Average Error: 18.0 → 0.1
Time: 22.6s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -166149344.053797900676727294921875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 89236118.13017952442169189453125:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -166149344.053797900676727294921875:\\
\;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\

\mathbf{elif}\;y \le 89236118.13017952442169189453125:\\
\;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\

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

\end{array}
double f(double x, double y) {
        double r18080537 = 1.0;
        double r18080538 = x;
        double r18080539 = y;
        double r18080540 = r18080538 - r18080539;
        double r18080541 = r18080537 - r18080539;
        double r18080542 = r18080540 / r18080541;
        double r18080543 = r18080537 - r18080542;
        double r18080544 = log(r18080543);
        double r18080545 = r18080537 - r18080544;
        return r18080545;
}
↓
double f(double x, double y) {
        double r18080546 = y;
        double r18080547 = -166149344.0537979;
        bool r18080548 = r18080546 <= r18080547;
        double r18080549 = 1.0;
        double r18080550 = 1.0;
        double r18080551 = r18080549 / r18080546;
        double r18080552 = r18080550 + r18080551;
        double r18080553 = x;
        double r18080554 = r18080553 / r18080546;
        double r18080555 = r18080552 * r18080554;
        double r18080556 = r18080555 - r18080551;
        double r18080557 = log(r18080556);
        double r18080558 = r18080549 - r18080557;
        double r18080559 = 89236118.13017952;
        bool r18080560 = r18080546 <= r18080559;
        double r18080561 = r18080553 - r18080546;
        double r18080562 = r18080549 - r18080546;
        double r18080563 = r18080561 / r18080562;
        double r18080564 = r18080549 - r18080563;
        double r18080565 = sqrt(r18080564);
        double r18080566 = log(r18080565);
        double r18080567 = r18080549 - r18080566;
        double r18080568 = log(r18080564);
        double r18080569 = 0.5;
        double r18080570 = r18080568 * r18080569;
        double r18080571 = r18080567 - r18080570;
        double r18080572 = r18080560 ? r18080571 : r18080558;
        double r18080573 = r18080548 ? r18080558 : r18080572;
        return r18080573;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.0
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 < -166149344.0537979 or 89236118.13017952 < y

    1. Initial program 46.3

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

    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

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

    if -166149344.0537979 < y < 89236118.13017952

    1. Initial program 0.1

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

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

    7. Applied pow1/20.1

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

    8. Applied log-pow0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -166149344.053797900676727294921875:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{elif}\;y \le 89236118.13017952442169189453125:\\ \;\;\;\;\left(1 - \log \left(\sqrt{1 - \frac{x - y}{1 - y}}\right)\right) - \log \left(1 - \frac{x - y}{1 - y}\right) \cdot \frac{1}{2}\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))))