Average Error: 3.5 → 2.2
Time: 4.5s
Precision: binary64
\[1.99 \leq x \land x \leq 2.01\]
\[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
↓
\[\begin{array}{l} t_0 := {\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5}\\ \cos x \cdot \left(t_0 \cdot \left(t_0 \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right)\right) \end{array} \]
\cos x \cdot e^{10 \cdot \left(x \cdot x\right)}
↓
\begin{array}{l}
t_0 := {\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5}\\
\cos x \cdot \left(t_0 \cdot \left(t_0 \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right)\right)
\end{array}

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))
↓
(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (pow (exp x) (/ x 2.0)) 5.0)))
   (* (cos x) (* t_0 (* t_0 (pow (pow (exp x) x) 5.0))))))
double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}
↓
double code(double x) {
	double t_0 = pow(pow(exp(x), (x / 2.0)), 5.0);
	return cos(x) * (t_0 * (t_0 * pow(pow(exp(x), x), 5.0)));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 3.5

    \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

  2. Applied add-sqr-sqrt_binary643.5

    \[\leadsto \cos x \cdot \color{blue}{\left(\sqrt{e^{10 \cdot \left(x \cdot x\right)}} \cdot \sqrt{e^{10 \cdot \left(x \cdot x\right)}}\right)} \]

  3. Simplified3.1

    \[\leadsto \cos x \cdot \left(\color{blue}{{\left(e^{x \cdot x}\right)}^{5}} \cdot \sqrt{e^{10 \cdot \left(x \cdot x\right)}}\right) \]

  4. Simplified3.0

    \[\leadsto \cos x \cdot \left({\left(e^{x \cdot x}\right)}^{5} \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{5}}\right) \]

  5. Applied add-log-exp_binary643.0

    \[\leadsto \cos x \cdot \left({\left(e^{x \cdot x}\right)}^{5} \cdot {\left(e^{\color{blue}{\log \left(e^{x}\right)} \cdot x}\right)}^{5}\right) \]

  6. Applied exp-to-pow_binary642.3

    \[\leadsto \cos x \cdot \left({\left(e^{x \cdot x}\right)}^{5} \cdot {\color{blue}{\left({\left(e^{x}\right)}^{x}\right)}}^{5}\right) \]

  7. Applied add-log-exp_binary642.3

    \[\leadsto \cos x \cdot \left({\left(e^{\color{blue}{\log \left(e^{x}\right)} \cdot x}\right)}^{5} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right) \]

  8. Applied exp-to-pow_binary642.1

    \[\leadsto \cos x \cdot \left({\color{blue}{\left({\left(e^{x}\right)}^{x}\right)}}^{5} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right) \]

  9. Applied sqr-pow_binary642.2

    \[\leadsto \cos x \cdot \left({\color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}}^{5} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right) \]

  10. Applied unpow-prod-down_binary642.2

    \[\leadsto \cos x \cdot \left(\color{blue}{\left({\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5} \cdot {\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5}\right)} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right) \]

  11. Applied associate-*l*_binary642.2

    \[\leadsto \cos x \cdot \color{blue}{\left({\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5} \cdot \left({\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right)\right)} \]

  12. Final simplification2.2

    \[\leadsto \cos x \cdot \left({\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5} \cdot \left({\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{5} \cdot {\left({\left(e^{x}\right)}^{x}\right)}^{5}\right)\right) \]

Reproduce

herbie shell --seed 2022125 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 1"
  :precision binary64
  :pre (and (<= 1.99 x) (<= x 2.01))
  (* (cos x) (exp (* 10.0 (* x x)))))