Average Error: 18.3 → 0.1
Time: 5.4s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -199754789.0343833863735198974609375 \lor \neg \left(y \le 32910417.669902421534061431884765625\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;y \le -199754789.0343833863735198974609375 \lor \neg \left(y \le 32910417.669902421534061431884765625\right):\\
\;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\

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

\end{array}
double f(double x, double y) {
        double r329944 = 1.0;
        double r329945 = x;
        double r329946 = y;
        double r329947 = r329945 - r329946;
        double r329948 = r329944 - r329946;
        double r329949 = r329947 / r329948;
        double r329950 = r329944 - r329949;
        double r329951 = log(r329950);
        double r329952 = r329944 - r329951;
        return r329952;
}
↓
double f(double x, double y) {
        double r329953 = y;
        double r329954 = -199754789.0343834;
        bool r329955 = r329953 <= r329954;
        double r329956 = 32910417.66990242;
        bool r329957 = r329953 <= r329956;
        double r329958 = !r329957;
        bool r329959 = r329955 || r329958;
        double r329960 = 1.0;
        double r329961 = exp(r329960);
        double r329962 = x;
        double r329963 = 2.0;
        double r329964 = pow(r329953, r329963);
        double r329965 = r329962 / r329964;
        double r329966 = 1.0;
        double r329967 = r329966 / r329953;
        double r329968 = r329965 - r329967;
        double r329969 = r329962 / r329953;
        double r329970 = fma(r329960, r329968, r329969);
        double r329971 = r329961 / r329970;
        double r329972 = log(r329971);
        double r329973 = r329962 - r329953;
        double r329974 = r329960 - r329953;
        double r329975 = r329973 / r329974;
        double r329976 = r329960 - r329975;
        double r329977 = r329961 / r329976;
        double r329978 = log(r329977);
        double r329979 = r329959 ? r329972 : r329978;
        return r329979;
}

Error

Bits error versus x

Bits error versus y

Target

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

Derivation

  1. Split input into 2 regimes

  2. if y < -199754789.0343834 or 32910417.66990242 < y

    1. Initial program 46.8

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

    3. Applied add-log-exp46.8

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

    4. Applied diff-log46.8

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

    5. Taylor expanded around inf 0.1

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

    6. Simplified0.1

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

    if -199754789.0343834 < y < 32910417.66990242

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied diff-log0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -199754789.0343833863735198974609375 \lor \neg \left(y \le 32910417.669902421534061431884765625\right):\\ \;\;\;\;\log \left(\frac{e^{1}}{\mathsf{fma}\left(1, \frac{x}{{y}^{2}} - \frac{1}{y}, \frac{x}{y}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))))