Average Error: 3.5 → 0.7
Time: 3.5s
Precision: binary64
\[1.99 \leq x \land x \leq 2.01\]
\[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
↓
\[\cos x \cdot {\left({\left(e^{10}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{e^{10}}\right)}^{x}\right)}^{x} \]
\cos x \cdot e^{10 \cdot \left(x \cdot x\right)}
↓
\cos x \cdot {\left({\left(e^{10}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{e^{10}}\right)}^{x}\right)}^{x}

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

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-log-exp_binary643.5

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

  3. Applied exp-to-pow_binary643.1

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

  4. Applied pow-unpow_binary641.3

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

  5. Applied add-sqr-sqrt_binary641.8

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

  6. Applied unpow-prod-down_binary641.1

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

  7. Applied sqrt-pow2_binary640.7

    \[\leadsto \cos x \cdot {\left(\color{blue}{{\left(e^{10}\right)}^{\left(\frac{x}{2}\right)}} \cdot {\left(\sqrt{e^{10}}\right)}^{x}\right)}^{x} \]

  8. Final simplification0.7

    \[\leadsto \cos x \cdot {\left({\left(e^{10}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{e^{10}}\right)}^{x}\right)}^{x} \]

Reproduce

herbie shell --seed 2022095 
(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)))))