Average Error: 18.6 → 0.1
Time: 24.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -240248535.9542663097381591796875:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 60168479.88061915338039398193359375:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -240248535.9542663097381591796875:\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

\mathbf{elif}\;y \le 60168479.88061915338039398193359375:\\
\;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r18522984 = 1.0;
        double r18522985 = x;
        double r18522986 = y;
        double r18522987 = r18522985 - r18522986;
        double r18522988 = r18522984 - r18522986;
        double r18522989 = r18522987 / r18522988;
        double r18522990 = r18522984 - r18522989;
        double r18522991 = log(r18522990);
        double r18522992 = r18522984 - r18522991;
        return r18522992;
}
↓
double f(double x, double y) {
        double r18522993 = y;
        double r18522994 = -240248535.9542663;
        bool r18522995 = r18522993 <= r18522994;
        double r18522996 = 1.0;
        double r18522997 = r18522996 / r18522993;
        double r18522998 = x;
        double r18522999 = r18522998 / r18522993;
        double r18523000 = r18522999 - r18522997;
        double r18523001 = fma(r18522997, r18522999, r18523000);
        double r18523002 = log(r18523001);
        double r18523003 = r18522996 - r18523002;
        double r18523004 = 60168479.88061915;
        bool r18523005 = r18522993 <= r18523004;
        double r18523006 = exp(r18522996);
        double r18523007 = r18522998 - r18522993;
        double r18523008 = r18522996 - r18522993;
        double r18523009 = r18523007 / r18523008;
        double r18523010 = r18522996 - r18523009;
        double r18523011 = r18523006 / r18523010;
        double r18523012 = log(r18523011);
        double r18523013 = r18523005 ? r18523012 : r18523003;
        double r18523014 = r18522995 ? r18523003 : r18523013;
        return r18523014;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.6
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 < -240248535.9542663 or 60168479.88061915 < y

    1. Initial program 47.0

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

    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

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)}\]

    if -240248535.9542663 < y < 60168479.88061915

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -240248535.9542663097381591796875:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 60168479.88061915338039398193359375:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))))))