Average Error: 18.8 → 0.1
Time: 8.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\right):\\
\;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

\end{array}
double f(double x, double y) {
        double r595273 = 1.0;
        double r595274 = x;
        double r595275 = y;
        double r595276 = r595274 - r595275;
        double r595277 = r595273 - r595275;
        double r595278 = r595276 / r595277;
        double r595279 = r595273 - r595278;
        double r595280 = log(r595279);
        double r595281 = r595273 - r595280;
        return r595281;
}
↓
double f(double x, double y) {
        double r595282 = y;
        double r595283 = -325586491895.52155;
        bool r595284 = r595282 <= r595283;
        double r595285 = 36683682.57368248;
        bool r595286 = r595282 <= r595285;
        double r595287 = !r595286;
        bool r595288 = r595284 || r595287;
        double r595289 = 1.0;
        double r595290 = x;
        double r595291 = r595290 / r595282;
        double r595292 = 2.0;
        double r595293 = pow(r595282, r595292);
        double r595294 = r595290 / r595293;
        double r595295 = r595289 * r595294;
        double r595296 = r595291 + r595295;
        double r595297 = 1.0;
        double r595298 = r595297 / r595282;
        double r595299 = r595289 * r595298;
        double r595300 = r595296 - r595299;
        double r595301 = log(r595300);
        double r595302 = r595289 - r595301;
        double r595303 = r595290 - r595282;
        double r595304 = r595289 - r595282;
        double r595305 = cbrt(r595304);
        double r595306 = r595305 * r595305;
        double r595307 = r595303 / r595306;
        double r595308 = r595307 / r595305;
        double r595309 = r595289 - r595308;
        double r595310 = log(r595309);
        double r595311 = r595289 - r595310;
        double r595312 = r595288 ? r595302 : r595311;
        return r595312;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.8
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -325586491895.52155 or 36683682.57368248 < y

    1. Initial program 47.2

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

    3. Applied add-cube-cbrt43.4

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied associate-/r*43.4

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]

    5. Taylor expanded around inf 0.0

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

    if -325586491895.52155 < y < 36683682.57368248

    1. Initial program 0.2

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

    3. Applied add-cube-cbrt0.2

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied associate-/r*0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -325586491895.521545 \lor \neg \left(y \le 36683682.5736824796\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020034 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.61947241) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))