Average Error: 2.0 → 0.4
Time: 38.7s
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 r122703 = x;
        double r122704 = y;
        double r122705 = z;
        double r122706 = log(r122705);
        double r122707 = t;
        double r122708 = r122706 - r122707;
        double r122709 = r122704 * r122708;
        double r122710 = a;
        double r122711 = 1.0;
        double r122712 = r122711 - r122705;
        double r122713 = log(r122712);
        double r122714 = b;
        double r122715 = r122713 - r122714;
        double r122716 = r122710 * r122715;
        double r122717 = r122709 + r122716;
        double r122718 = exp(r122717);
        double r122719 = r122703 * r122718;
        return r122719;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r122720 = 1.0;
        double r122721 = log(r122720);
        double r122722 = z;
        double r122723 = r122722 * r122720;
        double r122724 = r122721 - r122723;
        double r122725 = 0.5;
        double r122726 = r122725 / r122720;
        double r122727 = 2.0;
        double r122728 = pow(r122722, r122727);
        double r122729 = r122728 / r122720;
        double r122730 = r122726 * r122729;
        double r122731 = r122724 - r122730;
        double r122732 = b;
        double r122733 = r122731 - r122732;
        double r122734 = a;
        double r122735 = r122733 * r122734;
        double r122736 = log(r122722);
        double r122737 = t;
        double r122738 = r122736 - r122737;
        double r122739 = y;
        double r122740 = r122738 * r122739;
        double r122741 = r122735 + r122740;
        double r122742 = exp(r122741);
        double r122743 = x;
        double r122744 = r122742 * r122743;
        return r122744;
}

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 0.4

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

  3. Simplified0.4

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

  4. Final simplification0.4

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

Reproduce

herbie shell --seed 2019194 
(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))))))