Average Error: 18.5 → 0.1
Time: 9.3s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\

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

\end{array}
double f(double x, double y) {
        double r385936 = 1.0;
        double r385937 = x;
        double r385938 = y;
        double r385939 = r385937 - r385938;
        double r385940 = r385936 - r385938;
        double r385941 = r385939 / r385940;
        double r385942 = r385936 - r385941;
        double r385943 = log(r385942);
        double r385944 = r385936 - r385943;
        return r385944;
}
↓
double f(double x, double y) {
        double r385945 = y;
        double r385946 = -119577009.99898484;
        bool r385947 = r385945 <= r385946;
        double r385948 = 25589815.688386947;
        bool r385949 = r385945 <= r385948;
        double r385950 = !r385949;
        bool r385951 = r385947 || r385950;
        double r385952 = 1.0;
        double r385953 = exp(r385952);
        double r385954 = x;
        double r385955 = 2.0;
        double r385956 = pow(r385945, r385955);
        double r385957 = r385954 / r385956;
        double r385958 = r385954 / r385945;
        double r385959 = fma(r385952, r385957, r385958);
        double r385960 = r385952 / r385945;
        double r385961 = r385959 - r385960;
        double r385962 = r385953 / r385961;
        double r385963 = log(r385962);
        double r385964 = r385954 - r385945;
        double r385965 = 1.0;
        double r385966 = r385952 - r385945;
        double r385967 = r385965 / r385966;
        double r385968 = r385964 * r385967;
        double r385969 = r385952 - r385968;
        double r385970 = log(r385969);
        double r385971 = r385952 - r385970;
        double r385972 = r385951 ? r385963 : r385971;
        return r385972;
}

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 < -119577009.99898484 or 25589815.688386947 < y

    1. Initial program 46.9

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

    3. Applied add-log-exp46.9

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

    4. Applied diff-log46.9

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

    5. Taylor expanded around inf 0.2

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

    6. Simplified0.2

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

    if -119577009.99898484 < y < 25589815.688386947

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -119577009.99898484 \lor \neg \left(y \le 25589815.688386947\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}}, \frac{x}{y}\right) - \frac{1}{y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(1 - \left(x - y\right) \cdot \frac{1}{1 - y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +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))))))