Average Error: 18.1 → 0.1
Time: 21.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 758763659.74992859363555908203125\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 758763659.74992859363555908203125\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

\end{array}
double f(double x, double y) {
        double r238853 = 1.0;
        double r238854 = x;
        double r238855 = y;
        double r238856 = r238854 - r238855;
        double r238857 = r238853 - r238855;
        double r238858 = r238856 / r238857;
        double r238859 = r238853 - r238858;
        double r238860 = log(r238859);
        double r238861 = r238853 - r238860;
        return r238861;
}
↓
double f(double x, double y) {
        double r238862 = y;
        double r238863 = -212491914.19131115;
        bool r238864 = r238862 <= r238863;
        double r238865 = 758763659.7499286;
        bool r238866 = r238862 <= r238865;
        double r238867 = !r238866;
        bool r238868 = r238864 || r238867;
        double r238869 = 1.0;
        double r238870 = exp(r238869);
        double r238871 = x;
        double r238872 = 2.0;
        double r238873 = pow(r238862, r238872);
        double r238874 = r238871 / r238873;
        double r238875 = r238871 / r238862;
        double r238876 = fma(r238869, r238874, r238875);
        double r238877 = r238869 / r238862;
        double r238878 = r238876 - r238877;
        double r238879 = r238870 / r238878;
        double r238880 = log(r238879);
        double r238881 = r238871 - r238862;
        double r238882 = r238869 - r238862;
        double r238883 = r238881 / r238882;
        double r238884 = r238869 - r238883;
        double r238885 = r238870 / r238884;
        double r238886 = log(r238885);
        double r238887 = r238868 ? r238880 : r238886;
        return r238887;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.1
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -212491914.19131115 or 758763659.7499286 < y

    1. Initial program 46.8

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp46.8

      \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    4. Applied diff-log46.8

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]

    5. Taylor expanded around inf 0.2

      \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}}}\right)\]

    6. Simplified0.2

      \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}}\right)\]

    if -212491914.19131115 < y < 758763659.7499286

    1. Initial program 0.1

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp0.1

      \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    4. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 758763659.74992859363555908203125\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))