Average Error: 18.0 → 0.1
Time: 28.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -92144215.75918830931186676025390625:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 30353201.4492965638637542724609375:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\left(1 - \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right) + \mathsf{fma}\left(\frac{x - y}{\sqrt[3]{1 - y}}, \frac{\frac{-1}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}, \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -92144215.75918830931186676025390625:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 30353201.4492965638637542724609375:\\
\;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\left(1 - \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right) + \mathsf{fma}\left(\frac{x - y}{\sqrt[3]{1 - y}}, \frac{\frac{-1}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}, \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)}\right)\\

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

\end{array}
double f(double x, double y) {
        double r13545487 = 1.0;
        double r13545488 = x;
        double r13545489 = y;
        double r13545490 = r13545488 - r13545489;
        double r13545491 = r13545487 - r13545489;
        double r13545492 = r13545490 / r13545491;
        double r13545493 = r13545487 - r13545492;
        double r13545494 = log(r13545493);
        double r13545495 = r13545487 - r13545494;
        return r13545495;
}
↓
double f(double x, double y) {
        double r13545496 = y;
        double r13545497 = -92144215.75918831;
        bool r13545498 = r13545496 <= r13545497;
        double r13545499 = 1.0;
        double r13545500 = r13545499 / r13545496;
        double r13545501 = x;
        double r13545502 = r13545501 / r13545496;
        double r13545503 = r13545502 - r13545500;
        double r13545504 = fma(r13545500, r13545502, r13545503);
        double r13545505 = log(r13545504);
        double r13545506 = r13545499 - r13545505;
        double r13545507 = 30353201.449296564;
        bool r13545508 = r13545496 <= r13545507;
        double r13545509 = r13545501 - r13545496;
        double r13545510 = r13545499 - r13545496;
        double r13545511 = r13545509 / r13545510;
        double r13545512 = r13545499 - r13545511;
        double r13545513 = sqrt(r13545512);
        double r13545514 = log(r13545513);
        double r13545515 = r13545499 - r13545514;
        double r13545516 = cbrt(r13545510);
        double r13545517 = r13545509 / r13545516;
        double r13545518 = r13545517 / r13545516;
        double r13545519 = r13545518 / r13545516;
        double r13545520 = r13545499 - r13545519;
        double r13545521 = -1.0;
        double r13545522 = r13545521 / r13545516;
        double r13545523 = r13545522 / r13545516;
        double r13545524 = fma(r13545517, r13545523, r13545519);
        double r13545525 = r13545520 + r13545524;
        double r13545526 = sqrt(r13545525);
        double r13545527 = log(r13545526);
        double r13545528 = r13545515 - r13545527;
        double r13545529 = r13545508 ? r13545528 : r13545506;
        double r13545530 = r13545498 ? r13545506 : r13545529;
        return r13545530;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.0
Target0.1
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -92144215.75918831 or 30353201.449296564 < y

    1. Initial program 46.3

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -92144215.75918831 < y < 30353201.449296564

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

    5. Applied associate--r+0.1

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

    7. Applied add-cube-cbrt0.1

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

    8. Applied *-un-lft-identity0.1

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

    9. Applied times-frac0.1

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

    10. Applied add-sqr-sqrt0.1

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

    11. Applied prod-diff0.1

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

    12. Simplified0.1

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

    13. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -92144215.75918830931186676025390625:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 30353201.4492965638637542724609375:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\left(1 - \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right) + \mathsf{fma}\left(\frac{x - y}{\sqrt[3]{1 - y}}, \frac{\frac{-1}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}, \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))