Average Error: 18.0 → 0.2
Time: 23.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -123098519.9019673168659210205078125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 1.002831089821186738575420349661726504564:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(\frac{x - y}{\sqrt{1 - y}}, \frac{-1}{\sqrt{1 - y}}, \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right) + \left(1 - \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -123098519.9019673168659210205078125:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 1.002831089821186738575420349661726504564:\\
\;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(\frac{x - y}{\sqrt{1 - y}}, \frac{-1}{\sqrt{1 - y}}, \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right) + \left(1 - \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{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 r13533906 = 1.0;
        double r13533907 = x;
        double r13533908 = y;
        double r13533909 = r13533907 - r13533908;
        double r13533910 = r13533906 - r13533908;
        double r13533911 = r13533909 / r13533910;
        double r13533912 = r13533906 - r13533911;
        double r13533913 = log(r13533912);
        double r13533914 = r13533906 - r13533913;
        return r13533914;
}
↓
double f(double x, double y) {
        double r13533915 = y;
        double r13533916 = -123098519.90196732;
        bool r13533917 = r13533915 <= r13533916;
        double r13533918 = 1.0;
        double r13533919 = r13533918 / r13533915;
        double r13533920 = x;
        double r13533921 = r13533920 / r13533915;
        double r13533922 = r13533921 - r13533919;
        double r13533923 = fma(r13533919, r13533921, r13533922);
        double r13533924 = log(r13533923);
        double r13533925 = r13533918 - r13533924;
        double r13533926 = 1.0028310898211867;
        bool r13533927 = r13533915 <= r13533926;
        double r13533928 = r13533920 - r13533915;
        double r13533929 = r13533918 - r13533915;
        double r13533930 = r13533928 / r13533929;
        double r13533931 = r13533918 - r13533930;
        double r13533932 = sqrt(r13533931);
        double r13533933 = log(r13533932);
        double r13533934 = r13533918 - r13533933;
        double r13533935 = sqrt(r13533929);
        double r13533936 = r13533928 / r13533935;
        double r13533937 = -1.0;
        double r13533938 = r13533937 / r13533935;
        double r13533939 = r13533936 / r13533935;
        double r13533940 = fma(r13533936, r13533938, r13533939);
        double r13533941 = r13533918 - r13533939;
        double r13533942 = r13533940 + r13533941;
        double r13533943 = sqrt(r13533942);
        double r13533944 = log(r13533943);
        double r13533945 = r13533934 - r13533944;
        double r13533946 = r13533927 ? r13533945 : r13533925;
        double r13533947 = r13533917 ? r13533925 : r13533946;
        return r13533947;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.0
Target0.1
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 y < -123098519.90196732 or 1.0028310898211867 < y

    1. Initial program 46.1

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

    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 -123098519.90196732 < y < 1.0028310898211867

    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-sqr-sqrt0.1

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied times-frac0.1

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

    10. Applied add-sqr-sqrt0.1

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

    11. Applied prod-diff0.1

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

    12. Simplified0.1

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

    13. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -123098519.9019673168659210205078125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 1.002831089821186738575420349661726504564:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{\mathsf{fma}\left(\frac{x - y}{\sqrt{1 - y}}, \frac{-1}{\sqrt{1 - y}}, \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right) + \left(1 - \frac{\frac{x - y}{\sqrt{1 - y}}}{\sqrt{1 - y}}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

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))))))