Average Error: 2.0 → 0.5
Time: 11.4s
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 r106598 = x;
        double r106599 = y;
        double r106600 = z;
        double r106601 = log(r106600);
        double r106602 = t;
        double r106603 = r106601 - r106602;
        double r106604 = r106599 * r106603;
        double r106605 = a;
        double r106606 = 1.0;
        double r106607 = r106606 - r106600;
        double r106608 = log(r106607);
        double r106609 = b;
        double r106610 = r106608 - r106609;
        double r106611 = r106605 * r106610;
        double r106612 = r106604 + r106611;
        double r106613 = exp(r106612);
        double r106614 = r106598 * r106613;
        return r106614;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r106615 = x;
        double r106616 = y;
        double r106617 = z;
        double r106618 = log(r106617);
        double r106619 = t;
        double r106620 = r106618 - r106619;
        double r106621 = r106616 * r106620;
        double r106622 = a;
        double r106623 = 1.0;
        double r106624 = log(r106623);
        double r106625 = 0.5;
        double r106626 = 2.0;
        double r106627 = pow(r106617, r106626);
        double r106628 = pow(r106623, r106626);
        double r106629 = r106627 / r106628;
        double r106630 = r106625 * r106629;
        double r106631 = r106623 * r106617;
        double r106632 = r106630 + r106631;
        double r106633 = r106624 - r106632;
        double r106634 = b;
        double r106635 = r106633 - r106634;
        double r106636 = r106622 * r106635;
        double r106637 = r106621 + r106636;
        double r106638 = exp(r106637);
        double r106639 = r106615 * r106638;
        return r106639;
}

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 2.0

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

  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. Final simplification0.5

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

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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))))))