Average Error: 18.1 → 0.1
Time: 54.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -220425257.605022430419921875:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\ \;\;\;\;1 - \log \left(\frac{y}{1 - y} + \left(1 - \frac{x}{1 - y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -220425257.605022430419921875:\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\
\;\;\;\;1 - \log \left(\frac{y}{1 - y} + \left(1 - \frac{x}{1 - y}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r19139109 = 1.0;
        double r19139110 = x;
        double r19139111 = y;
        double r19139112 = r19139110 - r19139111;
        double r19139113 = r19139109 - r19139111;
        double r19139114 = r19139112 / r19139113;
        double r19139115 = r19139109 - r19139114;
        double r19139116 = log(r19139115);
        double r19139117 = r19139109 - r19139116;
        return r19139117;
}
↓
double f(double x, double y) {
        double r19139118 = y;
        double r19139119 = -220425257.60502243;
        bool r19139120 = r19139118 <= r19139119;
        double r19139121 = 1.0;
        double r19139122 = x;
        double r19139123 = r19139122 / r19139118;
        double r19139124 = r19139121 / r19139118;
        double r19139125 = r19139123 * r19139124;
        double r19139126 = r19139125 - r19139124;
        double r19139127 = r19139123 + r19139126;
        double r19139128 = log(r19139127);
        double r19139129 = r19139121 - r19139128;
        double r19139130 = 20833302.402704142;
        bool r19139131 = r19139118 <= r19139130;
        double r19139132 = r19139121 - r19139118;
        double r19139133 = r19139118 / r19139132;
        double r19139134 = r19139122 / r19139132;
        double r19139135 = r19139121 - r19139134;
        double r19139136 = r19139133 + r19139135;
        double r19139137 = log(r19139136);
        double r19139138 = r19139121 - r19139137;
        double r19139139 = r19139131 ? r19139138 : r19139129;
        double r19139140 = r19139120 ? r19139129 : r19139139;
        return r19139140;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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 < -220425257.60502243 or 20833302.402704142 < y

    1. Initial program 46.7

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -220425257.60502243 < y < 20833302.402704142

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate--r-0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -220425257.605022430419921875:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\ \;\;\;\;1 - \log \left(\frac{y}{1 - y} + \left(1 - \frac{x}{1 - y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(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))))))