Average Error: 2.1 → 0.5
Time: 29.7s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot {e}^{\left(y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(\frac{1}{2} \cdot \frac{{z}^{2}}{{1}^{2}} + 1 \cdot z\right)\right) - b\right)\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot {e}^{\left(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)\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r113899 = x;
        double r113900 = y;
        double r113901 = z;
        double r113902 = log(r113901);
        double r113903 = t;
        double r113904 = r113902 - r113903;
        double r113905 = r113900 * r113904;
        double r113906 = a;
        double r113907 = 1.0;
        double r113908 = r113907 - r113901;
        double r113909 = log(r113908);
        double r113910 = b;
        double r113911 = r113909 - r113910;
        double r113912 = r113906 * r113911;
        double r113913 = r113905 + r113912;
        double r113914 = exp(r113913);
        double r113915 = r113899 * r113914;
        return r113915;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r113916 = x;
        double r113917 = exp(1.0);
        double r113918 = y;
        double r113919 = z;
        double r113920 = log(r113919);
        double r113921 = t;
        double r113922 = r113920 - r113921;
        double r113923 = r113918 * r113922;
        double r113924 = a;
        double r113925 = 1.0;
        double r113926 = log(r113925);
        double r113927 = 0.5;
        double r113928 = 2.0;
        double r113929 = pow(r113919, r113928);
        double r113930 = pow(r113925, r113928);
        double r113931 = r113929 / r113930;
        double r113932 = r113927 * r113931;
        double r113933 = r113925 * r113919;
        double r113934 = r113932 + r113933;
        double r113935 = r113926 - r113934;
        double r113936 = b;
        double r113937 = r113935 - r113936;
        double r113938 = r113924 * r113937;
        double r113939 = r113923 + r113938;
        double r113940 = pow(r113917, r113939);
        double r113941 = r113916 * r113940;
        return r113941;
}

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. Using strategy rm

  4. Applied *-un-lft-identity0.5

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

  5. Applied exp-prod0.5

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

  6. Simplified0.5

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

  7. Final simplification0.5

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

Reproduce

herbie shell --seed 2019298 
(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))))))