ENA, Section 1.4, Exercise 1

Average Error: 3.5 → 0.4

Time: 8.7s

Precision: binary64

Cost: 26048

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

Math

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

↓

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

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 0.5))))

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 * 0.5));
}

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 * 0.5d0))
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 * 0.5));
}

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 * 0.5))

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 * 0.5)))
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 * 0.5));
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 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.5

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

2. Taylor expanded in x around inf 3.5

   \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot {x}^{2}}} \]

3. Simplified 2.1

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

   [Start] 3.5	\[ \cos x \cdot e^{10 \cdot {x}^{2}} \]
   *-commutative [=>] 3.5	\[ \cos x \cdot e^{\color{blue}{{x}^{2} \cdot 10}} \]
   unpow2 [=>] 3.5	\[ \cos x \cdot e^{\color{blue}{\left(x \cdot x\right)} \cdot 10} \]
   *-commutative [<=] 3.5	\[ \cos x \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}} \]
   associate-*r* [=>] 3.6	\[ \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]
   exp-prod [=>] 3.2	\[ \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]
   *-commutative [=>] 3.2	\[ \cos x \cdot {\left(e^{\color{blue}{x \cdot 10}}\right)}^{x} \]
   exp-prod [=>] 2.1	\[ \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{10}\right)}}^{x} \]

4. Applied egg-rr 63.1

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

5. Simplified 0.4

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

   [Start] 63.1	\[ \sqrt{{\cos x}^{2} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{x}} \]
   *-commutative [=>] 63.1	\[ \sqrt{\color{blue}{{\left({\left(e^{x}\right)}^{20}\right)}^{x} \cdot {\cos x}^{2}}} \]
   sqr-pow [=>] 63.1	\[ \sqrt{\color{blue}{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot {\cos x}^{2}} \]
   unpow2 [=>] 63.1	\[ \sqrt{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \color{blue}{\left(\cos x \cdot \cos x\right)}} \]
   unswap-sqr [=>] 63.1	\[ \sqrt{\color{blue}{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot \cos x\right) \cdot \left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot \cos x\right)}} \]
   rem-sqrt-square [=>] 63.1	\[ \color{blue}{\left|{\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot \cos x\right|} \]
   sqr-pow [=>] 63.1	\[ \left|\color{blue}{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}\right)} \cdot \cos x\right| \]
   unpow1 [<=] 63.1	\[ \left|\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}\right) \cdot \color{blue}{{\cos x}^{1}}\right| \]
   sqr-pow [=>] 64.0	\[ \left|\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}\right) \cdot \color{blue}{\left({\cos x}^{\left(\frac{1}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right)}\right| \]
   unswap-sqr [=>] 64.0	\[ \left|\color{blue}{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right)}\right| \]
   fabs-sqr [=>] 64.0	\[ \color{blue}{\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right)} \]
   *-commutative [=>] 64.0	\[ \color{blue}{\left({\cos x}^{\left(\frac{1}{2}\right)} \cdot {\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}\right)} \cdot \left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\cos x}^{\left(\frac{1}{2}\right)}\right) \]

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.3
Cost	25920

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

Alternative 2
Error	3.0
Cost	19712

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

Alternative 3
Error	3.0
Cost	19584

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

Alternative 4
Error	3.5
Cost	13248

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

Alternative 5
Error	57.8
Cost	6464

\[\cos x \]

Alternative 6
Error	63.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023060 
(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)))))