ENA, Section 1.4, Exercise 4b, n=2

Average Error: 16.0 → 0.0

Time: 34.7s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

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 = ((eps + x) + x) * eps
end function

Java

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 ((eps + x) + x) * eps;
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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[(eps + x), $MachinePrecision] + x), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 16.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right)} \]
   Proof

3. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(\varepsilon + x, \varepsilon, x \cdot \varepsilon\right)} \]

4. Taylor expanded in eps around 0 0.0

   \[\leadsto \color{blue}{{\varepsilon}^{2} + 2 \cdot \left(\varepsilon \cdot x\right)} \]

5. Simplified 0.0

   \[\leadsto \color{blue}{\left(\left(\varepsilon + x\right) + x\right) \cdot \varepsilon} \]
   Proof

Alternatives

Alternative 1
Error	5.9
Cost	584

\[\begin{array}{l} t_0 := \left(2 \cdot \varepsilon\right) \cdot x\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-107}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	17.7
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023033 
(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)))