Average Error: 18.0 → 0.1
Time: 6.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -142834896.66603637 \lor \neg \left(y \le 28334228.2028031461\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -142834896.66603637 \lor \neg \left(y \le 28334228.2028031461\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{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 r368292 = 1.0;
        double r368293 = x;
        double r368294 = y;
        double r368295 = r368293 - r368294;
        double r368296 = r368292 - r368294;
        double r368297 = r368295 / r368296;
        double r368298 = r368292 - r368297;
        double r368299 = log(r368298);
        double r368300 = r368292 - r368299;
        return r368300;
}
↓
double f(double x, double y) {
        double r368301 = y;
        double r368302 = -142834896.66603637;
        bool r368303 = r368301 <= r368302;
        double r368304 = 28334228.202803146;
        bool r368305 = r368301 <= r368304;
        double r368306 = !r368305;
        bool r368307 = r368303 || r368306;
        double r368308 = 1.0;
        double r368309 = x;
        double r368310 = 2.0;
        double r368311 = pow(r368301, r368310);
        double r368312 = r368309 / r368311;
        double r368313 = 1.0;
        double r368314 = r368313 / r368301;
        double r368315 = r368312 - r368314;
        double r368316 = r368308 * r368315;
        double r368317 = r368309 / r368301;
        double r368318 = r368316 + r368317;
        double r368319 = log(r368318);
        double r368320 = r368308 - r368319;
        double r368321 = r368308 - r368301;
        double r368322 = cbrt(r368321);
        double r368323 = r368322 * r368322;
        double r368324 = r368313 / r368323;
        double r368325 = r368309 - r368301;
        double r368326 = r368325 / r368322;
        double r368327 = r368324 * r368326;
        double r368328 = r368308 - r368327;
        double r368329 = log(r368328);
        double r368330 = r368308 - r368329;
        double r368331 = r368307 ? r368320 : r368330;
        return r368331;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.0
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 < -142834896.66603637 or 28334228.202803146 < y

    1. Initial program 46.3

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

    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

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

    if -142834896.66603637 < y < 28334228.202803146

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -142834896.66603637 \lor \neg \left(y \le 28334228.2028031461\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 
(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))))))