Average Error: 18.0 → 0.1
Time: 28.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -92144215.75918830931186676025390625:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 30353201.4492965638637542724609375:\\ \;\;\;\;\left(1 - \log \left(\sqrt{\mathsf{fma}\left(\frac{x - y}{\sqrt[3]{1 - y}}, \frac{\frac{-1}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}, \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right) + \left(1 - \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
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{\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) + \left(1 - \frac{\frac{\frac{x - y}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\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 r15305024 = 1.0;
        double r15305025 = x;
        double r15305026 = y;
        double r15305027 = r15305025 - r15305026;
        double r15305028 = r15305024 - r15305026;
        double r15305029 = r15305027 / r15305028;
        double r15305030 = r15305024 - r15305029;
        double r15305031 = log(r15305030);
        double r15305032 = r15305024 - r15305031;
        return r15305032;
}
↓
double f(double x, double y) {
        double r15305033 = y;
        double r15305034 = -92144215.75918831;
        bool r15305035 = r15305033 <= r15305034;
        double r15305036 = 1.0;
        double r15305037 = r15305036 / r15305033;
        double r15305038 = x;
        double r15305039 = r15305038 / r15305033;
        double r15305040 = r15305039 - r15305037;
        double r15305041 = fma(r15305037, r15305039, r15305040);
        double r15305042 = log(r15305041);
        double r15305043 = r15305036 - r15305042;
        double r15305044 = 30353201.449296564;
        bool r15305045 = r15305033 <= r15305044;
        double r15305046 = r15305038 - r15305033;
        double r15305047 = r15305036 - r15305033;
        double r15305048 = cbrt(r15305047);
        double r15305049 = r15305046 / r15305048;
        double r15305050 = -1.0;
        double r15305051 = r15305050 / r15305048;
        double r15305052 = r15305051 / r15305048;
        double r15305053 = r15305049 / r15305048;
        double r15305054 = r15305053 / r15305048;
        double r15305055 = fma(r15305049, r15305052, r15305054);
        double r15305056 = r15305036 - r15305054;
        double r15305057 = r15305055 + r15305056;
        double r15305058 = sqrt(r15305057);
        double r15305059 = log(r15305058);
        double r15305060 = r15305036 - r15305059;
        double r15305061 = r15305046 / r15305047;
        double r15305062 = r15305036 - r15305061;
        double r15305063 = sqrt(r15305062);
        double r15305064 = log(r15305063);
        double r15305065 = r15305060 - r15305064;
        double r15305066 = r15305045 ? r15305065 : r15305043;
        double r15305067 = r15305035 ? r15305043 : r15305066;
        return r15305067;
}

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

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

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

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

    9. Applied times-frac0.1

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

    10. Applied add-sqr-sqrt0.1

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

    11. Applied prod-diff0.1

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

    12. Simplified0.1

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

    13. Simplified0.1

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

  4. Final simplification0.1

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

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))))))