ENA, Section 1.4, Exercise 1

Average Error: 3.5 → 0.4

Time: 9.4s

Precision: binary64

Cost: 26048

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (* (pow (pow (exp 20.0) x) (* x 0.5)) (cos x)))

C

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

↓

double code(double x) {
	return pow(pow(exp(20.0), x), (x * 0.5)) * cos(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 = ((exp(20.0d0) ** x) ** (x * 0.5d0)) * cos(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.pow(Math.pow(Math.exp(20.0), x), (x * 0.5)) * Math.cos(x);
}

Python

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

↓

def code(x):
	return math.pow(math.pow(math.exp(20.0), x), (x * 0.5)) * math.cos(x)

Julia

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

↓

function code(x)
	return Float64(((exp(20.0) ^ x) ^ Float64(x * 0.5)) * cos(x))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((exp(20.0) ^ x) ^ (x * 0.5)) * cos(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[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $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

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

3. Simplified 1.3

   \[\leadsto \color{blue}{{\left({\left(e^{10}\right)}^{x}\right)}^{x} \cdot \cos x} \]
   Proof
   (*.f64 (pow.f64 (pow.f64 (exp.f64 10) x) x) (cos.f64 x)): 0 points increase in error, 0 points decrease in error
   (*.f64 (pow.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 10 x))) x) (cos.f64 x)): 216 points increase in error, 34 points decrease in error
   (*.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (*.f64 10 x) x))) (cos.f64 x)): 166 points increase in error, 82 points decrease in error
   (*.f64 (exp.f64 (Rewrite<= associate-*r*_binary64 (*.f64 10 (*.f64 x x)))) (cos.f64 x)): 40 points increase in error, 47 points decrease in error
   (*.f64 (exp.f64 (*.f64 10 (Rewrite<= unpow2_binary64 (pow.f64 x 2)))) (cos.f64 x)): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 0.5

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

5. Applied egg-rr 0.4

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

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	19584

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

Alternative 3
Error	3.9
Cost	13248

\[\cos x \cdot e^{x \cdot \frac{x}{0.1}} \]

Alternative 4
Error	3.6
Cost	13248

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

Alternative 5
Error	3.5
Cost	13248

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

Alternative 6
Error	57.8
Cost	6464

\[\cos x \]

Alternative 7
Error	63.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022294 
(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)))))