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

↓

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

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 r322288 = 1.0;
        double r322289 = x;
        double r322290 = y;
        double r322291 = r322289 - r322290;
        double r322292 = r322288 - r322290;
        double r322293 = r322291 / r322292;
        double r322294 = r322288 - r322293;
        double r322295 = log(r322294);
        double r322296 = r322288 - r322295;
        return r322296;
}

↓

double f(double x, double y) {
        double r322297 = y;
        double r322298 = -1630045945.922767;
        bool r322299 = r322297 <= r322298;
        double r322300 = 76447033.33081697;
        bool r322301 = r322297 <= r322300;
        double r322302 = !r322301;
        bool r322303 = r322299 || r322302;
        double r322304 = 1.0;
        double r322305 = x;
        double r322306 = r322305 / r322297;
        double r322307 = 2.0;
        double r322308 = pow(r322297, r322307);
        double r322309 = r322305 / r322308;
        double r322310 = r322304 * r322309;
        double r322311 = r322306 + r322310;
        double r322312 = 1.0;
        double r322313 = r322312 / r322297;
        double r322314 = r322304 * r322313;
        double r322315 = r322311 - r322314;
        double r322316 = log(r322315);
        double r322317 = r322304 - r322316;
        double r322318 = r322304 - r322297;
        double r322319 = cbrt(r322318);
        double r322320 = r322319 * r322319;
        double r322321 = r322312 / r322320;
        double r322322 = r322305 - r322297;
        double r322323 = r322322 / r322319;
        double r322324 = r322321 * r322323;
        double r322325 = r322304 - r322324;
        double r322326 = log(r322325);
        double r322327 = r322304 - r322326;
        double r322328 = r322303 ? r322317 : r322327;
        return r322328;
}

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

In
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