Average Error: 1.9 → 0.4
Time: 34.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r101888 = x;
        double r101889 = y;
        double r101890 = z;
        double r101891 = log(r101890);
        double r101892 = t;
        double r101893 = r101891 - r101892;
        double r101894 = r101889 * r101893;
        double r101895 = a;
        double r101896 = 1.0;
        double r101897 = r101896 - r101890;
        double r101898 = log(r101897);
        double r101899 = b;
        double r101900 = r101898 - r101899;
        double r101901 = r101895 * r101900;
        double r101902 = r101894 + r101901;
        double r101903 = exp(r101902);
        double r101904 = r101888 * r101903;
        return r101904;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r101905 = x;
        double r101906 = 1.0;
        double r101907 = log(r101906);
        double r101908 = 0.5;
        double r101909 = r101908 / r101906;
        double r101910 = z;
        double r101911 = 2.0;
        double r101912 = pow(r101910, r101911);
        double r101913 = r101912 / r101906;
        double r101914 = r101910 * r101906;
        double r101915 = fma(r101909, r101913, r101914);
        double r101916 = r101907 - r101915;
        double r101917 = b;
        double r101918 = r101916 - r101917;
        double r101919 = a;
        double r101920 = r101918 * r101919;
        double r101921 = log(r101910);
        double r101922 = t;
        double r101923 = r101921 - r101922;
        double r101924 = y;
        double r101925 = r101923 * r101924;
        double r101926 = r101920 + r101925;
        double r101927 = exp(r101926);
        double r101928 = r101905 * r101927;
        return r101928;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 1.9

    \[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]

  2. Taylor expanded around 0 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

    \[\leadsto x \cdot e^{\left(\left(\log 1 - \mathsf{fma}\left(\frac{\frac{1}{2}}{1}, \frac{{z}^{2}}{1}, z \cdot 1\right)\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))