ENA, Section 1.4, Exercise 1

Average Error: 3.5 → 0.4

Time: 9.0s

Precision: binary64

Cost: 32320

\[1.99 \leq x \land x \leq 2.01\]

Math

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

↓

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

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

↓

(FPCore (x) :precision binary64 (* (cos x) (sqrt (pow (pow (exp 20.0) x) x))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

↓

double code(double x) {
	return cos(x) * sqrt(pow(pow(exp(20.0), x), x));
}

Fortran

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) * sqrt(((exp(20.0d0) ** x) ** x))
end function

Java

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.sqrt(Math.pow(Math.pow(Math.exp(20.0), x), x));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

↓

def code(x):
	return math.cos(x) * math.sqrt(math.pow(math.pow(math.exp(20.0), x), x))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

↓

function code(x)
	return Float64(cos(x) * sqrt(((exp(20.0) ^ x) ^ x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

↓

function tmp = code(x)
	tmp = cos(x) * sqrt(((exp(20.0) ^ x) ^ x));
end

Wolfram

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[Sqrt[N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.5

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

2. Applied egg-rr 1.3

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

3. Applied egg-rr 1.3

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

4. Simplified 1.3

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

   [Start] 1.3	\[ \cos x \cdot \left({\left(\sqrt{{\left(e^{10}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{10}\right)}^{x}}\right)}^{x}\right) \]
   pow-sqr [=>] 1.3	\[ \cos x \cdot \color{blue}{{\left(\sqrt{{\left(e^{10}\right)}^{x}}\right)}^{\left(2 \cdot x\right)}} \]
   *-commutative [=>] 1.3	\[ \cos x \cdot {\left(\sqrt{{\left(e^{10}\right)}^{x}}\right)}^{\color{blue}{\left(x \cdot 2\right)}} \]

5. Applied egg-rr 0.4

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

6. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.1
Cost	26048

\[\cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x + x\right)} \]

Alternative 2
Error	0.4
Cost	26048

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

Alternative 3
Error	1.3
Cost	25920

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

Alternative 4
Error	3.0
Cost	19712

\[\cos x \cdot {\left(e^{20}\right)}^{\left(\left(x \cdot x\right) \cdot 0.5\right)} \]

Alternative 5
Error	3.0
Cost	19712

\[\cos x \cdot {\left(e^{40}\right)}^{\left(\left(x \cdot x\right) \cdot 0.25\right)} \]

Alternative 6
Error	3.0
Cost	19584

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

Alternative 7
Error	3.5
Cost	13248

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

Alternative 8
Error	57.8
Cost	6464

\[\cos x \]

Alternative 9
Error	63.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023039 
(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)))))