Average Error: 1.8 → 0.4
Time: 56.9s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot e^{\sqrt[3]{\log z - t} \cdot \left(y \cdot \left(\sqrt[3]{\log z - t} \cdot \sqrt[3]{\log z - t}\right)\right) + a \cdot \left(\left(\left(\log 1 - \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right) - 1 \cdot z\right) - b\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot e^{\sqrt[3]{\log z - t} \cdot \left(y \cdot \left(\sqrt[3]{\log z - t} \cdot \sqrt[3]{\log z - t}\right)\right) + a \cdot \left(\left(\left(\log 1 - \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right) - 1 \cdot z\right) - b\right)}
double f(double x, double y, double z, double t, double a, double b) {
        double r8359670 = x;
        double r8359671 = y;
        double r8359672 = z;
        double r8359673 = log(r8359672);
        double r8359674 = t;
        double r8359675 = r8359673 - r8359674;
        double r8359676 = r8359671 * r8359675;
        double r8359677 = a;
        double r8359678 = 1.0;
        double r8359679 = r8359678 - r8359672;
        double r8359680 = log(r8359679);
        double r8359681 = b;
        double r8359682 = r8359680 - r8359681;
        double r8359683 = r8359677 * r8359682;
        double r8359684 = r8359676 + r8359683;
        double r8359685 = exp(r8359684);
        double r8359686 = r8359670 * r8359685;
        return r8359686;
}
↓
double f(double x, double y, double z, double t, double a, double b) {
        double r8359687 = x;
        double r8359688 = z;
        double r8359689 = log(r8359688);
        double r8359690 = t;
        double r8359691 = r8359689 - r8359690;
        double r8359692 = cbrt(r8359691);
        double r8359693 = y;
        double r8359694 = r8359692 * r8359692;
        double r8359695 = r8359693 * r8359694;
        double r8359696 = r8359692 * r8359695;
        double r8359697 = a;
        double r8359698 = 1.0;
        double r8359699 = log(r8359698);
        double r8359700 = 0.5;
        double r8359701 = r8359688 / r8359698;
        double r8359702 = r8359701 * r8359701;
        double r8359703 = r8359700 * r8359702;
        double r8359704 = r8359699 - r8359703;
        double r8359705 = r8359698 * r8359688;
        double r8359706 = r8359704 - r8359705;
        double r8359707 = b;
        double r8359708 = r8359706 - r8359707;
        double r8359709 = r8359697 * r8359708;
        double r8359710 = r8359696 + r8359709;
        double r8359711 = exp(r8359710);
        double r8359712 = r8359687 * r8359711;
        return r8359712;
}

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 1.8

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

  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

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied associate-*r*0.4

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

  7. Final simplification0.4

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

Reproduce

herbie shell --seed 2019168 
(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))))))