Average Error: 18.3 → 0.2
Time: 21.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\
\;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{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 r340182 = 1.0;
        double r340183 = x;
        double r340184 = y;
        double r340185 = r340183 - r340184;
        double r340186 = r340182 - r340184;
        double r340187 = r340185 / r340186;
        double r340188 = r340182 - r340187;
        double r340189 = log(r340188);
        double r340190 = r340182 - r340189;
        return r340190;
}
↓
double f(double x, double y) {
        double r340191 = x;
        double r340192 = y;
        double r340193 = r340191 - r340192;
        double r340194 = 1.0;
        double r340195 = r340194 - r340192;
        double r340196 = r340193 / r340195;
        double r340197 = 0.9999998768371196;
        bool r340198 = r340196 <= r340197;
        double r340199 = 1.0;
        double r340200 = r340199 / r340195;
        double r340201 = r340193 * r340200;
        double r340202 = r340194 - r340201;
        double r340203 = log(r340202);
        double r340204 = r340194 - r340203;
        double r340205 = 2.0;
        double r340206 = pow(r340192, r340205);
        double r340207 = r340191 / r340206;
        double r340208 = r340191 / r340192;
        double r340209 = fma(r340207, r340194, r340208);
        double r340210 = r340194 / r340192;
        double r340211 = r340209 - r340210;
        double r340212 = log(r340211);
        double r340213 = r340194 - r340212;
        double r340214 = r340198 ? r340204 : r340213;
        return r340214;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.3
Target0.2
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999998768371196

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    if 0.9999998768371196 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.5

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

    3. Applied flip--62.5

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

    4. Simplified62.5

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

    5. Taylor expanded around inf 0.3

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

    6. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999998768371195501103443348256405442953:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{{y}^{2}}, 1, \frac{x}{y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +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))))))