Average Error: 18.9 → 1.5
Time: 6.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\end{array}
double f(double x, double y) {
        double r414107 = 1.0;
        double r414108 = x;
        double r414109 = y;
        double r414110 = r414108 - r414109;
        double r414111 = r414107 - r414109;
        double r414112 = r414110 / r414111;
        double r414113 = r414107 - r414112;
        double r414114 = log(r414113);
        double r414115 = r414107 - r414114;
        return r414115;
}
↓
double f(double x, double y) {
        double r414116 = x;
        double r414117 = y;
        double r414118 = r414116 - r414117;
        double r414119 = 1.0;
        double r414120 = r414119 - r414117;
        double r414121 = r414118 / r414120;
        double r414122 = 3.2716599026957865e-16;
        bool r414123 = r414121 <= r414122;
        double r414124 = r414119 - r414121;
        double r414125 = cbrt(r414124);
        double r414126 = r414125 * r414125;
        double r414127 = r414126 * r414125;
        double r414128 = sqrt(r414127);
        double r414129 = log(r414128);
        double r414130 = sqrt(r414124);
        double r414131 = log(r414130);
        double r414132 = r414129 + r414131;
        double r414133 = r414119 - r414132;
        double r414134 = 2.0;
        double r414135 = pow(r414117, r414134);
        double r414136 = r414116 / r414135;
        double r414137 = 1.0;
        double r414138 = r414137 / r414117;
        double r414139 = r414136 - r414138;
        double r414140 = r414116 / r414117;
        double r414141 = fma(r414119, r414139, r414140);
        double r414142 = log(r414141);
        double r414143 = r414119 - r414142;
        double r414144 = r414123 ? r414133 : r414143;
        return r414144;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.9
Target0.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 3.2716599026957865e-16

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied log-prod0.0

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

    6. Applied add-cube-cbrt0.1

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

    if 3.2716599026957865e-16 < (/ (- x y) (- 1.0 y))

    1. Initial program 57.6

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

    2. Taylor expanded around inf 4.4

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

    3. Simplified4.4

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

  4. Final simplification1.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 3.27165990269578651523296592754768024042 \cdot 10^{-16}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{\left(\sqrt[3]{1 - \frac{x - y}{1 - y}} \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}\right) \cdot \sqrt[3]{1 - \frac{x - y}{1 - y}}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))))))