Average Error: 17.6 → 0.1
Time: 29.2s
Precision: 64
\[1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -116643585.83872083:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]
1.0 - \log \left(1.0 - \frac{x - y}{1.0 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -116643585.83872083:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\mathbf{elif}\;y \le 51693858.328214146:\\
\;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r35754139 = 1.0;
        double r35754140 = x;
        double r35754141 = y;
        double r35754142 = r35754140 - r35754141;
        double r35754143 = r35754139 - r35754141;
        double r35754144 = r35754142 / r35754143;
        double r35754145 = r35754139 - r35754144;
        double r35754146 = log(r35754145);
        double r35754147 = r35754139 - r35754146;
        return r35754147;
}
↓
double f(double x, double y) {
        double r35754148 = y;
        double r35754149 = -116643585.83872083;
        bool r35754150 = r35754148 <= r35754149;
        double r35754151 = 1.0;
        double r35754152 = x;
        double r35754153 = r35754152 / r35754148;
        double r35754154 = r35754151 / r35754148;
        double r35754155 = r35754153 * r35754154;
        double r35754156 = r35754153 - r35754154;
        double r35754157 = r35754155 + r35754156;
        double r35754158 = log(r35754157);
        double r35754159 = r35754151 - r35754158;
        double r35754160 = 51693858.328214146;
        bool r35754161 = r35754148 <= r35754160;
        double r35754162 = r35754152 - r35754148;
        double r35754163 = 1.0;
        double r35754164 = r35754151 - r35754148;
        double r35754165 = r35754163 / r35754164;
        double r35754166 = r35754162 * r35754165;
        double r35754167 = r35754151 - r35754166;
        double r35754168 = log(r35754167);
        double r35754169 = r35754151 - r35754168;
        double r35754170 = r35754161 ? r35754169 : r35754159;
        double r35754171 = r35754150 ? r35754159 : r35754170;
        return r35754171;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.6
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.61947241:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{+25}:\\ \;\;\;\;\log \left(\frac{e^{1.0}}{1.0 - \frac{x - y}{1.0 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1.0}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -116643585.83872083 or 51693858.328214146 < y

    1. Initial program 46.1

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -116643585.83872083 < y < 51693858.328214146

    1. Initial program 0.1

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

    3. Applied div-inv0.1

      \[\leadsto 1.0 - \log \left(1.0 - \color{blue}{\left(x - y\right) \cdot \frac{1}{1.0 - y}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -116643585.83872083:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \mathbf{elif}\;y \le 51693858.328214146:\\ \;\;\;\;1.0 - \log \left(1.0 - \left(x - y\right) \cdot \frac{1}{1.0 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1.0 - \log \left(\frac{x}{y} \cdot \frac{1.0}{y} + \left(\frac{x}{y} - \frac{1.0}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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))))))