Average Error: 18.7 → 0.2
Time: 23.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999604026631461195506744843441992998:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \left(\frac{1}{y} + 1\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999604026631461195506744843441992998:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r17115486 = 1.0;
        double r17115487 = x;
        double r17115488 = y;
        double r17115489 = r17115487 - r17115488;
        double r17115490 = r17115486 - r17115488;
        double r17115491 = r17115489 / r17115490;
        double r17115492 = r17115486 - r17115491;
        double r17115493 = log(r17115492);
        double r17115494 = r17115486 - r17115493;
        return r17115494;
}
↓
double f(double x, double y) {
        double r17115495 = x;
        double r17115496 = y;
        double r17115497 = r17115495 - r17115496;
        double r17115498 = 1.0;
        double r17115499 = r17115498 - r17115496;
        double r17115500 = r17115497 / r17115499;
        double r17115501 = 0.9999999604026631;
        bool r17115502 = r17115500 <= r17115501;
        double r17115503 = exp(r17115498);
        double r17115504 = r17115498 - r17115500;
        double r17115505 = r17115503 / r17115504;
        double r17115506 = log(r17115505);
        double r17115507 = r17115495 / r17115496;
        double r17115508 = r17115498 / r17115496;
        double r17115509 = 1.0;
        double r17115510 = r17115508 + r17115509;
        double r17115511 = r17115507 * r17115510;
        double r17115512 = r17115511 - r17115508;
        double r17115513 = log(r17115512);
        double r17115514 = r17115498 - r17115513;
        double r17115515 = r17115502 ? r17115506 : r17115514;
        return r17115515;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.7
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 (/ (- x y) (- 1.0 y)) < 0.9999999604026631

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.2

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

    if 0.9999999604026631 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.5

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9999999604026631461195506744843441992998:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y} \cdot \left(\frac{1}{y} + 1\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(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))))))