Average Error: 1.9 → 0.5
Time: 29.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[e^{\left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{1} \cdot \frac{{z}^{2}}{1}\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
e^{\left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{1} \cdot \frac{{z}^{2}}{1}\right) - b\right) \cdot a + \left(\log z - t\right) \cdot y} \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r94587 = x;
        double r94588 = y;
        double r94589 = z;
        double r94590 = log(r94589);
        double r94591 = t;
        double r94592 = r94590 - r94591;
        double r94593 = r94588 * r94592;
        double r94594 = a;
        double r94595 = 1.0;
        double r94596 = r94595 - r94589;
        double r94597 = log(r94596);
        double r94598 = b;
        double r94599 = r94597 - r94598;
        double r94600 = r94594 * r94599;
        double r94601 = r94593 + r94600;
        double r94602 = exp(r94601);
        double r94603 = r94587 * r94602;
        return r94603;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r94604 = 1.0;
        double r94605 = log(r94604);
        double r94606 = z;
        double r94607 = r94606 * r94604;
        double r94608 = r94605 - r94607;
        double r94609 = 0.5;
        double r94610 = r94609 / r94604;
        double r94611 = 2.0;
        double r94612 = pow(r94606, r94611);
        double r94613 = r94612 / r94604;
        double r94614 = r94610 * r94613;
        double r94615 = r94608 - r94614;
        double r94616 = b;
        double r94617 = r94615 - r94616;
        double r94618 = a;
        double r94619 = r94617 * r94618;
        double r94620 = log(r94606);
        double r94621 = t;
        double r94622 = r94620 - r94621;
        double r94623 = y;
        double r94624 = r94622 * r94623;
        double r94625 = r94619 + r94624;
        double r94626 = exp(r94625);
        double r94627 = x;
        double r94628 = r94626 * r94627;
        return r94628;
}

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 1.9

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, B"
  (* x (exp (+ (* y (- (log z) t)) (* a (- (log (- 1.0 z)) b))))))