ENA, Section 1.4, Exercise 1

Average Accuracy: 94.5% → 99.5%

Time: 9.6s

Precision: binary64

Cost: 26048

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

Math

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

↓

\[\cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.125\right)} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 80.0) x) (* x 0.125))))

C

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

↓

double code(double x) {
	return cos(x) * pow(pow(exp(80.0), x), (x * 0.125));
}

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(80.0d0) ** x) ** (x * 0.125d0))
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(80.0), x), (x * 0.125));
}

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(80.0), x), (x * 0.125))

Julia

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

↓

function code(x)
	return Float64(cos(x) * ((exp(80.0) ^ x) ^ Float64(x * 0.125)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = cos(x) * ((exp(80.0) ^ x) ^ (x * 0.125));
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[80.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.125), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

2. Simplified 95.3%

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

   [Start] 94.5	\[ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
   exp-prod [=>] 95.3	\[ \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

3. Applied egg-rr 95.3%

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

   [Start] 95.3	\[ \cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]
   add-sqr-sqrt [=>] 95.3	\[ \cos x \cdot \color{blue}{\left(\sqrt{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \cdot \sqrt{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}\right)} \]
   sqrt-unprod [=>] 95.3	\[ \cos x \cdot \color{blue}{\sqrt{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]
   pow-prod-down [=>] 95.3	\[ \cos x \cdot \sqrt{\color{blue}{{\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot x\right)}}} \]
   prod-exp [=>] 95.3	\[ \cos x \cdot \sqrt{{\color{blue}{\left(e^{10 + 10}\right)}}^{\left(x \cdot x\right)}} \]
   metadata-eval [=>] 95.3	\[ \cos x \cdot \sqrt{{\left(e^{\color{blue}{20}}\right)}^{\left(x \cdot x\right)}} \]

4. Applied egg-rr 99.4%

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

   [Start] 95.3	\[ \cos x \cdot \sqrt{{\left(e^{20}\right)}^{\left(x \cdot x\right)}} \]
   pow-unpow [=>] 99.4	\[ \cos x \cdot \sqrt{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{x}}} \]
   sqrt-pow1 [=>] 99.4	\[ \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \cos x \cdot \color{blue}{\left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)}\right)} \]
   Proof

   [Start] 99.4	\[ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]
   sqr-pow [=>] 99.1	\[ \cos x \cdot \color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}\right)} \]
   pow-prod-down [=>] 99.4	\[ \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}} \]
   sqr-pow [=>] 99.1	\[ \cos x \cdot \color{blue}{\left({\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right)} \]
   pow-prod-down [=>] 99.2	\[ \cos x \cdot \left({\color{blue}{\left({\left(e^{20} \cdot e^{20}\right)}^{x}\right)}}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   prod-exp [=>] 99.2	\[ \cos x \cdot \left({\left({\color{blue}{\left(e^{20 + 20}\right)}}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   metadata-eval [=>] 99.2	\[ \cos x \cdot \left({\left({\left(e^{\color{blue}{40}}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   associate-/l/ [=>] 99.2	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{\frac{x}{2 \cdot 2}}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   associate-/l/ [=>] 99.2	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{2 \cdot \left(2 \cdot 2\right)}\right)}} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   metadata-eval [=>] 99.2	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{2 \cdot \color{blue}{4}}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   metadata-eval [=>] 99.2	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{8}}\right)} \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   pow-prod-down [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\color{blue}{\left({\left(e^{20} \cdot e^{20}\right)}^{x}\right)}}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   prod-exp [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\color{blue}{\left(e^{20 + 20}\right)}}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   metadata-eval [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{\color{blue}{40}}\right)}^{x}\right)}^{\left(\frac{\frac{\frac{x}{2}}{2}}{2}\right)}\right) \]
   associate-/l/ [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{\frac{x}{2 \cdot 2}}}{2}\right)}\right) \]
   associate-/l/ [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{2 \cdot \left(2 \cdot 2\right)}\right)}}\right) \]
   metadata-eval [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{2 \cdot \color{blue}{4}}\right)}\right) \]
   metadata-eval [=>] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{8}}\right)}\right) \]

6. Simplified 99.2%

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

   [Start] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)}\right) \]
   metadata-eval [<=] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{\frac{2}{0.25}}}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)}\right) \]
   associate-/l* [<=] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x \cdot 0.25}{2}\right)}} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)}\right) \]
   metadata-eval [<=] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x \cdot 0.25}{2}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{\frac{2}{0.25}}}\right)}\right) \]
   associate-/l* [<=] 99.0	\[ \cos x \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x \cdot 0.25}{2}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x \cdot 0.25}{2}\right)}}\right) \]
   sqr-pow [<=] 99.2	\[ \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]

7. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}\right)} + -1} \]
   Proof

   [Start] 99.2	\[ \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]
   expm1-log1p-u [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}\right)\right)} \]
   expm1-udef [=>] 0.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}\right)} - 1} \]
   sub-neg [=>] 0.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}\right)} + \left(-1\right)} \]

8. Simplified 99.5%

   \[\leadsto \color{blue}{\cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.125\right)}} \]
   Proof

   [Start] 0.0	\[ e^{\mathsf{log1p}\left(\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}\right)} + -1 \]
   metadata-eval [<=] 0.0	\[ e^{\mathsf{log1p}\left(\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}\right)} + \color{blue}{\left(-1\right)} \]
   sub-neg [<=] 0.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}\right)} - 1} \]
   expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}\right)\right)} \]
   expm1-log1p [=>] 95.1	\[ \color{blue}{\cos x \cdot {\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}} \]
   sqr-pow [=>] 95.1	\[ \cos x \cdot \color{blue}{\left({\left(e^{x \cdot 80}\right)}^{\left(\frac{x \cdot 0.125}{2}\right)} \cdot {\left(e^{x \cdot 80}\right)}^{\left(\frac{x \cdot 0.125}{2}\right)}\right)} \]
   sqr-pow [<=] 95.1	\[ \cos x \cdot \color{blue}{{\left(e^{x \cdot 80}\right)}^{\left(x \cdot 0.125\right)}} \]
   *-commutative [=>] 95.1	\[ \cos x \cdot {\left(e^{\color{blue}{80 \cdot x}}\right)}^{\left(x \cdot 0.125\right)} \]
   exp-prod [=>] 99.5	\[ \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{x}\right)}}^{\left(x \cdot 0.125\right)} \]

9. Final simplification 99.5%

   \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.125\right)} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	26048

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

Alternative 2
Accuracy	98.0%
Cost	25920

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

Alternative 3
Accuracy	95.3%
Cost	19712

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

Alternative 4
Accuracy	95.3%
Cost	19584

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

Alternative 5
Accuracy	94.5%
Cost	13248

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

Alternative 6
Accuracy	18.2%
Cost	7232

\[e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \]

Alternative 7
Accuracy	9.8%
Cost	6976

\[\cos x \cdot \left(10 \cdot \left(x \cdot x\right) + 1\right) \]

Alternative 8
Accuracy	9.7%
Cost	448

\[1 + \left(x \cdot x\right) \cdot -0.5 \]

Alternative 9
Accuracy	1.5%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023140 
(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)))))