Average Error: 18.3 → 0.1
Time: 7.9s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y} - 1 \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y} - 1 \cdot \frac{1}{y}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r340165 = 1.0;
        double r340166 = x;
        double r340167 = y;
        double r340168 = r340166 - r340167;
        double r340169 = r340165 - r340167;
        double r340170 = r340168 / r340169;
        double r340171 = r340165 - r340170;
        double r340172 = log(r340171);
        double r340173 = r340165 - r340172;
        return r340173;
}
↓
double f(double x, double y) {
        double r340174 = y;
        double r340175 = -1864702967354.4304;
        bool r340176 = r340174 <= r340175;
        double r340177 = 28223629.68475053;
        bool r340178 = r340174 <= r340177;
        double r340179 = !r340178;
        bool r340180 = r340176 || r340179;
        double r340181 = 1.0;
        double r340182 = x;
        double r340183 = 2.0;
        double r340184 = pow(r340174, r340183);
        double r340185 = r340182 / r340184;
        double r340186 = r340182 / r340174;
        double r340187 = 1.0;
        double r340188 = r340187 / r340174;
        double r340189 = r340181 * r340188;
        double r340190 = r340186 - r340189;
        double r340191 = fma(r340181, r340185, r340190);
        double r340192 = log(r340191);
        double r340193 = r340181 - r340192;
        double r340194 = r340181 - r340174;
        double r340195 = cbrt(r340194);
        double r340196 = r340195 * r340195;
        double r340197 = r340187 / r340196;
        double r340198 = r340182 - r340174;
        double r340199 = r340198 / r340195;
        double r340200 = r340197 * r340199;
        double r340201 = r340181 - r340200;
        double r340202 = log(r340201);
        double r340203 = r340181 - r340202;
        double r340204 = r340180 ? r340193 : r340203;
        return r340204;
}

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 < -1864702967354.4304 or 28223629.68475053 < y

    1. Initial program 46.6

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

    3. Applied add-cube-cbrt42.9

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

    4. Applied *-un-lft-identity42.9

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

    5. Applied times-frac42.8

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -1864702967354.4304 < y < 28223629.68475053

    1. Initial program 0.2

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    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 *-un-lft-identity0.2

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

    5. Applied times-frac0.2

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1864702967354.43042 \lor \neg \left(y \le 28223629.6847505309\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y} - 1 \cdot \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \frac{1}{\sqrt[3]{1 - y} \cdot \sqrt[3]{1 - y}} \cdot \frac{x - y}{\sqrt[3]{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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))))))