Average Error: 18.1 → 0.1
Time: 6.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -372431085.042961299 \lor \neg \left(y \le 44890028.239001065\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -372431085.042961299 \lor \neg \left(y \le 44890028.239001065\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\

\end{array}
double f(double x, double y) {
        double r394022 = 1.0;
        double r394023 = x;
        double r394024 = y;
        double r394025 = r394023 - r394024;
        double r394026 = r394022 - r394024;
        double r394027 = r394025 / r394026;
        double r394028 = r394022 - r394027;
        double r394029 = log(r394028);
        double r394030 = r394022 - r394029;
        return r394030;
}
↓
double f(double x, double y) {
        double r394031 = y;
        double r394032 = -372431085.0429613;
        bool r394033 = r394031 <= r394032;
        double r394034 = 44890028.239001065;
        bool r394035 = r394031 <= r394034;
        double r394036 = !r394035;
        bool r394037 = r394033 || r394036;
        double r394038 = 1.0;
        double r394039 = x;
        double r394040 = 2.0;
        double r394041 = pow(r394031, r394040);
        double r394042 = r394039 / r394041;
        double r394043 = 1.0;
        double r394044 = r394043 / r394031;
        double r394045 = r394042 - r394044;
        double r394046 = r394038 * r394045;
        double r394047 = r394039 / r394031;
        double r394048 = r394046 + r394047;
        double r394049 = log(r394048);
        double r394050 = r394038 - r394049;
        double r394051 = r394039 - r394031;
        double r394052 = cbrt(r394051);
        double r394053 = r394052 * r394052;
        double r394054 = r394038 - r394031;
        double r394055 = r394054 / r394052;
        double r394056 = r394053 / r394055;
        double r394057 = r394038 - r394056;
        double r394058 = log(r394057);
        double r394059 = r394038 - r394058;
        double r394060 = r394037 ? r394050 : r394059;
        return r394060;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.1
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 < -372431085.0429613 or 44890028.239001065 < y

    1. Initial program 46.4

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

    3. Applied add-cube-cbrt44.2

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

    4. Applied associate-/l*44.2

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -372431085.0429613 < y < 44890028.239001065

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/l*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -372431085.042961299 \lor \neg \left(y \le 44890028.239001065\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +o rules:numerics
(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))))))