Average Error: 18.5 → 0.1
Time: 9.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -32434980278.3643684 \lor \neg \left(y \le 47573649.5674214438\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -32434980278.3643684 \lor \neg \left(y \le 47573649.5674214438\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\right)\right)\\

\end{array}
double f(double x, double y) {
        double r348242 = 1.0;
        double r348243 = x;
        double r348244 = y;
        double r348245 = r348243 - r348244;
        double r348246 = r348242 - r348244;
        double r348247 = r348245 / r348246;
        double r348248 = r348242 - r348247;
        double r348249 = log(r348248);
        double r348250 = r348242 - r348249;
        return r348250;
}
↓
double f(double x, double y) {
        double r348251 = y;
        double r348252 = -32434980278.36437;
        bool r348253 = r348251 <= r348252;
        double r348254 = 47573649.567421444;
        bool r348255 = r348251 <= r348254;
        double r348256 = !r348255;
        bool r348257 = r348253 || r348256;
        double r348258 = 1.0;
        double r348259 = x;
        double r348260 = 2.0;
        double r348261 = pow(r348251, r348260);
        double r348262 = r348259 / r348261;
        double r348263 = 1.0;
        double r348264 = r348263 / r348251;
        double r348265 = r348262 - r348264;
        double r348266 = r348259 / r348251;
        double r348267 = fma(r348258, r348265, r348266);
        double r348268 = log(r348267);
        double r348269 = r348258 - r348268;
        double r348270 = r348259 - r348251;
        double r348271 = r348258 - r348251;
        double r348272 = cbrt(r348271);
        double r348273 = r348272 * r348272;
        double r348274 = r348270 / r348273;
        double r348275 = r348274 / r348272;
        double r348276 = r348258 - r348275;
        double r348277 = log1p(r348276);
        double r348278 = expm1(r348277);
        double r348279 = log(r348278);
        double r348280 = r348258 - r348279;
        double r348281 = r348257 ? r348269 : r348280;
        return r348281;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.5
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 < -32434980278.36437 or 47573649.567421444 < y

    1. Initial program 47.2

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

    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 -32434980278.36437 < y < 47573649.567421444

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.2

      \[\leadsto 1 - \log \left(1 - \frac{x - y}{\color{blue}{\left(\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}\right) \cdot \sqrt[3]{1 - y}}}\right)\]

    4. Applied associate-/r*0.2

      \[\leadsto 1 - \log \left(1 - \color{blue}{\frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}}\right)\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.2

      \[\leadsto 1 - \log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -32434980278.3643684 \lor \neg \left(y \le 47573649.5674214438\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \frac{\frac{x - y}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}}}{\sqrt[3]{1 - y}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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))))))