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

Average Error: 15.9 → 0.0

Time: 5.1s

Precision: binary64

Cost: 448

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps) :precision binary64 (* eps (+ eps (* x 2.0))))

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 * 2.0));
}

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 * 2.0d0))
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 * 2.0));
}

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 * 2.0))

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 * 2.0)))
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 * 2.0));
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 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.9

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

2. Simplified 0.0

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

   [Start] 15.9	\[ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   unpow2 [=>] 15.9	\[ \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]
   unpow2 [=>] 15.9	\[ \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]
   difference-of-squares [=>] 15.9	\[ \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]
   *-commutative [=>] 15.9	\[ \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]
   +-commutative [=>] 15.9	\[ \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   associate--l+ [=>] 0.0	\[ \color{blue}{\left(\varepsilon + \left(x - x\right)\right)} \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   +-inverses [=>] 0.0	\[ \left(\varepsilon + \color{blue}{0}\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   +-rgt-identity [=>] 0.0	\[ \color{blue}{\varepsilon} \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   +-commutative [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left(x + \left(x + \varepsilon\right)\right)} \]
   associate-+r+ [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\left(\left(x + x\right) + \varepsilon\right)} \]
   count-2 [=>] 0.0	\[ \varepsilon \cdot \left(\color{blue}{2 \cdot x} + \varepsilon\right) \]
   fma-def [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\mathsf{fma}\left(2, x, \varepsilon\right)} \]

3. Applied egg-rr 0.0

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

4. Applied egg-rr 10.2

   \[\leadsto \color{blue}{\frac{\left(4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon\right) \cdot \varepsilon}{2 \cdot x - \varepsilon}} \]

5. Applied egg-rr 10.3

   \[\leadsto \color{blue}{\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(\left(4 \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) + \frac{1}{x \cdot 2 - \varepsilon} \cdot \left(\left(-\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right)} \]

6. Simplified 0.0

   \[\leadsto \color{blue}{\varepsilon \cdot \frac{4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon}{2 \cdot x - \varepsilon}} \]
   Proof

   [Start] 10.3	\[ \frac{1}{x \cdot 2 - \varepsilon} \cdot \left(\left(4 \cdot \left(x \cdot x\right)\right) \cdot \varepsilon\right) + \frac{1}{x \cdot 2 - \varepsilon} \cdot \left(\left(-\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   associate-*r* [=>] 5.6	\[ \color{blue}{\left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(4 \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon} + \frac{1}{x \cdot 2 - \varepsilon} \cdot \left(\left(-\varepsilon \cdot \varepsilon\right) \cdot \varepsilon\right) \]
   associate-*r* [=>] 0.2	\[ \left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(4 \cdot \left(x \cdot x\right)\right)\right) \cdot \varepsilon + \color{blue}{\left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(-\varepsilon \cdot \varepsilon\right)\right) \cdot \varepsilon} \]
   distribute-rgt-out [=>] 0.2	\[ \color{blue}{\varepsilon \cdot \left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(4 \cdot \left(x \cdot x\right)\right) + \frac{1}{x \cdot 2 - \varepsilon} \cdot \left(-\varepsilon \cdot \varepsilon\right)\right)} \]
   distribute-lft-in [<=] 0.2	\[ \varepsilon \cdot \color{blue}{\left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \left(4 \cdot \left(x \cdot x\right) + \left(-\varepsilon \cdot \varepsilon\right)\right)\right)} \]
   sub-neg [<=] 0.2	\[ \varepsilon \cdot \left(\frac{1}{x \cdot 2 - \varepsilon} \cdot \color{blue}{\left(4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon\right)}\right) \]
   *-commutative [<=] 0.2	\[ \varepsilon \cdot \color{blue}{\left(\left(4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon\right) \cdot \frac{1}{x \cdot 2 - \varepsilon}\right)} \]
   associate-*r/ [=>] 0.0	\[ \varepsilon \cdot \color{blue}{\frac{\left(4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon\right) \cdot 1}{x \cdot 2 - \varepsilon}} \]
   *-rgt-identity [=>] 0.0	\[ \varepsilon \cdot \frac{\color{blue}{4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon}}{x \cdot 2 - \varepsilon} \]
   *-commutative [=>] 0.0	\[ \varepsilon \cdot \frac{4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon}{\color{blue}{2 \cdot x} - \varepsilon} \]

7. Applied egg-rr 0.0

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

8. Final simplification 0.0

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

Alternatives

Alternative 1
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-122} \lor \neg \left(x \leq 7 \cdot 10^{-109}\right):\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 2
Error	17.6
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023018 
(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)))