Average Error: 17.9 → 0.2
Time: 25.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\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 r230234 = 1.0;
        double r230235 = x;
        double r230236 = y;
        double r230237 = r230235 - r230236;
        double r230238 = r230234 - r230236;
        double r230239 = r230237 / r230238;
        double r230240 = r230234 - r230239;
        double r230241 = log(r230240);
        double r230242 = r230234 - r230241;
        return r230242;
}
↓
double f(double x, double y) {
        double r230243 = y;
        double r230244 = -1123308855491800.2;
        bool r230245 = r230243 <= r230244;
        double r230246 = 106063957.48696265;
        bool r230247 = r230243 <= r230246;
        double r230248 = !r230247;
        bool r230249 = r230245 || r230248;
        double r230250 = 1.0;
        double r230251 = x;
        double r230252 = r230251 / r230243;
        double r230253 = 2.0;
        double r230254 = pow(r230243, r230253);
        double r230255 = r230251 / r230254;
        double r230256 = r230250 * r230255;
        double r230257 = r230252 + r230256;
        double r230258 = r230250 / r230243;
        double r230259 = r230257 - r230258;
        double r230260 = log(r230259);
        double r230261 = r230250 - r230260;
        double r230262 = r230251 - r230243;
        double r230263 = 1.0;
        double r230264 = r230250 - r230243;
        double r230265 = r230263 / r230264;
        double r230266 = r230262 * r230265;
        double r230267 = r230250 - r230266;
        double r230268 = log(r230267);
        double r230269 = r230250 - r230268;
        double r230270 = r230249 ? r230261 : r230269;
        return r230270;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.9
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -1123308855491800.2 or 106063957.48696265 < y

    1. Initial program 46.6

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -1123308855491800.2 < y < 106063957.48696265

    1. Initial program 0.3

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

    3. Applied div-inv0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1123308855491800.2 \lor \neg \left(y \le 106063957.48696265\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 
(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))))))