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

Average Error: 15.8 → 0.0

Time: 5.0s

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 15.8

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

2. Simplified 0.0

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

   [Start] 15.8	\[ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   unpow2 [=>] 15.8	\[ \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]
   unpow2 [=>] 15.8	\[ \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]
   difference-of-squares [=>] 15.8	\[ \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]
   *-commutative [=>] 15.8	\[ \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]
   +-commutative [=>] 15.8	\[ \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   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 10.6

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

5. Simplified 0.2

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

   [Start] 10.6	\[ \frac{\varepsilon \cdot \left(4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon\right)}{2 \cdot x - \varepsilon} \]
   associate-/l* [=>] 0.2	\[ \color{blue}{\frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{4 \cdot \left(x \cdot x\right) - \varepsilon \cdot \varepsilon}}} \]

6. Taylor expanded in x around 0 17.4

   \[\leadsto \frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{\color{blue}{-1 \cdot {\varepsilon}^{2}}}} \]

7. Simplified 17.4

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

   [Start] 17.4	\[ \frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{-1 \cdot {\varepsilon}^{2}}} \]
   mul-1-neg [=>] 17.4	\[ \frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{\color{blue}{-{\varepsilon}^{2}}}} \]
   unpow2 [=>] 17.4	\[ \frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{-\color{blue}{\varepsilon \cdot \varepsilon}}} \]
   distribute-rgt-neg-out [<=] 17.4	\[ \frac{\varepsilon}{\frac{2 \cdot x - \varepsilon}{\color{blue}{\varepsilon \cdot \left(-\varepsilon\right)}}} \]

8. Taylor expanded in eps around inf 0.0

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

9. Simplified 0.0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(x \cdot 2 + \varepsilon\right)} \]
   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	\[ \varepsilon \cdot \varepsilon + 2 \cdot \color{blue}{\left(x \cdot \varepsilon\right)} \]
   associate-*r* [=>] 0.0	\[ \varepsilon \cdot \varepsilon + \color{blue}{\left(2 \cdot x\right) \cdot \varepsilon} \]
   distribute-rgt-out [=>] 0.0	\[ \color{blue}{\varepsilon \cdot \left(\varepsilon + 2 \cdot x\right)} \]
   +-commutative [<=] 0.0	\[ \varepsilon \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)} \]
   *-commutative [=>] 0.0	\[ \varepsilon \cdot \left(\color{blue}{x \cdot 2} + \varepsilon\right) \]

10. Final simplification 0.0

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

Alternatives

Alternative 1
Error	6.2
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4.65 \cdot 10^{-113} \lor \neg \left(x \leq 3.6 \cdot 10^{-121}\right):\\ \;\;\;\;\varepsilon \cdot \left(x \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 2
Error	17.5
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023031 
(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)))