Average Error: 2.2 → 0.4
Time: 12.6s
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 r139596 = x;
        double r139597 = y;
        double r139598 = z;
        double r139599 = log(r139598);
        double r139600 = t;
        double r139601 = r139599 - r139600;
        double r139602 = r139597 * r139601;
        double r139603 = a;
        double r139604 = 1.0;
        double r139605 = r139604 - r139598;
        double r139606 = log(r139605);
        double r139607 = b;
        double r139608 = r139606 - r139607;
        double r139609 = r139603 * r139608;
        double r139610 = r139602 + r139609;
        double r139611 = exp(r139610);
        double r139612 = r139596 * r139611;
        return r139612;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r139613 = x;
        double r139614 = y;
        double r139615 = z;
        double r139616 = log(r139615);
        double r139617 = t;
        double r139618 = r139616 - r139617;
        double r139619 = r139614 * r139618;
        double r139620 = a;
        double r139621 = 1.0;
        double r139622 = log(r139621);
        double r139623 = 0.5;
        double r139624 = 2.0;
        double r139625 = pow(r139615, r139624);
        double r139626 = pow(r139621, r139624);
        double r139627 = r139625 / r139626;
        double r139628 = r139623 * r139627;
        double r139629 = r139621 * r139615;
        double r139630 = r139628 + r139629;
        double r139631 = r139622 - r139630;
        double r139632 = b;
        double r139633 = r139631 - r139632;
        double r139634 = r139620 * r139633;
        double r139635 = r139619 + r139634;
        double r139636 = exp(r139635);
        double r139637 = r139613 * r139636;
        return r139637;
}

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 2.2

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

  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 2020065 
(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))))))