Average Error: 18.4 → 0.1
Time: 17.8s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -231188405.8267842233180999755859375 \lor \neg \left(y \le 6764068.6677209436893463134765625\right):\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -231188405.8267842233180999755859375 \lor \neg \left(y \le 6764068.6677209436893463134765625\right):\\
\;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\

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

\end{array}
double f(double x, double y) {
        double r327397 = 1.0;
        double r327398 = x;
        double r327399 = y;
        double r327400 = r327398 - r327399;
        double r327401 = r327397 - r327399;
        double r327402 = r327400 / r327401;
        double r327403 = r327397 - r327402;
        double r327404 = log(r327403);
        double r327405 = r327397 - r327404;
        return r327405;
}
↓
double f(double x, double y) {
        double r327406 = y;
        double r327407 = -231188405.82678422;
        bool r327408 = r327406 <= r327407;
        double r327409 = 6764068.667720944;
        bool r327410 = r327406 <= r327409;
        double r327411 = !r327410;
        bool r327412 = r327408 || r327411;
        double r327413 = 1.0;
        double r327414 = 1.0;
        double r327415 = r327413 / r327406;
        double r327416 = r327414 + r327415;
        double r327417 = x;
        double r327418 = r327417 / r327406;
        double r327419 = r327416 * r327418;
        double r327420 = r327419 - r327415;
        double r327421 = log(r327420);
        double r327422 = r327413 - r327421;
        double r327423 = exp(r327413);
        double r327424 = r327417 - r327406;
        double r327425 = r327413 - r327406;
        double r327426 = r327424 / r327425;
        double r327427 = r327413 - r327426;
        double r327428 = r327423 / r327427;
        double r327429 = log(r327428);
        double r327430 = r327412 ? r327422 : r327429;
        return r327430;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.4
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 < -231188405.82678422 or 6764068.667720944 < y

    1. Initial program 47.1

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

    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

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

    if -231188405.82678422 < y < 6764068.667720944

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -231188405.8267842233180999755859375 \lor \neg \left(y \le 6764068.6677209436893463134765625\right):\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))