ENA, Section 1.4, Exercise 1

Average Accuracy: 94.5% → 99.4%

Time: 10.0s

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 94.5%

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

2. Simplified 97.9%

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

   [Start] 94.5	\[ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
   associate-*r* [=>] 94.4	\[ \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]
   exp-prod [=>] 95.0	\[ \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]
   sqr-pow [=>] 95.0	\[ \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]
   sqr-pow [<=] 95.0	\[ \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]
   exp-prod [=>] 97.9	\[ \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Applied egg-rr 99.2%

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

   [Start] 97.9	\[ \cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \]
   add-sqr-sqrt [=>] 98.0	\[ \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{10}\right)}^{x}} \cdot \sqrt{{\left(e^{10}\right)}^{x}}\right)}}^{x} \]
   sqrt-unprod [=>] 97.9	\[ \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}}\right)}}^{x} \]
   pow-prod-down [=>] 98.0	\[ \cos x \cdot {\left(\sqrt{\color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{x}}}\right)}^{x} \]
   prod-exp [=>] 99.2	\[ \cos x \cdot {\left(\sqrt{{\color{blue}{\left(e^{10 + 10}\right)}}^{x}}\right)}^{x} \]
   metadata-eval [=>] 99.2	\[ \cos x \cdot {\left(\sqrt{{\left(e^{\color{blue}{20}}\right)}^{x}}\right)}^{x} \]

4. Applied egg-rr 99.2%

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

   [Start] 99.2	\[ \cos x \cdot {\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{x} \]
   pow1/2 [=>] 99.2	\[ \cos x \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{x} \]
   pow-pow [=>] 99.2	\[ \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \]

5. Applied egg-rr 95.4%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{10}\right)}^{x}\right)}^{\left(x \cdot 3\right)}}} \]
   Proof

   [Start] 99.2	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x} \]
   add-cbrt-cube [=>] 98.9	\[ \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}\right) \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}}} \]
   pow3 [=>] 98.9	\[ \cos x \cdot \sqrt[3]{\color{blue}{{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}\right)}^{3}}} \]
   pow-pow [=>] 95.7	\[ \cos x \cdot \sqrt[3]{\color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(x \cdot 3\right)}}} \]
   *-commutative [=>] 95.7	\[ \cos x \cdot \sqrt[3]{{\left({\left(e^{20}\right)}^{\color{blue}{\left(0.5 \cdot x\right)}}\right)}^{\left(x \cdot 3\right)}} \]
   pow-unpow [=>] 95.4	\[ \cos x \cdot \sqrt[3]{{\color{blue}{\left({\left({\left(e^{20}\right)}^{0.5}\right)}^{x}\right)}}^{\left(x \cdot 3\right)}} \]
   pow1/2 [<=] 95.4	\[ \cos x \cdot \sqrt[3]{{\left({\color{blue}{\left(\sqrt{e^{20}}\right)}}^{x}\right)}^{\left(x \cdot 3\right)}} \]
   pow1/2 [=>] 95.4	\[ \cos x \cdot \sqrt[3]{{\left({\color{blue}{\left({\left(e^{20}\right)}^{0.5}\right)}}^{x}\right)}^{\left(x \cdot 3\right)}} \]
   pow-exp [=>] 95.4	\[ \cos x \cdot \sqrt[3]{{\left({\color{blue}{\left(e^{20 \cdot 0.5}\right)}}^{x}\right)}^{\left(x \cdot 3\right)}} \]
   metadata-eval [=>] 95.4	\[ \cos x \cdot \sqrt[3]{{\left({\left(e^{\color{blue}{10}}\right)}^{x}\right)}^{\left(x \cdot 3\right)}} \]

6. Applied egg-rr 95.1%

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

   [Start] 95.4	\[ \cos x \cdot \sqrt[3]{{\left({\left(e^{10}\right)}^{x}\right)}^{\left(x \cdot 3\right)}} \]
   pow1/3 [=>] 94.3	\[ \cos x \cdot \color{blue}{{\left({\left({\left(e^{10}\right)}^{x}\right)}^{\left(x \cdot 3\right)}\right)}^{0.3333333333333333}} \]
   pow-unpow [=>] 94.1	\[ \cos x \cdot {\color{blue}{\left({\left({\left({\left(e^{10}\right)}^{x}\right)}^{x}\right)}^{3}\right)}}^{0.3333333333333333} \]
   pow-pow [=>] 97.9	\[ \cos x \cdot \color{blue}{{\left({\left({\left(e^{10}\right)}^{x}\right)}^{x}\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]
   metadata-eval [=>] 97.9	\[ \cos x \cdot {\left({\left({\left(e^{10}\right)}^{x}\right)}^{x}\right)}^{\color{blue}{1}} \]
   pow1 [<=] 97.9	\[ \cos x \cdot \color{blue}{{\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]
   sqr-pow [=>] 98.0	\[ \cos x \cdot \color{blue}{\left({\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]
   pow-prod-down [=>] 98.0	\[ \cos x \cdot \color{blue}{{\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}} \]
   add-exp-log [=>] 95.1	\[ \cos x \cdot {\color{blue}{\left(e^{\log \left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}\right)}}^{\left(\frac{x}{2}\right)} \]
   pow-prod-down [=>] 95.1	\[ \cos x \cdot {\left(e^{\log \color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}\right)}^{\left(\frac{x}{2}\right)} \]
   log-pow [=>] 95.1	\[ \cos x \cdot {\left(e^{\color{blue}{x \cdot \log \left(e^{10} \cdot e^{10}\right)}}\right)}^{\left(\frac{x}{2}\right)} \]
   log-prod [=>] 95.1	\[ \cos x \cdot {\left(e^{x \cdot \color{blue}{\left(\log \left(e^{10}\right) + \log \left(e^{10}\right)\right)}}\right)}^{\left(\frac{x}{2}\right)} \]
   add-log-exp [<=] 95.1	\[ \cos x \cdot {\left(e^{x \cdot \left(\color{blue}{10} + \log \left(e^{10}\right)\right)}\right)}^{\left(\frac{x}{2}\right)} \]
   add-log-exp [<=] 95.1	\[ \cos x \cdot {\left(e^{x \cdot \left(10 + \color{blue}{10}\right)}\right)}^{\left(\frac{x}{2}\right)} \]
   metadata-eval [=>] 95.1	\[ \cos x \cdot {\left(e^{x \cdot \color{blue}{20}}\right)}^{\left(\frac{x}{2}\right)} \]
   div-inv [=>] 95.1	\[ \cos x \cdot {\left(e^{x \cdot 20}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]
   metadata-eval [=>] 95.1	\[ \cos x \cdot {\left(e^{x \cdot 20}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \]

7. Simplified 99.4%

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

   [Start] 95.1	\[ \cos x \cdot {\left(e^{x \cdot 20}\right)}^{\left(x \cdot 0.5\right)} \]
   *-commutative [=>] 95.1	\[ \cos x \cdot {\left(e^{\color{blue}{20 \cdot x}}\right)}^{\left(x \cdot 0.5\right)} \]
   exp-prod [=>] 99.4	\[ \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{x}\right)}}^{\left(x \cdot 0.5\right)} \]

8. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	97.9%
Cost	25920

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

Alternative 2
Accuracy	95.3%
Cost	19712

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

Alternative 3
Accuracy	95.3%
Cost	19584

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

Alternative 4
Accuracy	94.5%
Cost	13376

\[\cos x \cdot \left(1 + \mathsf{expm1}\left(10 \cdot \left(x \cdot x\right)\right)\right) \]

Alternative 5
Accuracy	94.5%
Cost	13248

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

Alternative 6
Accuracy	9.6%
Cost	6464

\[\cos x \]

Alternative 7
Accuracy	1.5%
Cost	448

\[1 + \left(x \cdot x\right) \cdot 9.5 \]

Reproduce

herbie shell --seed 2023151 
(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)))))