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

↓

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

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):\\
\;\;\;\;1 - \log \left(\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 r2145 = 1.0;
        double r2146 = x;
        double r2147 = y;
        double r2148 = r2146 - r2147;
        double r2149 = r2145 - r2147;
        double r2150 = r2148 / r2149;
        double r2151 = r2145 - r2150;
        double r2152 = log(r2151);
        double r2153 = r2145 - r2152;
        return r2153;
}

↓

double f(double x, double y) {
        double r2154 = y;
        double r2155 = -150668776.17725816;
        bool r2156 = r2154 <= r2155;
        double r2157 = 15857199.25534847;
        bool r2158 = r2154 <= r2157;
        double r2159 = !r2158;
        bool r2160 = r2156 || r2159;
        double r2161 = 1.0;
        double r2162 = x;
        double r2163 = 2.0;
        double r2164 = pow(r2154, r2163);
        double r2165 = r2162 / r2164;
        double r2166 = 1.0;
        double r2167 = r2166 / r2154;
        double r2168 = r2165 - r2167;
        double r2169 = r2162 / r2154;
        double r2170 = fma(r2161, r2168, r2169);
        double r2171 = log(r2170);
        double r2172 = r2161 - r2171;
        double r2173 = exp(r2161);
        double r2174 = r2162 - r2154;
        double r2175 = r2161 - r2154;
        double r2176 = r2174 / r2175;
        double r2177 = r2161 - r2176;
        double r2178 = r2173 / r2177;
        double r2179 = log(r2178);
        double r2180 = r2160 ? r2172 : r2179;
        return r2180;
}

Original	18.3
Target	0.1
Herbie	0.1

Derivation

Split input into 2 regimes
if y < -150668776.17725816 or 15857199.25534847 < y
1. Initial program 46.5
  \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
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
  \[\leadsto 1 - \log \color{blue}{\left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)}\]
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
  \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;y \le -150668776.177258164 \lor \neg \left(y \le 15857199.25534847\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

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))))))

Error

Target

Derivation

`if y < -150668776.17725816 or 15857199.25534847 < y`

`if -150668776.17725816 < y < 15857199.25534847`

Reproduce