Average Error: 2.0 → 0.5
Time: 8.7s
Precision: binary64
\[x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}\]
↓
\[\left(x \cdot {e}^{\left(\frac{1}{2} \cdot \sqrt[3]{{\left(y \cdot \left(\log z - t\right) + a \cdot \left(\log 1 - \left(z \cdot 1 + \left(\frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right) + b\right)\right)\right)\right)}^{3}}\right)}\right) \cdot {e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(z \cdot 1 + \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right)\right) - b\right)}{2}\right)}\]
x \cdot e^{y \cdot \left(\log z - t\right) + a \cdot \left(\log \left(1 - z\right) - b\right)}
↓
\left(x \cdot {e}^{\left(\frac{1}{2} \cdot \sqrt[3]{{\left(y \cdot \left(\log z - t\right) + a \cdot \left(\log 1 - \left(z \cdot 1 + \left(\frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right) + b\right)\right)\right)\right)}^{3}}\right)}\right) \cdot {e}^{\left(\frac{y \cdot \left(\log z - t\right) + a \cdot \left(\left(\log 1 - \left(z \cdot 1 + \frac{1}{2} \cdot \left(\frac{z}{1} \cdot \frac{z}{1}\right)\right)\right) - b\right)}{2}\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) (((double) (x * ((double) pow(((double) M_E), ((double) (0.5 * ((double) cbrt(((double) pow(((double) (((double) (y * ((double) (((double) log(z)) - t)))) + ((double) (a * ((double) (((double) log(1.0)) - ((double) (((double) (z * 1.0)) + ((double) (((double) (0.5 * ((double) (((double) (z / 1.0)) * ((double) (z / 1.0)))))) + b)))))))))), 3.0)))))))))) * ((double) pow(((double) M_E), ((double) (((double) (((double) (y * ((double) (((double) log(z)) - t)))) + ((double) (a * ((double) (((double) (((double) log(1.0)) - ((double) (((double) (z * 1.0)) + ((double) (0.5 * ((double) (((double) (z / 1.0)) * ((double) (z / 1.0)))))))))) - b)))))) / 2.0))))));
}

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 0.5

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

  3. Simplified0.5

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

  5. Applied *-un-lft-identity0.5

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

  6. Applied exp-prod0.5

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

  7. Simplified0.5

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

  9. Applied sqr-pow0.5

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

  10. Applied associate-*r*0.5

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

  11. Simplified0.5

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

  13. Applied add-cbrt-cube0.5

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

  14. Simplified0.5

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

  15. Final simplification0.5

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

Reproduce

herbie shell --seed 2020179 
(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))))))