Average Error: 17.3 → 0.3
Time: 25.3s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 98095783.98664801:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\
\;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 98095783.98664801:\\
\;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r17214986 = 1.0;
        double r17214987 = x;
        double r17214988 = y;
        double r17214989 = r17214987 - r17214988;
        double r17214990 = r17214986 - r17214988;
        double r17214991 = r17214989 / r17214990;
        double r17214992 = r17214986 - r17214991;
        double r17214993 = log(r17214992);
        double r17214994 = r17214986 - r17214993;
        return r17214994;
}
↓
double f(double x, double y) {
        double r17214995 = y;
        double r17214996 = -3.475504821605494e+19;
        bool r17214997 = r17214995 <= r17214996;
        double r17214998 = 1.0;
        double r17214999 = r17214998 / r17214995;
        double r17215000 = x;
        double r17215001 = r17215000 / r17214995;
        double r17215002 = r17215001 - r17214999;
        double r17215003 = fma(r17214999, r17215001, r17215002);
        double r17215004 = log(r17215003);
        double r17215005 = r17214998 - r17215004;
        double r17215006 = 98095783.98664801;
        bool r17215007 = r17214995 <= r17215006;
        double r17215008 = r17215000 - r17214995;
        double r17215009 = r17214998 - r17214995;
        double r17215010 = r17215008 / r17215009;
        double r17215011 = r17214998 - r17215010;
        double r17215012 = sqrt(r17215011);
        double r17215013 = log(r17215012);
        double r17215014 = r17215013 + r17215013;
        double r17215015 = r17214998 - r17215014;
        double r17215016 = r17215007 ? r17215015 : r17215005;
        double r17215017 = r17214997 ? r17215005 : r17215016;
        return r17215017;
}

Error

Bits error versus x

Bits error versus y

Target

Original17.3
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -3.475504821605494e+19 or 98095783.98664801 < y

    1. Initial program 46.1

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]

    2. Taylor expanded around inf 0.0

      \[\leadsto 1.0 - \log \color{blue}{\left(\left(\frac{x}{y} + 1.0 \cdot \frac{x}{{y}^{2}}\right) - 1.0 \cdot \frac{1}{y}\right)}\]

    3. Simplified0.0

      \[\leadsto 1.0 - \log \color{blue}{\left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)}\]

    if -3.475504821605494e+19 < y < 98095783.98664801

    1. Initial program 0.5

      \[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.5

      \[\leadsto 1.0 - \log \color{blue}{\left(\sqrt{1.0 - \frac{x - y}{1.0 - y}} \cdot \sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)}\]

    4. Applied log-prod0.5

      \[\leadsto 1.0 - \color{blue}{\left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -3.475504821605494 \cdot 10^{+19}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 98095783.98664801:\\ \;\;\;\;1.0 - \left(\log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right) + \log \left(\sqrt{1.0 - \frac{x - y}{1.0 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\mathsf{fma}\left(\frac{1.0}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))