Average Error: 18.1 → 0.1
Time: 16.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -80228020.28782515227794647216796875 \lor \neg \left(y \le 42220580.41032813489437103271484375\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -80228020.28782515227794647216796875 \lor \neg \left(y \le 42220580.41032813489437103271484375\right):\\
\;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\

\end{array}
double f(double x, double y) {
        double r299185 = 1.0;
        double r299186 = x;
        double r299187 = y;
        double r299188 = r299186 - r299187;
        double r299189 = r299185 - r299187;
        double r299190 = r299188 / r299189;
        double r299191 = r299185 - r299190;
        double r299192 = log(r299191);
        double r299193 = r299185 - r299192;
        return r299193;
}
↓
double f(double x, double y) {
        double r299194 = y;
        double r299195 = -80228020.28782515;
        bool r299196 = r299194 <= r299195;
        double r299197 = 42220580.410328135;
        bool r299198 = r299194 <= r299197;
        double r299199 = !r299198;
        bool r299200 = r299196 || r299199;
        double r299201 = 1.0;
        double r299202 = x;
        double r299203 = r299202 / r299194;
        double r299204 = r299201 / r299194;
        double r299205 = r299203 * r299204;
        double r299206 = r299205 - r299204;
        double r299207 = r299203 + r299206;
        double r299208 = log(r299207);
        double r299209 = r299201 - r299208;
        double r299210 = r299202 - r299194;
        double r299211 = r299201 - r299194;
        double r299212 = cbrt(r299211);
        double r299213 = r299212 * r299212;
        double r299214 = r299210 / r299213;
        double r299215 = r299214 / r299212;
        double r299216 = r299201 - r299215;
        double r299217 = log(r299216);
        double r299218 = r299201 - r299217;
        double r299219 = r299200 ? r299209 : r299218;
        return r299219;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.1
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 < -80228020.28782515 or 42220580.410328135 < y

    1. Initial program 46.4

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

    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

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

    if -80228020.28782515 < y < 42220580.410328135

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -80228020.28782515227794647216796875 \lor \neg \left(y \le 42220580.41032813489437103271484375\right):\\ \;\;\;\;1 - \log \left(\frac{x}{y} + \left(\frac{x}{y} \cdot \frac{1}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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))))))