Average Error: 18.3 → 0.1
Time: 6.0s
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 r402013 = 1.0;
        double r402014 = x;
        double r402015 = y;
        double r402016 = r402014 - r402015;
        double r402017 = r402013 - r402015;
        double r402018 = r402016 / r402017;
        double r402019 = r402013 - r402018;
        double r402020 = log(r402019);
        double r402021 = r402013 - r402020;
        return r402021;
}
↓
double f(double x, double y) {
        double r402022 = y;
        double r402023 = -244170417.41587064;
        bool r402024 = r402022 <= r402023;
        double r402025 = 39316813.46217899;
        bool r402026 = r402022 <= r402025;
        double r402027 = !r402026;
        bool r402028 = r402024 || r402027;
        double r402029 = 1.0;
        double r402030 = x;
        double r402031 = 2.0;
        double r402032 = pow(r402022, r402031);
        double r402033 = r402030 / r402032;
        double r402034 = 1.0;
        double r402035 = r402034 / r402022;
        double r402036 = r402033 - r402035;
        double r402037 = r402030 / r402022;
        double r402038 = fma(r402029, r402036, r402037);
        double r402039 = log(r402038);
        double r402040 = r402029 - r402039;
        double r402041 = r402030 - r402022;
        double r402042 = r402029 - r402022;
        double r402043 = r402041 / r402042;
        double r402044 = r402029 - r402043;
        double r402045 = log1p(r402044);
        double r402046 = expm1(r402045);
        double r402047 = log(r402046);
        double r402048 = r402029 - r402047;
        double r402049 = r402028 ? r402040 : r402048;
        return r402049;
}

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))))))