Average Error: 18.5 → 0.1
Time: 7.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + \left(1 \cdot \frac{x}{{y}^{2}} - \frac{1}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + \left(1 \cdot \frac{x}{{y}^{2}} - \frac{1}{y}\right)}\right)\\

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

\end{array}
double f(double x, double y) {
        double r477539 = 1.0;
        double r477540 = x;
        double r477541 = y;
        double r477542 = r477540 - r477541;
        double r477543 = r477539 - r477541;
        double r477544 = r477542 / r477543;
        double r477545 = r477539 - r477544;
        double r477546 = log(r477545);
        double r477547 = r477539 - r477546;
        return r477547;
}
↓
double f(double x, double y) {
        double r477548 = y;
        double r477549 = -119577009.99898484;
        bool r477550 = r477548 <= r477549;
        double r477551 = 25589815.688386947;
        bool r477552 = r477548 <= r477551;
        double r477553 = !r477552;
        bool r477554 = r477550 || r477553;
        double r477555 = 1.0;
        double r477556 = exp(r477555);
        double r477557 = x;
        double r477558 = r477557 / r477548;
        double r477559 = 2.0;
        double r477560 = pow(r477548, r477559);
        double r477561 = r477557 / r477560;
        double r477562 = r477555 * r477561;
        double r477563 = r477555 / r477548;
        double r477564 = r477562 - r477563;
        double r477565 = r477558 + r477564;
        double r477566 = r477556 / r477565;
        double r477567 = log(r477566);
        double r477568 = r477557 - r477548;
        double r477569 = 1.0;
        double r477570 = r477555 - r477548;
        double r477571 = r477569 / r477570;
        double r477572 = r477568 * r477571;
        double r477573 = r477555 - r477572;
        double r477574 = log(r477573);
        double r477575 = r477555 - r477574;
        double r477576 = r477554 ? r477567 : r477575;
        return r477576;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.5
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 < -119577009.99898484 or 25589815.688386947 < y

    1. Initial program 46.9

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

    3. Applied add-log-exp46.9

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

    4. Applied diff-log46.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -119577009.99898484 < y < 25589815.688386947

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{x}{y} + \left(1 \cdot \frac{x}{{y}^{2}} - \frac{1}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))))))