Average Error: 18.4 → 0.3
Time: 4.7s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.3399084615053225566505545884865568950772:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.3399084615053225566505545884865568950772:\\
\;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - 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 r459940 = 1.0;
        double r459941 = x;
        double r459942 = y;
        double r459943 = r459941 - r459942;
        double r459944 = r459940 - r459942;
        double r459945 = r459943 / r459944;
        double r459946 = r459940 - r459945;
        double r459947 = log(r459946);
        double r459948 = r459940 - r459947;
        return r459948;
}
↓
double f(double x, double y) {
        double r459949 = x;
        double r459950 = y;
        double r459951 = r459949 - r459950;
        double r459952 = 1.0;
        double r459953 = r459952 - r459950;
        double r459954 = r459951 / r459953;
        double r459955 = 0.33990846150532256;
        bool r459956 = r459954 <= r459955;
        double r459957 = r459952 - r459954;
        double r459958 = log(r459957);
        double r459959 = r459952 - r459958;
        double r459960 = 2.0;
        double r459961 = pow(r459950, r459960);
        double r459962 = r459949 / r459961;
        double r459963 = 1.0;
        double r459964 = r459963 / r459950;
        double r459965 = r459962 - r459964;
        double r459966 = r459952 * r459965;
        double r459967 = r459949 / r459950;
        double r459968 = r459966 + r459967;
        double r459969 = log(r459968);
        double r459970 = r459952 - r459969;
        double r459971 = r459956 ? r459959 : r459970;
        return r459971;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original18.4
Target0.1
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.33990846150532256

    1. Initial program 0.0

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

    if 0.33990846150532256 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.2

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

    2. Taylor expanded around inf 0.8

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

    3. Simplified0.8

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.3399084615053225566505545884865568950772:\\ \;\;\;\;1 - \log \left(1 - \frac{x - y}{1 - y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 
(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))))))