Average Error: 2.2 → 3.4
Time: 37.5s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -1.436503598316956953340195468626916408539 \lor \neg \left(y \le 8.989039952438387375008659575680983873271 \cdot 10^{120}\right):\\ \;\;\;\;x \cdot e^{a \cdot \log 1 + \left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(a \cdot b + \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot {z}^{y}\right)\\ \end{array}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -1.436503598316956953340195468626916408539 \lor \neg \left(y \le 8.989039952438387375008659575680983873271 \cdot 10^{120}\right):\\
\;\;\;\;x \cdot e^{a \cdot \log 1 + \left(\log z - t\right) \cdot y}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(a \cdot b + \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot {z}^{y}\right)\\

\end{array}
double f(double x, double y, double z, double t, double a, double b) {
        double r99758 = x;
        double r99759 = y;
        double r99760 = z;
        double r99761 = log(r99760);
        double r99762 = t;
        double r99763 = r99761 - r99762;
        double r99764 = r99759 * r99763;
        double r99765 = a;
        double r99766 = 1.0;
        double r99767 = r99766 - r99760;
        double r99768 = log(r99767);
        double r99769 = b;
        double r99770 = r99768 - r99769;
        double r99771 = r99765 * r99770;
        double r99772 = r99764 + r99771;
        double r99773 = exp(r99772);
        double r99774 = r99758 * r99773;
        return r99774;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r99775 = y;
        double r99776 = -1.436503598316957;
        bool r99777 = r99775 <= r99776;
        double r99778 = 8.989039952438387e+120;
        bool r99779 = r99775 <= r99778;
        double r99780 = !r99779;
        bool r99781 = r99777 || r99780;
        double r99782 = x;
        double r99783 = a;
        double r99784 = 1.0;
        double r99785 = log(r99784);
        double r99786 = r99783 * r99785;
        double r99787 = z;
        double r99788 = log(r99787);
        double r99789 = t;
        double r99790 = r99788 - r99789;
        double r99791 = r99790 * r99775;
        double r99792 = r99786 + r99791;
        double r99793 = exp(r99792);
        double r99794 = r99782 * r99793;
        double r99795 = pow(r99784, r99783);
        double r99796 = 0.5;
        double r99797 = 2.0;
        double r99798 = pow(r99787, r99797);
        double r99799 = r99783 * r99798;
        double r99800 = pow(r99784, r99797);
        double r99801 = r99799 / r99800;
        double r99802 = r99796 * r99801;
        double r99803 = b;
        double r99804 = r99783 * r99803;
        double r99805 = r99789 * r99775;
        double r99806 = r99783 * r99787;
        double r99807 = r99784 * r99806;
        double r99808 = r99805 + r99807;
        double r99809 = r99804 + r99808;
        double r99810 = r99802 + r99809;
        double r99811 = exp(r99810);
        double r99812 = r99795 / r99811;
        double r99813 = pow(r99787, r99775);
        double r99814 = r99812 * r99813;
        double r99815 = r99782 * r99814;
        double r99816 = r99781 ? r99794 : r99815;
        return r99816;
}

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. Split input into 2 regimes

  2. if y < -1.436503598316957 or 8.989039952438387e+120 < y

    1. Initial program 1.7

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

    2. Taylor expanded around 0 8.9

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

    3. Simplified8.6

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

    if -1.436503598316957 < y < 8.989039952438387e+120

    1. Initial program 2.4

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

    2. Taylor expanded around 0 0.1

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.8

      \[\leadsto x \cdot \color{blue}{\left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(a \cdot b + \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot {z}^{y}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -1.436503598316956953340195468626916408539 \lor \neg \left(y \le 8.989039952438387375008659575680983873271 \cdot 10^{120}\right):\\ \;\;\;\;x \cdot e^{a \cdot \log 1 + \left(\log z - t\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{{1}^{a}}{e^{\frac{1}{2} \cdot \frac{a \cdot {z}^{2}}{{1}^{2}} + \left(a \cdot b + \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)\right)}} \cdot {z}^{y}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))))