Average Error: 26.9 → 0.9
Time: 8.4s
Precision: 64
\[\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.88601206440715126 \cdot 10^{43} \lor \neg \left(x \le 2.9045831949053032 \cdot 10^{50}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]
\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 -6.88601206440715126 \cdot 10^{43} \lor \neg \left(x \le 2.9045831949053032 \cdot 10^{50}\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 r457831 = x;
        double r457832 = 2.0;
        double r457833 = r457831 - r457832;
        double r457834 = 4.16438922228;
        double r457835 = r457831 * r457834;
        double r457836 = 78.6994924154;
        double r457837 = r457835 + r457836;
        double r457838 = r457837 * r457831;
        double r457839 = 137.519416416;
        double r457840 = r457838 + r457839;
        double r457841 = r457840 * r457831;
        double r457842 = y;
        double r457843 = r457841 + r457842;
        double r457844 = r457843 * r457831;
        double r457845 = z;
        double r457846 = r457844 + r457845;
        double r457847 = r457833 * r457846;
        double r457848 = 43.3400022514;
        double r457849 = r457831 + r457848;
        double r457850 = r457849 * r457831;
        double r457851 = 263.505074721;
        double r457852 = r457850 + r457851;
        double r457853 = r457852 * r457831;
        double r457854 = 313.399215894;
        double r457855 = r457853 + r457854;
        double r457856 = r457855 * r457831;
        double r457857 = 47.066876606;
        double r457858 = r457856 + r457857;
        double r457859 = r457847 / r457858;
        return r457859;
}
↓
double f(double x, double y, double z) {
        double r457860 = x;
        double r457861 = -6.886012064407151e+43;
        bool r457862 = r457860 <= r457861;
        double r457863 = 2.904583194905303e+50;
        bool r457864 = r457860 <= r457863;
        double r457865 = !r457864;
        bool r457866 = r457862 || r457865;
        double r457867 = 4.16438922228;
        double r457868 = y;
        double r457869 = 2.0;
        double r457870 = pow(r457860, r457869);
        double r457871 = r457868 / r457870;
        double r457872 = 110.1139242984811;
        double r457873 = r457871 - r457872;
        double r457874 = fma(r457860, r457867, r457873);
        double r457875 = 2.0;
        double r457876 = r457860 - r457875;
        double r457877 = r457860 * r457867;
        double r457878 = 78.6994924154;
        double r457879 = r457877 + r457878;
        double r457880 = r457879 * r457860;
        double r457881 = 137.519416416;
        double r457882 = r457880 + r457881;
        double r457883 = r457882 * r457860;
        double r457884 = r457883 + r457868;
        double r457885 = r457884 * r457860;
        double r457886 = z;
        double r457887 = r457885 + r457886;
        double r457888 = r457876 * r457887;
        double r457889 = 43.3400022514;
        double r457890 = r457860 + r457889;
        double r457891 = r457890 * r457860;
        double r457892 = 263.505074721;
        double r457893 = r457891 + r457892;
        double r457894 = r457893 * r457860;
        double r457895 = 313.399215894;
        double r457896 = r457894 + r457895;
        double r457897 = r457896 * r457860;
        double r457898 = 47.066876606;
        double r457899 = r457897 + r457898;
        double r457900 = r457888 / r457899;
        double r457901 = r457866 ? r457874 : r457900;
        return r457901;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original26.9
Target0.6
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -3.3261287258700048 \cdot 10^{62}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \mathbf{elif}\;x \lt 9.4299917145546727 \cdot 10^{55}:\\ \;\;\;\;\frac{x - 2}{1} \cdot \frac{\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z}{\left(\left(263.50507472100003 \cdot x + \left(43.3400022514000014 \cdot \left(x \cdot x\right) + x \cdot \left(x \cdot x\right)\right)\right) + 313.399215894\right) \cdot x + 47.066876606000001}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{y}{x \cdot x} + 4.16438922227999964 \cdot x\right) - 110.11392429848109\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6.886012064407151e+43 or 2.904583194905303e+50 < x

    1. Initial program 61.7

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

    2. Simplified58.0

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

    3. Taylor expanded around inf 0.7

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

    4. Simplified0.7

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

    if -6.886012064407151e+43 < x < 2.904583194905303e+50

    1. Initial program 1.0

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.88601206440715126 \cdot 10^{43} \lor \neg \left(x \le 2.9045831949053032 \cdot 10^{50}\right):\\ \;\;\;\;\mathsf{fma}\left(x, 4.16438922227999964, \frac{y}{{x}^{2}} - 110.11392429848109\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - 2\right) \cdot \left(\left(\left(\left(x \cdot 4.16438922227999964 + 78.6994924154000017\right) \cdot x + 137.51941641600001\right) \cdot x + y\right) \cdot x + z\right)}{\left(\left(\left(x + 43.3400022514000014\right) \cdot x + 263.50507472100003\right) \cdot x + 313.399215894\right) \cdot x + 47.066876606000001}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))