Average Error: 18.0 → 0.2
Time: 7.0s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999975065411717:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999975065411717:\\
\;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\

\mathbf{else}:\\
\;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\

\end{array}
double f(double x, double y) {
        double r359054 = 1.0;
        double r359055 = x;
        double r359056 = y;
        double r359057 = r359055 - r359056;
        double r359058 = r359054 - r359056;
        double r359059 = r359057 / r359058;
        double r359060 = r359054 - r359059;
        double r359061 = log(r359060);
        double r359062 = r359054 - r359061;
        return r359062;
}
↓
double f(double x, double y) {
        double r359063 = x;
        double r359064 = y;
        double r359065 = r359063 - r359064;
        double r359066 = 1.0;
        double r359067 = r359066 - r359064;
        double r359068 = r359065 / r359067;
        double r359069 = 0.9999997506541172;
        bool r359070 = r359068 <= r359069;
        double r359071 = cbrt(r359065);
        double r359072 = r359071 * r359071;
        double r359073 = r359067 / r359071;
        double r359074 = r359072 / r359073;
        double r359075 = r359066 - r359074;
        double r359076 = log(r359075);
        double r359077 = r359066 - r359076;
        double r359078 = 2.0;
        double r359079 = pow(r359064, r359078);
        double r359080 = r359063 / r359079;
        double r359081 = 1.0;
        double r359082 = r359081 / r359064;
        double r359083 = r359080 - r359082;
        double r359084 = r359066 * r359083;
        double r359085 = r359063 / r359064;
        double r359086 = r359084 + r359085;
        double r359087 = log(r359086);
        double r359088 = r359066 - r359087;
        double r359089 = r359070 ? r359077 : r359088;
        return r359089;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.0
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.619472414:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 3.0094271212461764 \cdot 10^{25}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9999997506541172

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/l*0.1

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

    if 0.9999997506541172 < (/ (- x y) (- 1.0 y))

    1. Initial program 62.3

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

    3. Applied add-cube-cbrt59.3

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

    4. Applied associate-/l*59.3

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

    5. Taylor expanded around inf 0.3

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

    6. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.99999975065411717:\\ \;\;\;\;1 - \log \left(1 - \frac{\sqrt[3]{x - y} \cdot \sqrt[3]{x - y}}{\frac{1 - y}{\sqrt[3]{x - y}}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 +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))))))