Average Error: 18.9 → 1.5
Time: 5.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
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{1 - \frac{x - y}{1 - y}}\right) + \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)\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 r416677 = 1.0;
        double r416678 = x;
        double r416679 = y;
        double r416680 = r416678 - r416679;
        double r416681 = r416677 - r416679;
        double r416682 = r416680 / r416681;
        double r416683 = r416677 - r416682;
        double r416684 = log(r416683);
        double r416685 = r416677 - r416684;
        return r416685;
}
↓
double f(double x, double y) {
        double r416686 = x;
        double r416687 = y;
        double r416688 = r416686 - r416687;
        double r416689 = 1.0;
        double r416690 = r416689 - r416687;
        double r416691 = r416688 / r416690;
        double r416692 = 3.2716599026957865e-16;
        bool r416693 = r416691 <= r416692;
        double r416694 = r416689 - r416691;
        double r416695 = sqrt(r416694);
        double r416696 = log(r416695);
        double r416697 = cbrt(r416694);
        double r416698 = r416697 * r416697;
        double r416699 = r416698 * r416697;
        double r416700 = sqrt(r416699);
        double r416701 = log(r416700);
        double r416702 = r416696 + r416701;
        double r416703 = r416689 - r416702;
        double r416704 = 2.0;
        double r416705 = pow(r416687, r416704);
        double r416706 = r416686 / r416705;
        double r416707 = 1.0;
        double r416708 = r416707 / r416687;
        double r416709 = r416706 - r416708;
        double r416710 = r416689 * r416709;
        double r416711 = r416686 / r416687;
        double r416712 = r416710 + r416711;
        double r416713 = log(r416712);
        double r416714 = r416689 - r416713;
        double r416715 = r416693 ? r416703 : r416714;
        return r416715;
}

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

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

    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

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

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))))))