Average Error: 18.7 → 0.9
Time: 18.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 1.537598818068412357583554970976054410148 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{1 \cdot x}{y \cdot y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 1.537598818068412357583554970976054410148 \cdot 10^{-9}:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r319536 = 1.0;
        double r319537 = x;
        double r319538 = y;
        double r319539 = r319537 - r319538;
        double r319540 = r319536 - r319538;
        double r319541 = r319539 / r319540;
        double r319542 = r319536 - r319541;
        double r319543 = log(r319542);
        double r319544 = r319536 - r319543;
        return r319544;
}
↓
double f(double x, double y) {
        double r319545 = x;
        double r319546 = y;
        double r319547 = r319545 - r319546;
        double r319548 = 1.0;
        double r319549 = r319548 - r319546;
        double r319550 = r319547 / r319549;
        double r319551 = 1.5375988180684124e-09;
        bool r319552 = r319550 <= r319551;
        double r319553 = exp(r319548);
        double r319554 = r319548 - r319550;
        double r319555 = r319553 / r319554;
        double r319556 = log(r319555);
        double r319557 = r319548 * r319545;
        double r319558 = r319546 * r319546;
        double r319559 = r319557 / r319558;
        double r319560 = r319545 / r319546;
        double r319561 = r319548 / r319546;
        double r319562 = r319560 - r319561;
        double r319563 = r319559 + r319562;
        double r319564 = r319553 / r319563;
        double r319565 = log(r319564);
        double r319566 = r319552 ? r319556 : r319565;
        return r319566;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.7
Target0.1
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 1.5375988180684124e-09

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied diff-log0.0

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

    if 1.5375988180684124e-09 < (/ (- x y) (- 1.0 y))

    1. Initial program 59.1

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

    3. Applied add-log-exp59.1

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

    4. Applied diff-log59.1

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

    5. Taylor expanded around inf 2.9

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

    6. Simplified2.9

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 1.537598818068412357583554970976054410148 \cdot 10^{-9}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{\frac{1 \cdot x}{y \cdot y} + \left(\frac{x}{y} - \frac{1}{y}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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))))))