Average Error: 16.4 → 0.0

Time: 8.8s

Precision: binary64

Cost: 576

\[\left(-1000000000 \leq x \land x \leq 1000000000\right) \land \left(-1 \leq \varepsilon \land \varepsilon \leq 1\right)\]

\[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]

↓

\[\left(x + x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon \]

(FPCore (x eps) :precision binary64 (- (pow (+ x eps) 2.0) (pow x 2.0)))

↓

(FPCore (x eps) :precision binary64 (+ (* (+ x x) eps) (* eps eps)))

double code(double x, double eps) {
	return pow((x + eps), 2.0) - pow(x, 2.0);
}

↓

double code(double x, double eps) {
	return ((x + x) * eps) + (eps * eps);
}

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + eps) ** 2.0d0) - (x ** 2.0d0)
end function

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((x + x) * eps) + (eps * eps)
end function

public static double code(double x, double eps) {
	return Math.pow((x + eps), 2.0) - Math.pow(x, 2.0);
}

↓

public static double code(double x, double eps) {
	return ((x + x) * eps) + (eps * eps);
}

def code(x, eps):
	return math.pow((x + eps), 2.0) - math.pow(x, 2.0)

↓

def code(x, eps):
	return ((x + x) * eps) + (eps * eps)

function code(x, eps)
	return Float64((Float64(x + eps) ^ 2.0) - (x ^ 2.0))
end

↓

function code(x, eps)
	return Float64(Float64(Float64(x + x) * eps) + Float64(eps * eps))
end

function tmp = code(x, eps)
	tmp = ((x + eps) ^ 2.0) - (x ^ 2.0);
end

↓

function tmp = code(x, eps)
	tmp = ((x + x) * eps) + (eps * eps);
end

code[x_, eps_] := N[(N[Power[N[(x + eps), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]

↓

code[x_, eps_] := N[(N[(N[(x + x), $MachinePrecision] * eps), $MachinePrecision] + N[(eps * eps), $MachinePrecision]), $MachinePrecision]

{\left(x + \varepsilon\right)}^{2} - {x}^{2}

↓

\left(x + x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon

Try it out

In
Out

Enter valid numbers for all inputs

Derivation

Initial program 16.4
\[{\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
Taylor expanded in x around 0 0.0
\[\leadsto \color{blue}{{\varepsilon}^{2} + 2 \cdot \left(\varepsilon \cdot x\right)} \]
Simplified0.0
\[\leadsto \color{blue}{\left(\left(x + x\right) + \varepsilon\right) \cdot \varepsilon} \]
Proof
Applied egg-rr0.0
\[\leadsto \color{blue}{\left(x + x\right) \cdot \varepsilon + \varepsilon \cdot \varepsilon} \]

Alternatives

Alternative 1
Error	5.9
Cost	584

\[\begin{array}{l} t_0 := \left(x + x\right) \cdot \varepsilon\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-110}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

\[\varepsilon \cdot \left(x + \left(x + \varepsilon\right)\right) \]

Alternative 3
Error	18.0
Cost	192

\[\varepsilon \cdot \varepsilon \]

Alternative 4
Error	39.6
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023010 
(FPCore (x eps)
  :name "ENA, Section 1.4, Exercise 4b, n=2"
  :precision binary64
  :pre (and (and (<= -1000000000.0 x) (<= x 1000000000.0)) (and (<= -1.0 eps) (<= eps 1.0)))
  (- (pow (+ x eps) 2.0) (pow x 2.0)))