Average Error: 18.3 → 0.1
Time: 4.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r363576 = 1.0;
        double r363577 = x;
        double r363578 = y;
        double r363579 = r363577 - r363578;
        double r363580 = r363576 - r363578;
        double r363581 = r363579 / r363580;
        double r363582 = r363576 - r363581;
        double r363583 = log(r363582);
        double r363584 = r363576 - r363583;
        return r363584;
}
↓
double f(double x, double y) {
        double r363585 = y;
        double r363586 = -150668776.17725816;
        bool r363587 = r363585 <= r363586;
        double r363588 = 15857199.25534847;
        bool r363589 = r363585 <= r363588;
        double r363590 = !r363589;
        bool r363591 = r363587 || r363590;
        double r363592 = 1.0;
        double r363593 = x;
        double r363594 = 2.0;
        double r363595 = pow(r363585, r363594);
        double r363596 = r363593 / r363595;
        double r363597 = 1.0;
        double r363598 = r363597 / r363585;
        double r363599 = r363596 - r363598;
        double r363600 = r363592 * r363599;
        double r363601 = r363593 / r363585;
        double r363602 = r363600 + r363601;
        double r363603 = log(r363602);
        double r363604 = r363592 - r363603;
        double r363605 = exp(r363592);
        double r363606 = r363593 - r363585;
        double r363607 = r363592 - r363585;
        double r363608 = r363606 / r363607;
        double r363609 = r363592 - r363608;
        double r363610 = r363605 / r363609;
        double r363611 = log(r363610);
        double r363612 = r363591 ? r363604 : r363611;
        return r363612;
}

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 < -150668776.17725816 or 15857199.25534847 < y

    1. Initial program 46.5

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

    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 -150668776.17725816 < y < 15857199.25534847

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(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))))))