ENA, Section 1.4, Exercise 1

Average Error: 3.5 → 0.4

Time: 6.6s

Precision: binary64

Cost: 32448

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (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(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.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.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[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], (-x)], $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 3.5

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

2. Simplified 3.1

   \[\leadsto \color{blue}{\cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]
   Proof
   (*.f64 (cos.f64 x) (pow.f64 (exp.f64 10) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (*.f64 (cos.f64 x) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 10 (*.f64 x x))))): 164 points increase in error, 87 points decrease in error

3. Applied egg-rr 3.0

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

4. Applied egg-rr 0.4

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

5. Applied egg-rr 0.4

   \[\leadsto \cos x \cdot \color{blue}{\frac{1}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{-2}\right)}}} \]

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Alternatives

Alternative 1
Error	1.2
Cost	26048

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

Alternative 2
Error	0.4
Cost	26048

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

Alternative 3
Error	1.3
Cost	25920

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

Alternative 4
Error	3.1
Cost	19584

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

Alternative 5
Error	3.1
Cost	19584

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

Alternative 6
Error	3.5
Cost	13376

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

Alternative 7
Error	3.5
Cost	13248

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

Alternative 8
Error	57.8
Cost	6464

\[\cos x \]

Alternative 9
Error	63.0
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022332 
(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)))))