Average Error: 18.9 → 0.1
Time: 43.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999625628327448723098314076196402311:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \left(\frac{1}{y} + 1\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999625628327448723098314076196402311:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r19284156 = 1.0;
        double r19284157 = x;
        double r19284158 = y;
        double r19284159 = r19284157 - r19284158;
        double r19284160 = r19284156 - r19284158;
        double r19284161 = r19284159 / r19284160;
        double r19284162 = r19284156 - r19284161;
        double r19284163 = log(r19284162);
        double r19284164 = r19284156 - r19284163;
        return r19284164;
}
↓
double f(double x, double y) {
        double r19284165 = x;
        double r19284166 = y;
        double r19284167 = r19284165 - r19284166;
        double r19284168 = 1.0;
        double r19284169 = r19284168 - r19284166;
        double r19284170 = r19284167 / r19284169;
        double r19284171 = 0.9999999625628327;
        bool r19284172 = r19284170 <= r19284171;
        double r19284173 = exp(r19284168);
        double r19284174 = r19284168 - r19284170;
        double r19284175 = r19284173 / r19284174;
        double r19284176 = log(r19284175);
        double r19284177 = r19284165 / r19284166;
        double r19284178 = r19284168 / r19284166;
        double r19284179 = 1.0;
        double r19284180 = r19284178 + r19284179;
        double r19284181 = r19284177 * r19284180;
        double r19284182 = r19284181 - r19284178;
        double r19284183 = log(r19284182);
        double r19284184 = r19284168 - r19284183;
        double r19284185 = r19284172 ? r19284176 : r19284184;
        return r19284185;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.9
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 (/ (- x y) (- 1.0 y)) < 0.9999999625628327

    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

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

    if 0.9999999625628327 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.8

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999625628327448723098314076196402311:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \left(\frac{1}{y} + 1\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(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))))))