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

Average Error: 15.8 → 0.0

Time: 3.6s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

function tmp = code(x, eps)
	tmp = eps * (eps + (x + x));
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[(eps * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.8

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

2. Simplified 15.8

   \[\leadsto \color{blue}{{\left(x + \varepsilon\right)}^{2} - x \cdot x} \]
   Proof
   (-.f64 (pow.f64 (+.f64 x eps) 2) (*.f64 x x)): 0 points increase in error, 0 points decrease in error
   (-.f64 (pow.f64 (+.f64 x eps) 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in x around 0 0.0

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

4. Simplified 0.0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon + \left(x + x\right)\right)} \]
   Proof
   (*.f64 eps (+.f64 eps (+.f64 x x))): 0 points increase in error, 0 points decrease in error
   (*.f64 eps (+.f64 eps (Rewrite=> count-2_binary64 (*.f64 2 x)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-out_binary64 (+.f64 (*.f64 eps eps) (*.f64 (*.f64 2 x) eps))): 3 points increase in error, 2 points decrease in error
   (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 eps 2)) (*.f64 (*.f64 2 x) eps)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 eps 2) (Rewrite<= associate-*r*_binary64 (*.f64 2 (*.f64 x eps)))): 3 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 eps 2) (*.f64 2 (Rewrite<= *-commutative_binary64 (*.f64 eps x)))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	5.9
Cost	584

\[\begin{array}{l} t_0 := \varepsilon \cdot \left(x \cdot 2\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-125}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-98}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	17.4
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2022321 
(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)))