sqrt B (should all be same)

Percentage Accurate: 54.6% → 100.0%

Time: 5.7s

Precision: binary64

Cost: 6528

• Unsound rule application detected in e-graph. Results may not be sound.

Math

\[\sqrt{\left(2 \cdot x\right) \cdot x} \]

↓

\[\mathsf{hypot}\left(x, x\right) \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))

↓

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return sqrt(((2.0 * x) * x));
}

↓

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}

↓

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.sqrt(((2.0 * x) * x))

↓

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end

↓

function code(x)
	return hypot(x, x)
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((2.0 * x) * x));
end

↓

function tmp = code(x)
	tmp = hypot(x, x);
end

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]

↓

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 50.2%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]

2. Applied egg-rr 46.2%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]
   Step-by-step derivation

   [Start] 50.2%	\[ \sqrt{\left(2 \cdot x\right) \cdot x} \]
   sqrt-prod [=>] 46.2%	\[ \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]

3. Taylor expanded in x around 0 47.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

4. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
   Step-by-step derivation

   [Start] 47.4%	\[ \sqrt{2} \cdot x \]
   rem-square-sqrt [<=] 45.9%	\[ \color{blue}{\sqrt{\sqrt{2} \cdot x} \cdot \sqrt{\sqrt{2} \cdot x}} \]
   fabs-sqr [<=] 45.9%	\[ \color{blue}{\left|\sqrt{\sqrt{2} \cdot x} \cdot \sqrt{\sqrt{2} \cdot x}\right|} \]
   rem-square-sqrt [=>] 99.3%	\[ \left|\color{blue}{\sqrt{2} \cdot x}\right| \]
   rem-sqrt-square [<=] 50.0%	\[ \color{blue}{\sqrt{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]
   swap-sqr [=>] 49.8%	\[ \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}} \]
   rem-square-sqrt [=>] 50.2%	\[ \sqrt{\color{blue}{2} \cdot \left(x \cdot x\right)} \]
   count-2 [<=] 50.2%	\[ \sqrt{\color{blue}{x \cdot x + x \cdot x}} \]
   hypot-def [=>] 100.0%	\[ \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{hypot}\left(x, x\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	6528

\[\mathsf{hypot}\left(x, x\right) \]

Alternative 2
Accuracy	13.6%
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;4 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]

Alternative 3
Accuracy	6.4%
Cost	192

\[x + -2 \]

Alternative 4
Accuracy	11.2%
Cost	192

\[x + x \]

Alternative 5
Accuracy	1.7%
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2023256 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))