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

Average Accuracy: 74.9% → 100.0%

Time: 4.9s

Precision: binary64

Cost: 6848

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

Math

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

↓

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

FPCore

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

↓

(FPCore (x eps) :precision binary64 (fma eps 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 fma(eps, 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 fma(eps, eps, Float64(eps * Float64(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 * eps + N[(eps * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

2. Simplified 74.9%

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

   [Start] 74.9	\[ {\left(x + \varepsilon\right)}^{2} - {x}^{2} \]
   unpow2 [=>] 74.9	\[ {\left(x + \varepsilon\right)}^{2} - \color{blue}{x \cdot x} \]

3. Taylor expanded in x around 0 100.0%

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

4. Simplified 100.0%

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

   [Start] 100.0	\[ {\varepsilon}^{2} + 2 \cdot \left(\varepsilon \cdot x\right) \]
   unpow2 [=>] 100.0	\[ \color{blue}{\varepsilon \cdot \varepsilon} + 2 \cdot \left(\varepsilon \cdot x\right) \]
   fma-def [=>] 100.0	\[ \color{blue}{\mathsf{fma}\left(\varepsilon, \varepsilon, 2 \cdot \left(\varepsilon \cdot x\right)\right)} \]
   *-commutative [=>] 100.0	\[ \mathsf{fma}\left(\varepsilon, \varepsilon, \color{blue}{\left(\varepsilon \cdot x\right) \cdot 2}\right) \]
   associate-*l* [=>] 100.0	\[ \mathsf{fma}\left(\varepsilon, \varepsilon, \color{blue}{\varepsilon \cdot \left(x \cdot 2\right)}\right) \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	91.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{-112} \lor \neg \left(x \leq 2.8 \cdot 10^{-130}\right):\\ \;\;\;\;\varepsilon \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \varepsilon\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	448

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

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	72.7%
Cost	192

\[\varepsilon \cdot \varepsilon \]

Reproduce

herbie shell --seed 2023151 
(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)))