Average Error: 1.9 → 0.4
Time: 24.4s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[e^{a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
e^{a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + \left(\log z - t\right) \cdot y} \cdot x
double f(double x, double y, double z, double t, double a, double b) {
        double r5186674 = x;
        double r5186675 = y;
        double r5186676 = z;
        double r5186677 = log(r5186676);
        double r5186678 = t;
        double r5186679 = r5186677 - r5186678;
        double r5186680 = r5186675 * r5186679;
        double r5186681 = a;
        double r5186682 = 1.0;
        double r5186683 = r5186682 - r5186676;
        double r5186684 = log(r5186683);
        double r5186685 = b;
        double r5186686 = r5186684 - r5186685;
        double r5186687 = r5186681 * r5186686;
        double r5186688 = r5186680 + r5186687;
        double r5186689 = exp(r5186688);
        double r5186690 = r5186674 * r5186689;
        return r5186690;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r5186691 = a;
        double r5186692 = 1.0;
        double r5186693 = log(r5186692);
        double r5186694 = z;
        double r5186695 = r5186694 * r5186692;
        double r5186696 = r5186693 - r5186695;
        double r5186697 = 0.5;
        double r5186698 = r5186692 / r5186694;
        double r5186699 = r5186698 * r5186698;
        double r5186700 = r5186697 / r5186699;
        double r5186701 = r5186696 - r5186700;
        double r5186702 = b;
        double r5186703 = r5186701 - r5186702;
        double r5186704 = r5186691 * r5186703;
        double r5186705 = log(r5186694);
        double r5186706 = t;
        double r5186707 = r5186705 - r5186706;
        double r5186708 = y;
        double r5186709 = r5186707 * r5186708;
        double r5186710 = r5186704 + r5186709;
        double r5186711 = exp(r5186710);
        double r5186712 = x;
        double r5186713 = r5186711 * r5186712;
        return r5186713;
}

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 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(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))))))