Average Error: 18.2 → 0.1
Time: 9.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -173788009.49411574 \lor \neg \left(y \le 98105251.590079397\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -173788009.49411574 \lor \neg \left(y \le 98105251.590079397\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r372413 = 1.0;
        double r372414 = x;
        double r372415 = y;
        double r372416 = r372414 - r372415;
        double r372417 = r372413 - r372415;
        double r372418 = r372416 / r372417;
        double r372419 = r372413 - r372418;
        double r372420 = log(r372419);
        double r372421 = r372413 - r372420;
        return r372421;
}
↓
double f(double x, double y) {
        double r372422 = y;
        double r372423 = -173788009.49411574;
        bool r372424 = r372422 <= r372423;
        double r372425 = 98105251.5900794;
        bool r372426 = r372422 <= r372425;
        double r372427 = !r372426;
        bool r372428 = r372424 || r372427;
        double r372429 = 1.0;
        double r372430 = exp(r372429);
        double r372431 = x;
        double r372432 = 2.0;
        double r372433 = pow(r372422, r372432);
        double r372434 = r372431 / r372433;
        double r372435 = r372431 / r372422;
        double r372436 = fma(r372429, r372434, r372435);
        double r372437 = r372429 / r372422;
        double r372438 = r372436 - r372437;
        double r372439 = r372430 / r372438;
        double r372440 = log(r372439);
        double r372441 = r372431 - r372422;
        double r372442 = r372429 - r372422;
        double r372443 = r372441 / r372442;
        double r372444 = r372429 - r372443;
        double r372445 = log(r372444);
        double r372446 = r372429 - r372445;
        double r372447 = r372428 ? r372440 : r372446;
        return r372447;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.2
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 < -173788009.49411574 or 98105251.5900794 < y

    1. Initial program 47.1

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

    3. Applied add-log-exp47.1

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

    4. Applied diff-log47.1

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -173788009.49411574 < y < 98105251.5900794

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
    5. Using strategy rm

    6. Applied log-div0.1

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -173788009.49411574 \lor \neg \left(y \le 98105251.590079397\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 +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))))))