Average Error: 18.9 → 1.5
Time: 6.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r333365 = 1.0;
        double r333366 = x;
        double r333367 = y;
        double r333368 = r333366 - r333367;
        double r333369 = r333365 - r333367;
        double r333370 = r333368 / r333369;
        double r333371 = r333365 - r333370;
        double r333372 = log(r333371);
        double r333373 = r333365 - r333372;
        return r333373;
}
↓
double f(double x, double y) {
        double r333374 = x;
        double r333375 = y;
        double r333376 = r333374 - r333375;
        double r333377 = 1.0;
        double r333378 = r333377 - r333375;
        double r333379 = r333376 / r333378;
        double r333380 = 3.2716599026957865e-16;
        bool r333381 = r333379 <= r333380;
        double r333382 = r333377 - r333379;
        double r333383 = cbrt(r333382);
        double r333384 = r333383 * r333383;
        double r333385 = r333384 * r333383;
        double r333386 = sqrt(r333385);
        double r333387 = log(r333386);
        double r333388 = sqrt(r333382);
        double r333389 = log(r333388);
        double r333390 = r333387 + r333389;
        double r333391 = r333377 - r333390;
        double r333392 = 2.0;
        double r333393 = pow(r333375, r333392);
        double r333394 = r333374 / r333393;
        double r333395 = 1.0;
        double r333396 = r333395 / r333375;
        double r333397 = r333394 - r333396;
        double r333398 = r333374 / r333375;
        double r333399 = fma(r333377, r333397, r333398);
        double r333400 = log(r333399);
        double r333401 = r333377 - r333400;
        double r333402 = r333381 ? r333391 : r333401;
        return r333402;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.9
Target0.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 3.2716599026957865e-16

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied log-prod0.0

      \[\leadsto 1 - \color{blue}{\left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.1

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

    if 3.2716599026957865e-16 < (/ (- x y) (- 1.0 y))

    1. Initial program 57.6

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))))