Average Error: 1.8 → 0.4
Time: 9.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r86858 = x;
        double r86859 = y;
        double r86860 = z;
        double r86861 = log(r86860);
        double r86862 = t;
        double r86863 = r86861 - r86862;
        double r86864 = r86859 * r86863;
        double r86865 = a;
        double r86866 = 1.0;
        double r86867 = r86866 - r86860;
        double r86868 = log(r86867);
        double r86869 = b;
        double r86870 = r86868 - r86869;
        double r86871 = r86865 * r86870;
        double r86872 = r86864 + r86871;
        double r86873 = exp(r86872);
        double r86874 = r86858 * r86873;
        return r86874;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r86875 = x;
        double r86876 = y;
        double r86877 = z;
        double r86878 = log(r86877);
        double r86879 = t;
        double r86880 = r86878 - r86879;
        double r86881 = r86876 * r86880;
        double r86882 = a;
        double r86883 = 1.0;
        double r86884 = log(r86883);
        double r86885 = 0.5;
        double r86886 = 2.0;
        double r86887 = pow(r86877, r86886);
        double r86888 = pow(r86883, r86886);
        double r86889 = r86887 / r86888;
        double r86890 = r86885 * r86889;
        double r86891 = r86883 * r86877;
        double r86892 = r86890 + r86891;
        double r86893 = r86884 - r86892;
        double r86894 = b;
        double r86895 = r86893 - r86894;
        double r86896 = r86882 * r86895;
        double r86897 = r86881 + r86896;
        double r86898 = exp(r86897);
        double r86899 = r86875 * r86898;
        return r86899;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.8

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

  2. Taylor expanded around 0 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  :precision binary64
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1 z)) b))))))