Average Error: 1.9 → 0.4
Time: 11.6s
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 r151720 = x;
        double r151721 = y;
        double r151722 = z;
        double r151723 = log(r151722);
        double r151724 = t;
        double r151725 = r151723 - r151724;
        double r151726 = r151721 * r151725;
        double r151727 = a;
        double r151728 = 1.0;
        double r151729 = r151728 - r151722;
        double r151730 = log(r151729);
        double r151731 = b;
        double r151732 = r151730 - r151731;
        double r151733 = r151727 * r151732;
        double r151734 = r151726 + r151733;
        double r151735 = exp(r151734);
        double r151736 = r151720 * r151735;
        return r151736;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r151737 = x;
        double r151738 = y;
        double r151739 = z;
        double r151740 = log(r151739);
        double r151741 = t;
        double r151742 = r151740 - r151741;
        double r151743 = r151738 * r151742;
        double r151744 = a;
        double r151745 = 1.0;
        double r151746 = log(r151745);
        double r151747 = 0.5;
        double r151748 = 2.0;
        double r151749 = pow(r151739, r151748);
        double r151750 = pow(r151745, r151748);
        double r151751 = r151749 / r151750;
        double r151752 = r151747 * r151751;
        double r151753 = r151745 * r151739;
        double r151754 = r151752 + r151753;
        double r151755 = r151746 - r151754;
        double r151756 = b;
        double r151757 = r151755 - r151756;
        double r151758 = r151744 * r151757;
        double r151759 = r151743 + r151758;
        double r151760 = exp(r151759);
        double r151761 = r151737 * r151760;
        return r151761;
}

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

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

  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 2020039 
(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))))))