Average Error: 2.1 → 0.6
Time: 21.0s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{y \cdot \left(\log z - t\right) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}\]
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) + \left(a \cdot \left(\log 1 - b\right) - 1 \cdot \left(a \cdot z\right)\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r80901 = x;
        double r80902 = y;
        double r80903 = z;
        double r80904 = log(r80903);
        double r80905 = t;
        double r80906 = r80904 - r80905;
        double r80907 = r80902 * r80906;
        double r80908 = a;
        double r80909 = 1.0;
        double r80910 = r80909 - r80903;
        double r80911 = log(r80910);
        double r80912 = b;
        double r80913 = r80911 - r80912;
        double r80914 = r80908 * r80913;
        double r80915 = r80907 + r80914;
        double r80916 = exp(r80915);
        double r80917 = r80901 * r80916;
        return r80917;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r80918 = x;
        double r80919 = y;
        double r80920 = z;
        double r80921 = log(r80920);
        double r80922 = t;
        double r80923 = r80921 - r80922;
        double r80924 = r80919 * r80923;
        double r80925 = a;
        double r80926 = 1.0;
        double r80927 = log(r80926);
        double r80928 = b;
        double r80929 = r80927 - r80928;
        double r80930 = r80925 * r80929;
        double r80931 = r80925 * r80920;
        double r80932 = r80926 * r80931;
        double r80933 = r80930 - r80932;
        double r80934 = r80924 + r80933;
        double r80935 = exp(r80934);
        double r80936 = r80918 * r80935;
        return r80936;
}

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 0.6

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

  3. Simplified0.6

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

  4. Final simplification0.6

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

Reproduce

herbie shell --seed 2019326 
(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))))))