Average Error: 18.5 → 0.1
Time: 11.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r494737 = 1.0;
        double r494738 = x;
        double r494739 = y;
        double r494740 = r494738 - r494739;
        double r494741 = r494737 - r494739;
        double r494742 = r494740 / r494741;
        double r494743 = r494737 - r494742;
        double r494744 = log(r494743);
        double r494745 = r494737 - r494744;
        return r494745;
}
↓
double f(double x, double y) {
        double r494746 = y;
        double r494747 = -119577009.99898484;
        bool r494748 = r494746 <= r494747;
        double r494749 = 25589815.688386947;
        bool r494750 = r494746 <= r494749;
        double r494751 = !r494750;
        bool r494752 = r494748 || r494751;
        double r494753 = 1.0;
        double r494754 = exp(r494753);
        double r494755 = 1.0;
        double r494756 = r494753 / r494746;
        double r494757 = r494755 + r494756;
        double r494758 = x;
        double r494759 = r494758 / r494746;
        double r494760 = r494757 * r494759;
        double r494761 = r494760 - r494756;
        double r494762 = r494754 / r494761;
        double r494763 = log(r494762);
        double r494764 = r494758 - r494746;
        double r494765 = r494753 - r494746;
        double r494766 = r494755 / r494765;
        double r494767 = r494764 * r494766;
        double r494768 = r494753 - r494767;
        double r494769 = log(r494768);
        double r494770 = r494753 - r494769;
        double r494771 = r494752 ? r494763 : r494770;
        return r494771;
}

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
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -119577009.99898484 or 25589815.688386947 < y

    1. Initial program 46.9

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

    3. Applied add-log-exp46.9

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

    4. Applied diff-log46.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -119577009.99898484 < y < 25589815.688386947

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))))