Average Error: 18.5 → 0.1
Time: 9.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r351796 = 1.0;
        double r351797 = x;
        double r351798 = y;
        double r351799 = r351797 - r351798;
        double r351800 = r351796 - r351798;
        double r351801 = r351799 / r351800;
        double r351802 = r351796 - r351801;
        double r351803 = log(r351802);
        double r351804 = r351796 - r351803;
        return r351804;
}
↓
double f(double x, double y) {
        double r351805 = y;
        double r351806 = -119577009.99898484;
        bool r351807 = r351805 <= r351806;
        double r351808 = 25589815.688386947;
        bool r351809 = r351805 <= r351808;
        double r351810 = !r351809;
        bool r351811 = r351807 || r351810;
        double r351812 = 1.0;
        double r351813 = exp(r351812);
        double r351814 = x;
        double r351815 = 2.0;
        double r351816 = pow(r351805, r351815);
        double r351817 = r351814 / r351816;
        double r351818 = r351814 / r351805;
        double r351819 = fma(r351817, r351812, r351818);
        double r351820 = r351812 / r351805;
        double r351821 = r351819 - r351820;
        double r351822 = r351813 / r351821;
        double r351823 = log(r351822);
        double r351824 = r351814 - r351805;
        double r351825 = 1.0;
        double r351826 = r351812 - r351805;
        double r351827 = r351825 / r351826;
        double r351828 = r351824 * r351827;
        double r351829 = r351812 - r351828;
        double r351830 = log(r351829);
        double r351831 = r351812 - r351830;
        double r351832 = r351811 ? r351823 : r351831;
        return r351832;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.5
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -119577009.99898484 or 25589815.688386947 < y

    1. Initial program 46.9

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

    3. Applied add-log-exp46.9

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

    4. Applied diff-log46.9

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

    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

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

    if -119577009.99898484 < y < 25589815.688386947

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))))