Average Error: 18.3 → 0.1
Time: 54.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -60638491.817888684570789337158203125:\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 4239884631250423709580859711422464:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r24098264 = 1.0;
        double r24098265 = x;
        double r24098266 = y;
        double r24098267 = r24098265 - r24098266;
        double r24098268 = r24098264 - r24098266;
        double r24098269 = r24098267 / r24098268;
        double r24098270 = r24098264 - r24098269;
        double r24098271 = log(r24098270);
        double r24098272 = r24098264 - r24098271;
        return r24098272;
}
↓
double f(double x, double y) {
        double r24098273 = y;
        double r24098274 = -60638491.817888685;
        bool r24098275 = r24098273 <= r24098274;
        double r24098276 = 1.0;
        double r24098277 = x;
        double r24098278 = r24098277 / r24098273;
        double r24098279 = r24098276 / r24098273;
        double r24098280 = r24098278 * r24098279;
        double r24098281 = r24098280 - r24098279;
        double r24098282 = r24098278 + r24098281;
        double r24098283 = log(r24098282);
        double r24098284 = r24098276 - r24098283;
        double r24098285 = 4.2398846312504237e+33;
        bool r24098286 = r24098273 <= r24098285;
        double r24098287 = r24098277 - r24098273;
        double r24098288 = r24098276 - r24098273;
        double r24098289 = cbrt(r24098288);
        double r24098290 = r24098289 * r24098289;
        double r24098291 = r24098287 / r24098290;
        double r24098292 = r24098291 / r24098289;
        double r24098293 = r24098276 - r24098292;
        double r24098294 = log(r24098293);
        double r24098295 = r24098276 - r24098294;
        double r24098296 = r24098286 ? r24098295 : r24098284;
        double r24098297 = r24098275 ? r24098284 : r24098296;
        return r24098297;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.3
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -60638491.817888685 or 4.2398846312504237e+33 < y

    1. Initial program 47.4

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

    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

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

    if -60638491.817888685 < y < 4.2398846312504237e+33

    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 associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -60638491.817888684570789337158203125:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 4239884631250423709580859711422464:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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))))))