Average Error: 17.8 → 0.2
Time: 15.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9976104054069743209964826746727339923382:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9976104054069743209964826746727339923382:\\
\;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r317457 = 1.0;
        double r317458 = x;
        double r317459 = y;
        double r317460 = r317458 - r317459;
        double r317461 = r317457 - r317459;
        double r317462 = r317460 / r317461;
        double r317463 = r317457 - r317462;
        double r317464 = log(r317463);
        double r317465 = r317457 - r317464;
        return r317465;
}
↓
double f(double x, double y) {
        double r317466 = x;
        double r317467 = y;
        double r317468 = r317466 - r317467;
        double r317469 = 1.0;
        double r317470 = r317469 - r317467;
        double r317471 = r317468 / r317470;
        double r317472 = 0.9976104054069743;
        bool r317473 = r317471 <= r317472;
        double r317474 = r317469 - r317471;
        double r317475 = sqrt(r317474);
        double r317476 = log(r317475);
        double r317477 = r317469 - r317476;
        double r317478 = r317477 - r317476;
        double r317479 = r317466 / r317467;
        double r317480 = r317469 * r317466;
        double r317481 = r317467 * r317467;
        double r317482 = r317480 / r317481;
        double r317483 = r317479 + r317482;
        double r317484 = r317469 / r317467;
        double r317485 = r317483 - r317484;
        double r317486 = log(r317485);
        double r317487 = r317469 - r317486;
        double r317488 = r317473 ? r317478 : r317487;
        return r317488;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.8
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)) < 0.9976104054069743

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied log-prod0.1

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

    5. Applied associate--r+0.1

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

    if 0.9976104054069743 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.6

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

    3. Applied flip--57.0

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

    4. Applied associate-/r/56.3

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

    5. Simplified60.1

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

    6. Taylor expanded around inf 0.5

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

    7. Simplified0.5

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9976104054069743209964826746727339923382:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(\frac{x}{y} + \frac{1 \cdot x}{y \cdot y}\right) - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))))))