Average Error: 18.4 → 0.2
Time: 20.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -47727873.56991197168827056884765625 \lor \neg \left(y \le 0.996418055055319551449599657644284889102\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{\sqrt[3]{x - y}}{\sqrt{1 - y}}, -\frac{\sqrt[3]{x - y}}{\frac{\sqrt{1 - y}}{\sqrt[3]{x - y}}}, 1\right) + \left(\frac{\sqrt[3]{x - y}}{\sqrt{1 - y}} + \frac{-\sqrt[3]{x - y}}{\sqrt{1 - y}}\right) \cdot \left(\sqrt[3]{x - y} \cdot \frac{\sqrt[3]{x - y}}{\sqrt{1 - y}}\right)\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -47727873.56991197168827056884765625 \lor \neg \left(y \le 0.996418055055319551449599657644284889102\right):\\
\;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\

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

\end{array}
double f(double x, double y) {
        double r362177 = 1.0;
        double r362178 = x;
        double r362179 = y;
        double r362180 = r362178 - r362179;
        double r362181 = r362177 - r362179;
        double r362182 = r362180 / r362181;
        double r362183 = r362177 - r362182;
        double r362184 = log(r362183);
        double r362185 = r362177 - r362184;
        return r362185;
}
↓
double f(double x, double y) {
        double r362186 = y;
        double r362187 = -47727873.56991197;
        bool r362188 = r362186 <= r362187;
        double r362189 = 0.9964180550553196;
        bool r362190 = r362186 <= r362189;
        double r362191 = !r362190;
        bool r362192 = r362188 || r362191;
        double r362193 = 1.0;
        double r362194 = r362193 / r362186;
        double r362195 = x;
        double r362196 = r362195 / r362186;
        double r362197 = r362196 - r362194;
        double r362198 = fma(r362194, r362196, r362197);
        double r362199 = log(r362198);
        double r362200 = r362193 - r362199;
        double r362201 = r362195 - r362186;
        double r362202 = cbrt(r362201);
        double r362203 = r362193 - r362186;
        double r362204 = sqrt(r362203);
        double r362205 = r362202 / r362204;
        double r362206 = r362204 / r362202;
        double r362207 = r362202 / r362206;
        double r362208 = -r362207;
        double r362209 = fma(r362205, r362208, r362193);
        double r362210 = -r362202;
        double r362211 = r362210 / r362204;
        double r362212 = r362205 + r362211;
        double r362213 = r362202 * r362205;
        double r362214 = r362212 * r362213;
        double r362215 = r362209 + r362214;
        double r362216 = log(r362215);
        double r362217 = r362193 - r362216;
        double r362218 = r362192 ? r362200 : r362217;
        return r362218;
}

Error

Bits error versus x

Bits error versus y

Target

Original18.4
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -47727873.56991197 or 0.9964180550553196 < y

    1. Initial program 47.2

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

    if -47727873.56991197 < y < 0.9964180550553196

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied times-frac0.1

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

    6. Applied add-sqr-sqrt0.1

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

    7. Applied prod-diff0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -47727873.56991197168827056884765625 \lor \neg \left(y \le 0.996418055055319551449599657644284889102\right):\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{1}{y}, \frac{x}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{\sqrt[3]{x - y}}{\sqrt{1 - y}}, -\frac{\sqrt[3]{x - y}}{\frac{\sqrt{1 - y}}{\sqrt[3]{x - y}}}, 1\right) + \left(\frac{\sqrt[3]{x - y}}{\sqrt{1 - y}} + \frac{-\sqrt[3]{x - y}}{\sqrt{1 - y}}\right) \cdot \left(\sqrt[3]{x - y} \cdot \frac{\sqrt[3]{x - y}}{\sqrt{1 - y}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"

  :herbie-target
  (if (< y -81284752.61947241) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y))))) (if (< y 3.0094271212461764e+25) (log (/ (exp 1.0) (- 1.0 (/ (- x y) (- 1.0 y))))) (- 1.0 (log (- (/ x (* y y)) (- (/ 1.0 y) (/ x y)))))))

  (- 1.0 (log (- 1.0 (/ (- x y) (- 1.0 y))))))