Average Error: 18.3 → 0.1
Time: 5.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -244170417.415870637 \lor \neg \left(y \le 39316813.46217899\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{x - y}{1 - y}\right)\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -244170417.415870637 \lor \neg \left(y \le 39316813.46217899\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{x - y}{1 - y}\right)\right)\right)\\

\end{array}
double f(double x, double y) {
        double r406531 = 1.0;
        double r406532 = x;
        double r406533 = y;
        double r406534 = r406532 - r406533;
        double r406535 = r406531 - r406533;
        double r406536 = r406534 / r406535;
        double r406537 = r406531 - r406536;
        double r406538 = log(r406537);
        double r406539 = r406531 - r406538;
        return r406539;
}
↓
double f(double x, double y) {
        double r406540 = y;
        double r406541 = -244170417.41587064;
        bool r406542 = r406540 <= r406541;
        double r406543 = 39316813.46217899;
        bool r406544 = r406540 <= r406543;
        double r406545 = !r406544;
        bool r406546 = r406542 || r406545;
        double r406547 = 1.0;
        double r406548 = x;
        double r406549 = 2.0;
        double r406550 = pow(r406540, r406549);
        double r406551 = r406548 / r406550;
        double r406552 = 1.0;
        double r406553 = r406552 / r406540;
        double r406554 = r406551 - r406553;
        double r406555 = r406548 / r406540;
        double r406556 = fma(r406547, r406554, r406555);
        double r406557 = log(r406556);
        double r406558 = r406547 - r406557;
        double r406559 = r406548 - r406540;
        double r406560 = r406547 - r406540;
        double r406561 = r406559 / r406560;
        double r406562 = r406547 - r406561;
        double r406563 = log1p(r406562);
        double r406564 = expm1(r406563);
        double r406565 = log(r406564);
        double r406566 = r406547 - r406565;
        double r406567 = r406546 ? r406558 : r406566;
        return r406567;
}

Error

Bits error versus x

Bits error versus y

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 < -244170417.41587064 or 39316813.46217899 < y

    1. Initial program 46.4

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

    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

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

    if -244170417.41587064 < y < 39316813.46217899

    1. Initial program 0.1

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

    3. Applied expm1-log1p-u0.1

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{x - y}{1 - y}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -244170417.415870637 \lor \neg \left(y \le 39316813.46217899\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{x - y}{1 - y}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))))