Average Error: 18.5 → 0.3
Time: 4.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -79025762686.63672:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{elif}\;y \le 4.14547202165829592 \cdot 10^{23}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \frac{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -79025762686.63672:\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

\mathbf{elif}\;y \le 4.14547202165829592 \cdot 10^{23}:\\
\;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 \cdot \frac{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r460147 = 1.0;
        double r460148 = x;
        double r460149 = y;
        double r460150 = r460148 - r460149;
        double r460151 = r460147 - r460149;
        double r460152 = r460150 / r460151;
        double r460153 = r460147 - r460152;
        double r460154 = log(r460153);
        double r460155 = r460147 - r460154;
        return r460155;
}
↓
double f(double x, double y) {
        double r460156 = y;
        double r460157 = -79025762686.63672;
        bool r460158 = r460156 <= r460157;
        double r460159 = 1.0;
        double r460160 = x;
        double r460161 = 2.0;
        double r460162 = pow(r460156, r460161);
        double r460163 = r460160 / r460162;
        double r460164 = 1.0;
        double r460165 = r460164 / r460156;
        double r460166 = r460163 - r460165;
        double r460167 = r460159 * r460166;
        double r460168 = r460160 / r460156;
        double r460169 = r460167 + r460168;
        double r460170 = log(r460169);
        double r460171 = r460159 - r460170;
        double r460172 = 4.145472021658296e+23;
        bool r460173 = r460156 <= r460172;
        double r460174 = r460160 - r460156;
        double r460175 = r460159 - r460156;
        double r460176 = r460174 / r460175;
        double r460177 = r460159 - r460176;
        double r460178 = sqrt(r460177);
        double r460179 = log(r460178);
        double r460180 = r460179 + r460179;
        double r460181 = r460159 - r460180;
        double r460182 = -r460165;
        double r460183 = r460182 / r460156;
        double r460184 = 4.0;
        double r460185 = pow(r460156, r460184);
        double r460186 = r460185 / r460160;
        double r460187 = r460160 / r460186;
        double r460188 = r460183 + r460187;
        double r460189 = r460163 + r460165;
        double r460190 = r460188 / r460189;
        double r460191 = r460159 * r460190;
        double r460192 = r460191 + r460168;
        double r460193 = log(r460192);
        double r460194 = r460159 - r460193;
        double r460195 = r460173 ? r460181 : r460194;
        double r460196 = r460158 ? r460171 : r460195;
        return r460196;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if y < -79025762686.63672

    1. Initial program 52.6

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

    if -79025762686.63672 < y < 4.145472021658296e+23

    1. Initial program 0.2

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

    3. Applied add-sqr-sqrt0.2

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

    4. Applied log-prod0.2

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

    if 4.145472021658296e+23 < y

    1. Initial program 32.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    5. Applied flip--1.8

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

    6. Simplified1.8

      \[\leadsto 1 - \log \left(1 \cdot \frac{\color{blue}{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -79025762686.63672:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{elif}\;y \le 4.14547202165829592 \cdot 10^{23}:\\ \;\;\;\;1 - \left(\log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right) + \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \frac{\frac{-\frac{1}{y}}{y} + \frac{x}{\frac{{y}^{4}}{x}}}{\frac{x}{{y}^{2}} + \frac{1}{y}} + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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))))))