ENA, Section 1.4, Exercise 1

Average Error: 94.5% → 99.4%

Time: 17.1s

Precision: binary64

Cost: 26048.00

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

Math

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

↓

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

FPCore

(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))))

C

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));
}

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) * ((exp(20.0d0) ** x) ** (x / 2.0d0))
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.pow(Math.pow(Math.exp(20.0), x), (x / 2.0));
}

Python

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))

Julia

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

MATLAB

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

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[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5

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

2. Applied egg-rr 98.0

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

3. Applied egg-rr 99.2

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

4. Applied egg-rr 94.2

   \[\leadsto \cos x \cdot \color{blue}{\left({\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3}\right)} \]

5. Simplified 97.4

   \[\leadsto \cos x \cdot \color{blue}{{\left(\sqrt[3]{{\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}}\right)}^{3}} \]
   Proof

   [Start] 94.2	\[ \cos x \cdot \left({\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3} \cdot {\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3}\right) \]
   cube-mult [=>] 94.2	\[ \cos x \cdot \left(\color{blue}{\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot \left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)\right)} \cdot {\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3}\right) \]
   cube-mult [=>] 94.2	\[ \cos x \cdot \left(\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot \left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)\right) \cdot \color{blue}{\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot \left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)\right)}\right) \]
   swap-sqr [=>] 94.2	\[ \cos x \cdot \color{blue}{\left(\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right) \cdot \left(\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right) \cdot \left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)\right)\right)} \]
   cube-unmult [=>] 94.2	\[ \cos x \cdot \color{blue}{{\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666} \cdot {\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{0.16666666666666666}\right)}^{3}} \]
   pow-sqr [=>] 94.1	\[ \cos x \cdot {\color{blue}{\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{\left(2 \cdot 0.16666666666666666\right)}\right)}}^{3} \]
   metadata-eval [=>] 94.1	\[ \cos x \cdot {\left({\left({\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}\right)}^{\color{blue}{0.3333333333333333}}\right)}^{3} \]
   unpow1/3 [=>] 97.4	\[ \cos x \cdot {\color{blue}{\left(\sqrt[3]{{\left({\left(\sqrt{e^{20}}\right)}^{x}\right)}^{x}}\right)}}^{3} \]

6. Applied egg-rr 99.1

   \[\leadsto \cos x \cdot \color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{4}\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{4}\right)}\right)} \]

7. Simplified 99.4

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

   [Start] 99.1	\[ \cos x \cdot \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{4}\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{4}\right)}\right) \]
   pow-sqr [=>] 99.4	\[ \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(2 \cdot \frac{x}{4}\right)}} \]
   associate-*r/ [=>] 99.4	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(\frac{2 \cdot x}{4}\right)}} \]
   *-commutative [=>] 99.4	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot 2}}{4}\right)} \]
   associate-/l* [=>] 99.4	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{\frac{4}{2}}\right)}} \]
   metadata-eval [=>] 99.4	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{2}}\right)} \]

8. Final simplification 99.4

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

Alternatives

Alternative 1
Error	98.4%
Cost	26048.00

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

Alternative 2
Error	98.0%
Cost	25920.00

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

Alternative 3
Error	95.3%
Cost	19584.00

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

Alternative 4
Error	95.2%
Cost	19584.00

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

Alternative 5
Error	94.5%
Cost	13248.00

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

Alternative 6
Error	9.6%
Cost	6464.00

\[\cos x \]

Alternative 7
Error	1.5%
Cost	64.00

\[1 \]

Reproduce

herbie shell --seed 2023093 
(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)))))