Average Error: 18.5 → 0.1
Time: 14.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}}\right) \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -34474611802046.8828125:\\
\;\;\;\;1 - \log \left(\left(\sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}}\right) \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} - \frac{1}{y}\right)\\

\mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r22783681 = 1.0;
        double r22783682 = x;
        double r22783683 = y;
        double r22783684 = r22783682 - r22783683;
        double r22783685 = r22783681 - r22783683;
        double r22783686 = r22783684 / r22783685;
        double r22783687 = r22783681 - r22783686;
        double r22783688 = log(r22783687);
        double r22783689 = r22783681 - r22783688;
        return r22783689;
}
↓
double f(double x, double y) {
        double r22783690 = y;
        double r22783691 = -34474611802046.883;
        bool r22783692 = r22783690 <= r22783691;
        double r22783693 = 1.0;
        double r22783694 = x;
        double r22783695 = r22783694 / r22783690;
        double r22783696 = r22783693 * r22783694;
        double r22783697 = r22783690 * r22783690;
        double r22783698 = r22783696 / r22783697;
        double r22783699 = r22783695 + r22783698;
        double r22783700 = cbrt(r22783699);
        double r22783701 = r22783700 * r22783700;
        double r22783702 = r22783701 * r22783700;
        double r22783703 = r22783693 / r22783690;
        double r22783704 = r22783702 - r22783703;
        double r22783705 = log(r22783704);
        double r22783706 = r22783693 - r22783705;
        double r22783707 = 43744445.70071104;
        bool r22783708 = r22783690 <= r22783707;
        double r22783709 = r22783694 - r22783690;
        double r22783710 = r22783693 - r22783690;
        double r22783711 = r22783709 / r22783710;
        double r22783712 = r22783693 - r22783711;
        double r22783713 = sqrt(r22783712);
        double r22783714 = log(r22783713);
        double r22783715 = r22783714 + r22783714;
        double r22783716 = r22783693 - r22783715;
        double r22783717 = r22783699 - r22783703;
        double r22783718 = sqrt(r22783717);
        double r22783719 = log(r22783718);
        double r22783720 = r22783719 + r22783719;
        double r22783721 = r22783693 - r22783720;
        double r22783722 = r22783708 ? r22783716 : r22783721;
        double r22783723 = r22783692 ? r22783706 : r22783722;
        return r22783723;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
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 3 regimes

  2. if y < -34474611802046.883

    1. Initial program 53.9

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied add-cube-cbrt0.0

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

    if -34474611802046.883 < y < 43744445.70071104

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

    if 43744445.70071104 < y

    1. Initial program 29.8

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied add-sqr-sqrt0.0

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

    6. Applied log-prod0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\left(\sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}}\right) \cdot \sqrt[3]{\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))))