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

Average Error: 16.0 → 0.0

Time: 6.1s

Precision: binary64

Cost: 6848

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps) :precision binary64 (fma (* 2.0 x) eps (* eps 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 fma((2.0 * x), eps, (eps * eps));
}

Julia

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

↓

function code(x, eps)
	return fma(Float64(2.0 * x), eps, Float64(eps * 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[(2.0 * x), $MachinePrecision] * eps + N[(eps * eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.0

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

2. Simplified 0.0

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

   [Start] 16.0	\[ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   unpow2 [=>] 16.0	\[ \color{blue}{\left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right)} - {x}^{2} \]
   unpow2 [=>] 16.0	\[ \left(x + \varepsilon\right) \cdot \left(x + \varepsilon\right) - \color{blue}{x \cdot x} \]
   difference-of-squares [=>] 16.0	\[ \color{blue}{\left(\left(x + \varepsilon\right) + x\right) \cdot \left(\left(x + \varepsilon\right) - x\right)} \]
   *-commutative [=>] 16.0	\[ \color{blue}{\left(\left(x + \varepsilon\right) - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right)} \]
   +-commutative [=>] 16.0	\[ \left(\color{blue}{\left(\varepsilon + x\right)} - x\right) \cdot \left(\left(x + \varepsilon\right) + x\right) \]
   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 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	5.5
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-106} \lor \neg \left(x \leq 10^{-103}\right):\\ \;\;\;\;\left(2 \cdot x\right) \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 2
Error	0.0
Cost	448

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

Alternative 3
Error	17.5
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023039 
(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)))