Average Error: 18.1 → 0.1
Time: 14.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -212491914.1913111507892608642578125 \lor \neg \left(y \le 54137734.38243345916271209716796875\right):\\ \;\;\;\;1 - \log \left(\left(1 + \frac{1}{y}\right) \cdot \frac{x}{y} - \frac{1}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
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):\\
\;\;\;\;1 - \log \left(\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 r282121 = 1.0;
        double r282122 = x;
        double r282123 = y;
        double r282124 = r282122 - r282123;
        double r282125 = r282121 - r282123;
        double r282126 = r282124 / r282125;
        double r282127 = r282121 - r282126;
        double r282128 = log(r282127);
        double r282129 = r282121 - r282128;
        return r282129;
}
↓
double f(double x, double y) {
        double r282130 = y;
        double r282131 = -212491914.19131115;
        bool r282132 = r282130 <= r282131;
        double r282133 = 54137734.38243346;
        bool r282134 = r282130 <= r282133;
        double r282135 = !r282134;
        bool r282136 = r282132 || r282135;
        double r282137 = 1.0;
        double r282138 = 1.0;
        double r282139 = r282137 / r282130;
        double r282140 = r282138 + r282139;
        double r282141 = x;
        double r282142 = r282141 / r282130;
        double r282143 = r282140 * r282142;
        double r282144 = r282143 - r282139;
        double r282145 = log(r282144);
        double r282146 = r282137 - r282145;
        double r282147 = exp(r282137);
        double r282148 = r282141 - r282130;
        double r282149 = r282137 - r282130;
        double r282150 = r282148 / r282149;
        double r282151 = r282137 - r282150;
        double r282152 = r282147 / r282151;
        double r282153 = log(r282152);
        double r282154 = r282136 ? r282146 : r282153;
        return r282154;
}

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. 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

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

    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

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

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))))))