Average Error: 18.3 → 0.1
Time: 5.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -589339982.767086387 \lor \neg \left(y \le 146835981.479329318\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\sqrt{1 - \frac{x - y}{1 - y}} \cdot \sqrt{1 - \frac{x - y}{1 - y}}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -589339982.767086387 \lor \neg \left(y \le 146835981.479329318\right):\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r418338 = 1.0;
        double r418339 = x;
        double r418340 = y;
        double r418341 = r418339 - r418340;
        double r418342 = r418338 - r418340;
        double r418343 = r418341 / r418342;
        double r418344 = r418338 - r418343;
        double r418345 = log(r418344);
        double r418346 = r418338 - r418345;
        return r418346;
}
↓
double f(double x, double y) {
        double r418347 = y;
        double r418348 = -589339982.7670864;
        bool r418349 = r418347 <= r418348;
        double r418350 = 146835981.47932932;
        bool r418351 = r418347 <= r418350;
        double r418352 = !r418351;
        bool r418353 = r418349 || r418352;
        double r418354 = 1.0;
        double r418355 = x;
        double r418356 = 2.0;
        double r418357 = pow(r418347, r418356);
        double r418358 = r418355 / r418357;
        double r418359 = 1.0;
        double r418360 = r418359 / r418347;
        double r418361 = r418358 - r418360;
        double r418362 = r418354 * r418361;
        double r418363 = r418355 / r418347;
        double r418364 = r418362 + r418363;
        double r418365 = log(r418364);
        double r418366 = r418354 - r418365;
        double r418367 = exp(r418354);
        double r418368 = r418355 - r418347;
        double r418369 = r418354 - r418347;
        double r418370 = r418368 / r418369;
        double r418371 = r418354 - r418370;
        double r418372 = sqrt(r418371);
        double r418373 = r418372 * r418372;
        double r418374 = r418367 / r418373;
        double r418375 = log(r418374);
        double r418376 = r418353 ? r418366 : r418375;
        return r418376;
}

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

Derivation

  1. Split input into 2 regimes

  2. if y < -589339982.7670864 or 146835981.47932932 < y

    1. Initial program 46.9

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

    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 -589339982.7670864 < y < 146835981.47932932

    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-log0.1

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

    6. Applied add-sqr-sqrt0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -589339982.767086387 \lor \neg \left(y \le 146835981.479329318\right):\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\sqrt{1 - \frac{x - y}{1 - y}} \cdot \sqrt{1 - \frac{x - y}{1 - y}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 
(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))))))