Average Error: 18.1 → 0.8
Time: 21.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.960614984267581169834128874268630582378 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 3.960614984267581169834128874268630582378 \cdot 10^{-8}:\\
\;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r290168 = 1.0;
        double r290169 = x;
        double r290170 = y;
        double r290171 = r290169 - r290170;
        double r290172 = r290168 - r290170;
        double r290173 = r290171 / r290172;
        double r290174 = r290168 - r290173;
        double r290175 = log(r290174);
        double r290176 = r290168 - r290175;
        return r290176;
}
↓
double f(double x, double y) {
        double r290177 = x;
        double r290178 = y;
        double r290179 = r290177 - r290178;
        double r290180 = 1.0;
        double r290181 = r290180 - r290178;
        double r290182 = r290179 / r290181;
        double r290183 = 3.960614984267581e-08;
        bool r290184 = r290182 <= r290183;
        double r290185 = 1.0;
        double r290186 = cbrt(r290181);
        double r290187 = r290186 * r290186;
        double r290188 = r290185 / r290187;
        double r290189 = r290179 / r290186;
        double r290190 = r290188 * r290189;
        double r290191 = r290180 - r290190;
        double r290192 = log(r290191);
        double r290193 = r290180 - r290192;
        double r290194 = 2.0;
        double r290195 = pow(r290178, r290194);
        double r290196 = r290177 / r290195;
        double r290197 = r290177 / r290178;
        double r290198 = fma(r290196, r290180, r290197);
        double r290199 = r290180 / r290178;
        double r290200 = r290198 - r290199;
        double r290201 = log(r290200);
        double r290202 = r290180 - r290201;
        double r290203 = r290184 ? r290193 : r290202;
        return r290203;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.1
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)) < 3.960614984267581e-08

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.0

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

    4. Applied *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    if 3.960614984267581e-08 < (/ (- x y) (- 1.0 y))

    1. Initial program 58.9

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

    2. Taylor expanded around inf 2.6

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

    3. Simplified2.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.960614984267581169834128874268630582378 \cdot 10^{-8}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +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))))))