 = z;
        double r367870 = r367868 + r367869;
        double r367871 = r367857 * r367870;
        double r367872 = 43.3400022514;
        double r367873 = r367855 + r367872;
        double r367874 = r367873 * r367855;
        double r367875 = 263.505074721;
        double r367876 = r367874 + r367875;
        double r367877 = r367876 * r367855;
        double r367878 = 313.399215894;
        double r367879 = r367877 + r367878;
        double r367880 = r367879 * r367855;
        double r367881 = 47.066876606;
        double r367882 = r367880 + r367881;
        double r367883 = r367871 / r367882;
        return r367883;
}
↓
double f(double x, double y, double z) {
        double r367884 = x;
        double r367885 = -1.367602166453584e+20;
        bool r367886 = r367884 <= r367885;
        double r367887 = 5.449213609141282e+18;
        bool r367888 = r367884 <= r367887;
        double r367889 = !r367888;
        bool r367890 = r367886 || r367889;
        double r367891 = 4.16438922228;
        double r367892 = y;
        double r367893 = 2.0;
        double r367894 = pow(r367884, r367893);
        double r367895 = r367892 / r367894;
        double r367896 = 110.1139242984811;
        double r367897 = r367895 - r367896;
        double r367898 = fma(r367884, r367891, r367897);
        double r367899 = 2.0;
        double r367900 = r367884 - r367899;
        double r367901 = r367884 * r367891;
        double r367902 = 78.6994924154;
        double r367903 = r367901 + r367902;
        double r367904 = r367903 * r367884;
        double r367905 = 137.519416416;
        double r367906 = r367904 + r367905;
        double r367907 = r367906 * r367884;
        double r367908 = r367907 + r367892;
        double r367909 = r367908 * r367884;
        double r367910 = z;
        double r367911 = r367909 + r367910;
        double r367912 = r367900 * r367911;
        double r367913 = 43.3400022514;
        double r367914 = r367884 + r367913;
        double r367915 = r367914 * r367884;
        double r367916 = 263.505074721;
        double r367917 = r367915 + r367916;
        double r367918 = r367917 * r367884;
        double r367919 = 313.399215894;
        double r367920 = r367918 + r367919;
        double r367921 = r367920 * r367884;
        double r367922 = 47.066876606;
        double r367923 = r367921 + r367922;
        double r367924 = r367912 / r367923;
        double r367925 = r367890 ? r367898 : r367924;
        return r367925;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.6
Target0.5
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.367602166453584e+20 or 5.449213609141282e+18 < x

    1. Initial program 56.2

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]

    2. Simplified52.4

      \[\leadsto \color{blue}{\frac{x - 2}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x + 43.3400022514000014, x, 263.50507472100003\right), x, 313.399215894\right), x, 47.066876606000001\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, 4.16438922227999964, 78.6994924154000017\right), x, 137.51941641600001\right), x, y\right), x, z\right)}}}\]

    3. Taylor expanded around inf 2.0

      \[\leadsto \color{blue}{\left(\frac{y}{{x}^{2}} + 4.16438922227999964 \cdot x\right) - 110.11392429848109}\]

    4. Simplified2.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)}\]

    if -1.367602166453584e+20 < x < 5.449213609141282e+18

    1. Initial program 0.4

      \[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -136760216645358404000 \lor \neg \left(x \le 5449213609141281790\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< x -3.326128725870005e+62) (- (+ (/ y (* x x)) (* 4.16438922228 x)) 110.1139242984811) (if (< x 9.429991714554673e+55) (* (/ (- x 2) 1) (/ (+ (* (+ (* (+ (* (+ (* 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) (+ (* (+ (* (+ (* (+ (* x 4.16438922228) 78.6994924154) x) 137.519416416) x) y) x) z)) (+ (* (+ (* (+ (* (+ x 43.3400022514) x) 263.505074721) x) 313.399215894) x) 47.066876606)))