Average Error: 2.0 → 0.4
Time: 11.1s
Precision: 64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[x \cdot \log \left({\left(e^{{z}^{y}}\right)}^{\left(e^{a \cdot \left(\log 1 - b\right) - \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)}\right)}\right)\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
x \cdot \log \left({\left(e^{{z}^{y}}\right)}^{\left(e^{a \cdot \left(\log 1 - b\right) - \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)}\right)}\right)
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) exp(((double) (((double) (y * ((double) (((double) log(z)) - t)))) + ((double) (a * ((double) (((double) log(((double) (1.0 - z)))) - b))))))))));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return ((double) (x * ((double) log(((double) pow(((double) exp(((double) pow(z, y)))), ((double) exp(((double) (((double) (a * ((double) (((double) log(1.0)) - b)))) - ((double) (((double) (t * y)) + ((double) (1.0 * ((double) (a * z))))))))))))))));
}

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

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

  4. Applied add-cbrt-cube0.5

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

  5. Simplified12.4

    \[\leadsto x \cdot \sqrt[3]{\color{blue}{{\left({z}^{y} \cdot e^{a \cdot \left(\log 1 - b\right) - \left(t \cdot y + 1 \cdot \left(a \cdot z\right)\right)}\right)}^{3}}}\]
  6. Using strategy rm

  7. Applied add-log-exp12.6

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020140 
(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 (- 1.0 z)) b))))))