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

Average Error: 15.9 → 0.0

Time: 4.5s

Precision: binary64

Cost: 448

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

Math

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

↓

\[\varepsilon \cdot \left(\varepsilon + 2 \cdot x\right) \]

FPCore

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

↓

(FPCore (x eps) :precision binary64 (* eps (+ eps (* 2.0 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 + (2.0 * 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 + (2.0d0 * 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 + (2.0 * 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 + (2.0 * 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(2.0 * 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 + (2.0 * 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[(2.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.9

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

2. Simplified 15.9

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

   [Start] 15.9	\[ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   unpow2 [=>] 15.9	\[ {\left(x + \varepsilon\right)}^{2} - \color{blue}{x \cdot x} \]

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

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

   [Start] 0.0	\[ {\varepsilon}^{2} + 2 \cdot \left(\varepsilon \cdot x\right) \]
   +-commutative [=>] 0.0	\[ \color{blue}{2 \cdot \left(\varepsilon \cdot x\right) + {\varepsilon}^{2}} \]
   unpow2 [=>] 0.0	\[ 2 \cdot \left(\varepsilon \cdot x\right) + \color{blue}{\varepsilon \cdot \varepsilon} \]

5. Taylor expanded in eps around 0 0.0

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

6. Simplified 0.0

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

   [Start] 0.0	\[ {\varepsilon}^{2} + 2 \cdot \left(\varepsilon \cdot x\right) \]
   unpow2 [=>] 0.0	\[ \color{blue}{\varepsilon \cdot \varepsilon} + 2 \cdot \left(\varepsilon \cdot x\right) \]
   *-commutative [=>] 0.0	\[ \varepsilon \cdot \varepsilon + \color{blue}{\left(\varepsilon \cdot x\right) \cdot 2} \]
   associate-*l* [=>] 0.0	\[ \varepsilon \cdot \varepsilon + \color{blue}{\varepsilon \cdot \left(x \cdot 2\right)} \]
   distribute-lft-out [=>] 0.0	\[ \color{blue}{\varepsilon \cdot \left(\varepsilon + x \cdot 2\right)} \]
   +-commutative [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left(x \cdot 2 + \varepsilon\right)} \]
   *-commutative [<=] 0.0	\[ \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]

7. Final simplification 0.0

   \[\leadsto \varepsilon \cdot \left(\varepsilon + 2 \cdot x\right) \]

Alternatives

Alternative 1
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-99} \lor \neg \left(x \leq 2.2 \cdot 10^{-69}\right):\\ \;\;\;\;2 \cdot \left(\varepsilon \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 2
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-99} \lor \neg \left(x \leq 2.2 \cdot 10^{-69}\right):\\ \;\;\;\;\varepsilon \cdot \left(2 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 3
Error	17.3
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2022364 
(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)))