Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B

Average Error: 18.5 → 0.1

Time: 25.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, -\left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right) + \mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

C

double f(double x, double y) {
        double r21838950 = 1.0;
        double r21838951 = x;
        double r21838952 = y;
        double r21838953 = r21838951 - r21838952;
        double r21838954 = r21838950 - r21838952;
        double r21838955 = r21838953 / r21838954;
        double r21838956 = r21838950 - r21838955;
        double r21838957 = log(r21838956);
        double r21838958 = r21838950 - r21838957;
        return r21838958;
}

↓

double f(double x, double y) {
        double r21838959 = y;
        double r21838960 = -34474611802046.883;
        bool r21838961 = r21838959 <= r21838960;
        double r21838962 = 1.0;
        double r21838963 = x;
        double r21838964 = r21838963 / r21838959;
        double r21838965 = r21838962 / r21838959;
        double r21838966 = r21838964 - r21838965;
        double r21838967 = fma(r21838964, r21838965, r21838966);
        double r21838968 = log(r21838967);
        double r21838969 = r21838962 - r21838968;
        double r21838970 = 43744445.70071104;
        bool r21838971 = r21838959 <= r21838970;
        double r21838972 = 1.0;
        double r21838973 = r21838962 + r21838959;
        double r21838974 = r21838963 - r21838959;
        double r21838975 = r21838962 * r21838962;
        double r21838976 = r21838959 * r21838959;
        double r21838977 = r21838975 - r21838976;
        double r21838978 = r21838974 / r21838977;
        double r21838979 = r21838973 * r21838978;
        double r21838980 = -r21838979;
        double r21838981 = fma(r21838972, r21838962, r21838980);
        double r21838982 = -r21838973;
        double r21838983 = fma(r21838982, r21838978, r21838979);
        double r21838984 = r21838981 + r21838983;
        double r21838985 = log(r21838984);
        double r21838986 = r21838962 - r21838985;
        double r21838987 = r21838971 ? r21838986 : r21838969;
        double r21838988 = r21838961 ? r21838969 : r21838987;
        return r21838988;
}

Error

Bits error versus x

Bits error versus y

Target

Original	18.5
Target	0.1
Herbie	0.1

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

Derivation

1. Split input into 2 regimes

2. if y < -34474611802046.883 or 43744445.70071104 < y

   1. Initial program 47.4

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

   2. Taylor expanded around inf 0.0

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

   3. Simplified 0.0

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

   if -34474611802046.883 < y < 43744445.70071104

   1. Initial program 0.2

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

   3. Applied flip-- 0.2

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

   4. Applied associate-/r/ 0.2

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

   5. Applied *-un-lft-identity 0.2

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

   6. Applied prod-diff 0.2

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

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -34474611802046.8828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \mathbf{elif}\;y \le 43744445.7007110416889190673828125:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(1, 1, -\left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right) + \mathsf{fma}\left(-\left(1 + y\right), \frac{x - y}{1 \cdot 1 - y \cdot y}, \left(1 + y\right) \cdot \frac{x - y}{1 \cdot 1 - y \cdot y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\mathsf{fma}\left(\frac{x}{y}, \frac{1}{y}, \frac{x}{y} - \frac{1}{y}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +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))))))