Average Error: 18.1 → 0.1
Time: 16.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 54137734.38243345916271209716796875\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 54137734.38243345916271209716796875\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\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 r305925 = 1.0;
        double r305926 = x;
        double r305927 = y;
        double r305928 = r305926 - r305927;
        double r305929 = r305925 - r305927;
        double r305930 = r305928 / r305929;
        double r305931 = r305925 - r305930;
        double r305932 = log(r305931);
        double r305933 = r305925 - r305932;
        return r305933;
}
↓
double f(double x, double y) {
        double r305934 = y;
        double r305935 = -212491914.19131115;
        bool r305936 = r305934 <= r305935;
        double r305937 = 54137734.38243346;
        bool r305938 = r305934 <= r305937;
        double r305939 = !r305938;
        bool r305940 = r305936 || r305939;
        double r305941 = 1.0;
        double r305942 = exp(r305941);
        double r305943 = 1.0;
        double r305944 = r305941 / r305934;
        double r305945 = r305943 + r305944;
        double r305946 = x;
        double r305947 = r305946 / r305934;
        double r305948 = r305945 * r305947;
        double r305949 = r305948 - r305944;
        double r305950 = r305942 / r305949;
        double r305951 = log(r305950);
        double r305952 = r305946 - r305934;
        double r305953 = r305941 - r305934;
        double r305954 = r305952 / r305953;
        double r305955 = r305941 - r305954;
        double r305956 = r305942 / r305955;
        double r305957 = log(r305956);
        double r305958 = r305940 ? r305951 : r305957;
        return r305958;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.1
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 < -212491914.19131115 or 54137734.38243346 < y

    1. Initial program 46.7

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

    3. Applied add-log-exp46.7

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

    4. Applied diff-log46.7

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

    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

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

    if -212491914.19131115 < y < 54137734.38243346

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 54137734.38243345916271209716796875\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))