Average Error: 18.2 → 0.8
Time: 22.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 1.545842237750884217971380531209124598035 \cdot 10^{-7}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\right)\\ \end{array}\]
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{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right) + \log \left(\sqrt{\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r219601 = 1.0;
        double r219602 = x;
        double r219603 = y;
        double r219604 = r219602 - r219603;
        double r219605 = r219601 - r219603;
        double r219606 = r219604 / r219605;
        double r219607 = r219601 - r219606;
        double r219608 = log(r219607);
        double r219609 = r219601 - r219608;
        return r219609;
}
↓
double f(double x, double y) {
        double r219610 = x;
        double r219611 = y;
        double r219612 = r219610 - r219611;
        double r219613 = 1.0;
        double r219614 = r219613 - r219611;
        double r219615 = r219612 / r219614;
        double r219616 = 1.5458422377508842e-07;
        bool r219617 = r219615 <= r219616;
        double r219618 = 1.0;
        double r219619 = r219618 / r219614;
        double r219620 = r219612 * r219619;
        double r219621 = r219613 - r219620;
        double r219622 = log(r219621);
        double r219623 = r219613 - r219622;
        double r219624 = 2.0;
        double r219625 = pow(r219611, r219624);
        double r219626 = r219610 / r219625;
        double r219627 = r219610 / r219611;
        double r219628 = fma(r219626, r219613, r219627);
        double r219629 = r219613 / r219611;
        double r219630 = r219628 - r219629;
        double r219631 = sqrt(r219630);
        double r219632 = log(r219631);
        double r219633 = r219632 + r219632;
        double r219634 = r219613 - r219633;
        double r219635 = r219617 ? r219623 : r219634;
        return r219635;
}

Error

Bits error versus x

Bits error versus y

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

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

    5. Applied add-sqr-sqrt2.4

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

    6. Applied log-prod2.4

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

  4. Final simplification0.8

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

Reproduce

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

  :herbie-target
  (if (< y -81284752.619472414) (- 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))))))