Average Error: 2.1 → 0.5
Time: 14.9s
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 r144836 = x;
        double r144837 = y;
        double r144838 = z;
        double r144839 = log(r144838);
        double r144840 = t;
        double r144841 = r144839 - r144840;
        double r144842 = r144837 * r144841;
        double r144843 = a;
        double r144844 = 1.0;
        double r144845 = r144844 - r144838;
        double r144846 = log(r144845);
        double r144847 = b;
        double r144848 = r144846 - r144847;
        double r144849 = r144843 * r144848;
        double r144850 = r144842 + r144849;
        double r144851 = exp(r144850);
        double r144852 = r144836 * r144851;
        return r144852;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r144853 = x;
        double r144854 = y;
        double r144855 = z;
        double r144856 = log(r144855);
        double r144857 = t;
        double r144858 = r144856 - r144857;
        double r144859 = r144854 * r144858;
        double r144860 = a;
        double r144861 = 1.0;
        double r144862 = log(r144861);
        double r144863 = 0.5;
        double r144864 = 2.0;
        double r144865 = pow(r144855, r144864);
        double r144866 = pow(r144861, r144864);
        double r144867 = r144865 / r144866;
        double r144868 = r144863 * r144867;
        double r144869 = r144861 * r144855;
        double r144870 = r144868 + r144869;
        double r144871 = r144862 - r144870;
        double r144872 = b;
        double r144873 = r144871 - r144872;
        double r144874 = r144860 * r144873;
        double r144875 = r144859 + r144874;
        double r144876 = exp(r144875);
        double r144877 = r144853 * r144876;
        return r144877;
}

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 2.1

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

  2. Taylor expanded around 0 0.5

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

  3. Final simplification0.5

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

Reproduce

herbie shell --seed 2020003 
(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))))))