Average Error: 18.3 → 0.1
Time: 5.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -150492446.473063856 \lor \neg \left(y \le 533944797.16773558\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -150492446.473063856 \lor \neg \left(y \le 533944797.16773558\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 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

\end{array}
double f(double x, double y) {
        double r378547 = 1.0;
        double r378548 = x;
        double r378549 = y;
        double r378550 = r378548 - r378549;
        double r378551 = r378547 - r378549;
        double r378552 = r378550 / r378551;
        double r378553 = r378547 - r378552;
        double r378554 = log(r378553);
        double r378555 = r378547 - r378554;
        return r378555;
}
↓
double f(double x, double y) {
        double r378556 = y;
        double r378557 = -150492446.47306386;
        bool r378558 = r378556 <= r378557;
        double r378559 = 533944797.1677356;
        bool r378560 = r378556 <= r378559;
        double r378561 = !r378560;
        bool r378562 = r378558 || r378561;
        double r378563 = 1.0;
        double r378564 = x;
        double r378565 = 2.0;
        double r378566 = pow(r378556, r378565);
        double r378567 = r378564 / r378566;
        double r378568 = 1.0;
        double r378569 = r378568 / r378556;
        double r378570 = r378567 - r378569;
        double r378571 = r378563 * r378570;
        double r378572 = r378564 / r378556;
        double r378573 = r378571 + r378572;
        double r378574 = log(r378573);
        double r378575 = r378563 - r378574;
        double r378576 = r378564 - r378556;
        double r378577 = r378563 - r378556;
        double r378578 = r378568 / r378577;
        double r378579 = r378576 * r378578;
        double r378580 = r378563 - r378579;
        double r378581 = log(r378580);
        double r378582 = r378563 - r378581;
        double r378583 = r378562 ? r378575 : r378582;
        return r378583;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.3
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 < -150492446.47306386 or 533944797.1677356 < y

    1. Initial program 47.4

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -150492446.47306386 < y < 533944797.1677356

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150492446.473063856 \lor \neg \left(y \le 533944797.16773558\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 
(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))))))