Average Error: 18.9 → 1.5
Time: 5.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r467484 = 1.0;
        double r467485 = x;
        double r467486 = y;
        double r467487 = r467485 - r467486;
        double r467488 = r467484 - r467486;
        double r467489 = r467487 / r467488;
        double r467490 = r467484 - r467489;
        double r467491 = log(r467490);
        double r467492 = r467484 - r467491;
        return r467492;
}
↓
double f(double x, double y) {
        double r467493 = x;
        double r467494 = y;
        double r467495 = r467493 - r467494;
        double r467496 = 1.0;
        double r467497 = r467496 - r467494;
        double r467498 = r467495 / r467497;
        double r467499 = 3.2716599026957865e-16;
        bool r467500 = r467498 <= r467499;
        double r467501 = r467496 - r467498;
        double r467502 = cbrt(r467501);
        double r467503 = r467502 * r467502;
        double r467504 = r467503 * r467502;
        double r467505 = sqrt(r467504);
        double r467506 = log(r467505);
        double r467507 = sqrt(r467501);
        double r467508 = log(r467507);
        double r467509 = r467506 + r467508;
        double r467510 = r467496 - r467509;
        double r467511 = 2.0;
        double r467512 = pow(r467494, r467511);
        double r467513 = r467493 / r467512;
        double r467514 = 1.0;
        double r467515 = r467514 / r467494;
        double r467516 = r467513 - r467515;
        double r467517 = r467496 * r467516;
        double r467518 = r467493 / r467494;
        double r467519 = r467517 + r467518;
        double r467520 = log(r467519);
        double r467521 = r467496 - r467520;
        double r467522 = r467500 ? r467510 : r467521;
        return r467522;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.9
Target0.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 3.2716599026957865e-16

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied log-prod0.0

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.1

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

    if 3.2716599026957865e-16 < (/ (- x y) (- 1.0 y))

    1. Initial program 57.6

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))))))