Average Error: 18.4 → 0.2
Time: 10.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r343576 = 1.0;
        double r343577 = x;
        double r343578 = y;
        double r343579 = r343577 - r343578;
        double r343580 = r343576 - r343578;
        double r343581 = r343579 / r343580;
        double r343582 = r343576 - r343581;
        double r343583 = log(r343582);
        double r343584 = r343576 - r343583;
        return r343584;
}
↓
double f(double x, double y) {
        double r343585 = x;
        double r343586 = y;
        double r343587 = r343585 - r343586;
        double r343588 = 1.0;
        double r343589 = r343588 - r343586;
        double r343590 = r343587 / r343589;
        double r343591 = 0.9411461947580387;
        bool r343592 = r343590 <= r343591;
        double r343593 = 1.0;
        double r343594 = r343593 / r343589;
        double r343595 = r343587 * r343594;
        double r343596 = r343588 - r343595;
        double r343597 = log(r343596);
        double r343598 = r343588 - r343597;
        double r343599 = 2.0;
        double r343600 = pow(r343586, r343599);
        double r343601 = r343585 / r343600;
        double r343602 = r343585 / r343586;
        double r343603 = fma(r343588, r343601, r343602);
        double r343604 = r343588 / r343586;
        double r343605 = r343603 - r343604;
        double r343606 = log(r343605);
        double r343607 = r343588 - r343606;
        double r343608 = r343592 ? r343598 : r343607;
        return r343608;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.4
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9411461947580387

    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 0.9411461947580387 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.4

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

    3. Applied div-inv60.0

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

    4. Taylor expanded around inf 0.7

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

    5. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.941146194758038734:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +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))))))