Average Error: 18.3 → 0.2
Time: 5.1s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99781943200392309:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.99781943200392309:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r413192 = 1.0;
        double r413193 = x;
        double r413194 = y;
        double r413195 = r413193 - r413194;
        double r413196 = r413192 - r413194;
        double r413197 = r413195 / r413196;
        double r413198 = r413192 - r413197;
        double r413199 = log(r413198);
        double r413200 = r413192 - r413199;
        return r413200;
}
↓
double f(double x, double y) {
        double r413201 = x;
        double r413202 = y;
        double r413203 = r413201 - r413202;
        double r413204 = 1.0;
        double r413205 = r413204 - r413202;
        double r413206 = r413203 / r413205;
        double r413207 = 0.9978194320039231;
        bool r413208 = r413206 <= r413207;
        double r413209 = exp(r413204);
        double r413210 = r413204 - r413206;
        double r413211 = r413209 / r413210;
        double r413212 = log(r413211);
        double r413213 = 2.0;
        double r413214 = pow(r413202, r413213);
        double r413215 = r413201 / r413214;
        double r413216 = 1.0;
        double r413217 = r413216 / r413202;
        double r413218 = r413215 - r413217;
        double r413219 = r413204 * r413218;
        double r413220 = r413201 / r413202;
        double r413221 = r413219 + r413220;
        double r413222 = r413209 / r413221;
        double r413223 = log(r413222);
        double r413224 = r413208 ? r413212 : r413223;
        return r413224;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.3
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9978194320039231

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied diff-log0.0

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

    if 0.9978194320039231 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.5

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

    3. Applied add-log-exp61.5

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

    4. Applied diff-log61.5

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

    5. Taylor expanded around inf 0.7

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

    6. Simplified0.7

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99781943200392309:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))))))