sqrt D (should all be same)

Average Error: 30.4 → 0.0

Time: 3.0s

Precision: binary64

Cost: 6528

Math

\[\sqrt{2 \cdot {x}^{2}} \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 30.4

   \[\sqrt{2 \cdot {x}^{2}} \]

2. Simplified 30.4

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

   [Start] 30.4	\[ \sqrt{2 \cdot {x}^{2}} \]
   unpow2 [=>] 30.4	\[ \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

3. Taylor expanded in x around 0 31.0

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

4. Simplified 0.0

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

   [Start] 31.0	\[ \sqrt{2} \cdot x \]
   *-commutative [=>] 31.0	\[ \color{blue}{x \cdot \sqrt{2}} \]
   rem-square-sqrt [<=] 31.7	\[ \color{blue}{\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}} \]
   fabs-sqr [<=] 31.7	\[ \color{blue}{\left|\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}\right|} \]
   rem-square-sqrt [=>] 0.4	\[ \left|\color{blue}{x \cdot \sqrt{2}}\right| \]
   rem-sqrt-square [<=] 30.5	\[ \color{blue}{\sqrt{\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right)}} \]
   swap-sqr [=>] 30.7	\[ \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}} \]
   rem-square-sqrt [=>] 30.4	\[ \sqrt{\left(x \cdot x\right) \cdot \color{blue}{2}} \]
   associate-*l* [=>] 30.4	\[ \sqrt{\color{blue}{x \cdot \left(x \cdot 2\right)}} \]
   *-commutative [<=] 30.4	\[ \sqrt{x \cdot \color{blue}{\left(2 \cdot x\right)}} \]
   count-2 [<=] 30.4	\[ \sqrt{x \cdot \color{blue}{\left(x + x\right)}} \]
   distribute-lft-in [=>] 30.4	\[ \sqrt{\color{blue}{x \cdot x + x \cdot x}} \]
   hypot-def [=>] 0.0	\[ \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2023041 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))