Average Error: 18.3 → 0.1
Time: 5.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\

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

\end{array}
double f(double x, double y) {
        double r370237 = 1.0;
        double r370238 = x;
        double r370239 = y;
        double r370240 = r370238 - r370239;
        double r370241 = r370237 - r370239;
        double r370242 = r370240 / r370241;
        double r370243 = r370237 - r370242;
        double r370244 = log(r370243);
        double r370245 = r370237 - r370244;
        return r370245;
}
↓
double f(double x, double y) {
        double r370246 = y;
        double r370247 = -150668776.17725816;
        bool r370248 = r370246 <= r370247;
        double r370249 = 15857199.25534847;
        bool r370250 = r370246 <= r370249;
        double r370251 = !r370250;
        bool r370252 = r370248 || r370251;
        double r370253 = 1.0;
        double r370254 = exp(r370253);
        double r370255 = x;
        double r370256 = 2.0;
        double r370257 = pow(r370246, r370256);
        double r370258 = r370255 / r370257;
        double r370259 = 1.0;
        double r370260 = r370259 / r370246;
        double r370261 = r370258 - r370260;
        double r370262 = r370255 / r370246;
        double r370263 = fma(r370253, r370261, r370262);
        double r370264 = r370254 / r370263;
        double r370265 = log(r370264);
        double r370266 = r370255 - r370246;
        double r370267 = r370253 - r370246;
        double r370268 = r370266 / r370267;
        double r370269 = r370253 - r370268;
        double r370270 = r370254 / r370269;
        double r370271 = log(r370270);
        double r370272 = r370252 ? r370265 : r370271;
        return r370272;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.3
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 < -150668776.17725816 or 15857199.25534847 < y

    1. Initial program 46.5

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

    3. Applied add-log-exp46.5

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

    4. Applied diff-log46.5

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -150668776.17725816 < y < 15857199.25534847

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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))))))