Average Error: 18.1 → 0.1
Time: 19.2s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -220425257.605022430419921875:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -220425257.605022430419921875:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y}\right) - \frac{1}{y}\right)\\

\mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\
\;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r15702452 = 1.0;
        double r15702453 = x;
        double r15702454 = y;
        double r15702455 = r15702453 - r15702454;
        double r15702456 = r15702452 - r15702454;
        double r15702457 = r15702455 / r15702456;
        double r15702458 = r15702452 - r15702457;
        double r15702459 = log(r15702458);
        double r15702460 = r15702452 - r15702459;
        return r15702460;
}
↓
double f(double x, double y) {
        double r15702461 = y;
        double r15702462 = -220425257.60502243;
        bool r15702463 = r15702461 <= r15702462;
        double r15702464 = 1.0;
        double r15702465 = x;
        double r15702466 = r15702465 / r15702461;
        double r15702467 = r15702464 / r15702461;
        double r15702468 = fma(r15702466, r15702467, r15702466);
        double r15702469 = r15702468 - r15702467;
        double r15702470 = log(r15702469);
        double r15702471 = r15702464 - r15702470;
        double r15702472 = 20833302.402704142;
        bool r15702473 = r15702461 <= r15702472;
        double r15702474 = r15702464 - r15702461;
        double r15702475 = r15702465 / r15702474;
        double r15702476 = r15702464 - r15702475;
        double r15702477 = r15702461 / r15702474;
        double r15702478 = r15702476 + r15702477;
        double r15702479 = log(r15702478);
        double r15702480 = r15702464 - r15702479;
        double r15702481 = r15702473 ? r15702480 : r15702471;
        double r15702482 = r15702463 ? r15702471 : r15702481;
        return r15702482;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.1
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -220425257.60502243 or 20833302.402704142 < y

    1. Initial program 46.7

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

    3. Applied div-sub46.7

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

    4. Applied associate--r-46.7

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -220425257.60502243 < y < 20833302.402704142

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate--r-0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -220425257.605022430419921875:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 20833302.402704142034053802490234375:\\ \;\;\;\;1 - \log \left(\left(1 - \frac{x}{1 - y}\right) + \frac{y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))