Average Error: 3.5 → 0.4
Time: 8.3s
Precision: binary64
Cost: 26048
\[1.99 \leq x \land x \leq 2.01\]
\[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
↓
\[\cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]
(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))
↓
(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 20.0) x) (/ x 2.0))))
double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}
↓
double code(double x) {
	return cos(x) * pow(pow(exp(20.0), x), (x / 2.0));
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** x) ** (x / 2.0d0))
end function

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}
↓
public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), x), (x / 2.0));
}

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))
↓
def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), x), (x / 2.0))

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end
↓
function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ x) ^ Float64(x / 2.0)))
end

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end
↓
function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ x) ^ (x / 2.0));
end

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\cos x \cdot e^{10 \cdot \left(x \cdot x\right)}
↓
\cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}

Error

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. Simplified3.1

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

    [Start]3.5

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

    exp-prod [=>]3.1

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

  3. Applied egg-rr3.0

    \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left(e^{20}\right)}^{\left(x \cdot x\right)}}} \]

  4. Applied egg-rr0.4

    \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}} \]

  5. Final simplification0.4

    \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

Alternatives

Alternative 1
Error1.1
Cost26048
\[\cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x \cdot 2\right)} \]
Alternative 2
Error1.3
Cost25920
\[\cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \]
Alternative 3
Error3.0
Cost19712
\[\cos x \cdot {\left(e^{20}\right)}^{\left(\left(x \cdot x\right) \cdot 0.5\right)} \]
Alternative 4
Error3.1
Cost19584
\[\cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]
Alternative 5
Error3.5
Cost13248
\[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
Alternative 6
Error52.3
Cost7232
\[e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \]
Alternative 7
Error57.7
Cost6976
\[\cos x \cdot \left(10 \cdot \left(x \cdot x\right) + 1\right) \]
Alternative 8
Error57.8
Cost448
\[1 + \left(x \cdot x\right) \cdot -0.5 \]
Alternative 9
Error63.0
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022364 
(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)))))