Average Error: 2.1 → 0.5
Time: 33.3s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\left(a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\left(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) + y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}
double f(double x, double y, double z, double t, double a, double b) {
        double r5351574 = x;
        double r5351575 = y;
        double r5351576 = z;
        double r5351577 = log(r5351576);
        double r5351578 = t;
        double r5351579 = r5351577 - r5351578;
        double r5351580 = r5351575 * r5351579;
        double r5351581 = a;
        double r5351582 = 1.0;
        double r5351583 = r5351582 - r5351576;
        double r5351584 = log(r5351583);
        double r5351585 = b;
        double r5351586 = r5351584 - r5351585;
        double r5351587 = r5351581 * r5351586;
        double r5351588 = r5351580 + r5351587;
        double r5351589 = exp(r5351588);
        double r5351590 = r5351574 * r5351589;
        return r5351590;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r5351591 = x;
        double r5351592 = a;
        double r5351593 = 1.0;
        double r5351594 = log(r5351593);
        double r5351595 = z;
        double r5351596 = r5351595 * r5351593;
        double r5351597 = r5351594 - r5351596;
        double r5351598 = 0.5;
        double r5351599 = r5351593 / r5351595;
        double r5351600 = r5351599 * r5351599;
        double r5351601 = r5351598 / r5351600;
        double r5351602 = r5351597 - r5351601;
        double r5351603 = b;
        double r5351604 = r5351602 - r5351603;
        double r5351605 = r5351592 * r5351604;
        double r5351606 = y;
        double r5351607 = cbrt(r5351595);
        double r5351608 = log(r5351607);
        double r5351609 = t;
        double r5351610 = r5351608 - r5351609;
        double r5351611 = r5351606 * r5351610;
        double r5351612 = r5351605 + r5351611;
        double r5351613 = r5351607 * r5351607;
        double r5351614 = log(r5351613);
        double r5351615 = r5351614 * r5351606;
        double r5351616 = r5351612 + r5351615;
        double r5351617 = exp(r5351616);
        double r5351618 = r5351591 * r5351617;
        return r5351618;
}

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 2.1

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

  2. Taylor expanded around 0 0.5

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

  3. Simplified0.5

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

  5. Applied add-cube-cbrt0.5

    \[\leadsto x \cdot e^{y \cdot \left(\log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)} - t\right) + a \cdot \left(\left(\left(\log 1 - 1 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right)}\]

  6. Applied log-prod0.5

    \[\leadsto x \cdot e^{y \cdot \left(\color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)} - t\right) + a \cdot \left(\left(\left(\log 1 - 1 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right)}\]

  7. Applied associate--l+0.5

    \[\leadsto x \cdot e^{y \cdot \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)} + a \cdot \left(\left(\left(\log 1 - 1 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right)}\]

  8. Applied distribute-lft-in0.5

    \[\leadsto x \cdot e^{\color{blue}{\left(y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right)} + a \cdot \left(\left(\left(\log 1 - 1 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right)}\]

  9. Applied associate-+l+0.5

    \[\leadsto x \cdot e^{\color{blue}{y \cdot \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \left(y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right) + a \cdot \left(\left(\left(\log 1 - 1 \cdot z\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right)\right)}}\]

  10. Final simplification0.5

    \[\leadsto x \cdot e^{\left(a \cdot \left(\left(\left(\log 1 - z \cdot 1\right) - \frac{\frac{1}{2}}{\frac{1}{z} \cdot \frac{1}{z}}\right) - b\right) + y \cdot \left(\log \left(\sqrt[3]{z}\right) - t\right)\right) + \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot y}\]

Reproduce

herbie shell --seed 2019171 
(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))))))