Average Error: 18.2 → 0.8
Time: 18.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 1.545842237750884217971380531209124598035 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 1.545842237750884217971380531209124598035 \cdot 10^{-7}:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r233115 = 1.0;
        double r233116 = x;
        double r233117 = y;
        double r233118 = r233116 - r233117;
        double r233119 = r233115 - r233117;
        double r233120 = r233118 / r233119;
        double r233121 = r233115 - r233120;
        double r233122 = log(r233121);
        double r233123 = r233115 - r233122;
        return r233123;
}
↓
double f(double x, double y) {
        double r233124 = x;
        double r233125 = y;
        double r233126 = r233124 - r233125;
        double r233127 = 1.0;
        double r233128 = r233127 - r233125;
        double r233129 = r233126 / r233128;
        double r233130 = 1.5458422377508842e-07;
        bool r233131 = r233129 <= r233130;
        double r233132 = 1.0;
        double r233133 = r233132 / r233128;
        double r233134 = r233126 * r233133;
        double r233135 = r233127 - r233134;
        double r233136 = log(r233135);
        double r233137 = r233127 - r233136;
        double r233138 = r233124 / r233125;
        double r233139 = 2.0;
        double r233140 = pow(r233125, r233139);
        double r233141 = r233124 / r233140;
        double r233142 = r233127 * r233141;
        double r233143 = r233138 + r233142;
        double r233144 = r233127 / r233125;
        double r233145 = r233143 - r233144;
        double r233146 = sqrt(r233145);
        double r233147 = log(r233146);
        double r233148 = r233147 + r233147;
        double r233149 = r233127 - r233148;
        double r233150 = r233131 ? r233137 : r233149;
        return r233150;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.2
Target0.1
Herbie0.8
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 1.5458422377508842e-07

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    if 1.5458422377508842e-07 < (/ (- x y) (- 1.0 y))

    1. Initial program 59.8

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

    2. Taylor expanded around inf 2.4

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

    3. Simplified2.4

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

    5. Applied add-sqr-sqrt2.4

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

    6. Applied log-prod2.4

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 1.545842237750884217971380531209124598035 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - \frac{1}{y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))