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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\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{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\

\end{array}

double f(double x, double y) {
        double r412356 = 1.0;
        double r412357 = x;
        double r412358 = y;
        double r412359 = r412357 - r412358;
        double r412360 = r412356 - r412358;
        double r412361 = r412359 / r412360;
        double r412362 = r412356 - r412361;
        double r412363 = log(r412362);
        double r412364 = r412356 - r412363;
        return r412364;
}

↓

double f(double x, double y) {
        double r412365 = y;
        double r412366 = -1630045945.922767;
        bool r412367 = r412365 <= r412366;
        double r412368 = 76447033.33081697;
        bool r412369 = r412365 <= r412368;
        double r412370 = !r412369;
        bool r412371 = r412367 || r412370;
        double r412372 = 1.0;
        double r412373 = x;
        double r412374 = r412373 / r412365;
        double r412375 = 2.0;
        double r412376 = pow(r412365, r412375);
        double r412377 = r412373 / r412376;
        double r412378 = r412372 * r412377;
        double r412379 = r412374 + r412378;
        double r412380 = 1.0;
        double r412381 = r412380 / r412365;
        double r412382 = r412372 * r412381;
        double r412383 = r412379 - r412382;
        double r412384 = log(r412383);
        double r412385 = r412372 - r412384;
        double r412386 = r412372 - r412365;
        double r412387 = cbrt(r412386);
        double r412388 = r412387 * r412387;
        double r412389 = r412380 / r412388;
        double r412390 = r412373 - r412365;
        double r412391 = r412390 / r412387;
        double r412392 = r412389 * r412391;
        double r412393 = r412372 - r412392;
        double r412394 = log(r412393);
        double r412395 = r412372 - r412394;
        double r412396 = r412371 ? r412385 : r412395;
        return r412396;
}

Original	17.8
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -1630045945.922767 or 76447033.33081697 < y
1. Initial program 46.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied flip3--53.3
  \[\leadsto 1 - \log \color{blue}{\left(\frac{{1}^{3} - {\left(\frac{x - y}{1 - y}\right)}^{3}}{1 \cdot 1 + \left(\frac{x - y}{1 - y} \cdot \frac{x - y}{1 - y} + 1 \cdot \frac{x - y}{1 - y}\right)}\right)}\]
4. Simplified53.3
  \[\leadsto 1 - \log \left(\frac{{1}^{3} - {\left(\frac{x - y}{1 - y}\right)}^{3}}{\color{blue}{\mathsf{fma}\left(\frac{x - y}{1 - y}, 1 + \frac{x - y}{1 - y}, 1 \cdot 1\right)}}\right)\]
5. Taylor expanded around inf 0.1
  \[\leadsto 1 - \log \color{blue}{\left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)}\]
if -1630045945.922767 < y < 76447033.33081697
1. Initial program 0.1
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
2. Using strategy rm
3. Applied add-cube-cbrt0.1
  \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]
4. Applied *-un-lft-identity0.1
  \[\leadsto 1 - \log \left(1 - \frac{\color{blue}{1 \cdot \left(x - y\right)}}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}\right)\]
5. Applied times-frac0.1
  \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}}\right)\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1630045945.92276692 \lor \neg \left(y \le 76447033.3308169693\right):\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +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))))))

Error

Try it out

Target

Derivation

`if y < -1630045945.922767 or 76447033.33081697 < y`

`if -1630045945.922767 < y < 76447033.33081697`

Reproduce

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