sqrt D (should all be same)

Average Error: 30.4 → 0.0

Time: 2.8s

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. Applied egg-rr 45.8

   \[\leadsto \color{blue}{\left(1 + \sqrt{2 \cdot \left(x \cdot x\right)}\right) - 1} \]

4. Simplified 0.0

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

   [Start] 45.8	\[ \left(1 + \sqrt{2 \cdot \left(x \cdot x\right)}\right) - 1 \]
   +-commutative [=>] 45.8	\[ \color{blue}{\left(\sqrt{2 \cdot \left(x \cdot x\right)} + 1\right)} - 1 \]
   associate--l+ [=>] 30.4	\[ \color{blue}{\sqrt{2 \cdot \left(x \cdot x\right)} + \left(1 - 1\right)} \]
   metadata-eval [=>] 30.4	\[ \sqrt{2 \cdot \left(x \cdot x\right)} + \color{blue}{0} \]
   +-rgt-identity [=>] 30.4	\[ \color{blue}{\sqrt{2 \cdot \left(x \cdot x\right)}} \]
   associate-*r* [=>] 30.4	\[ \sqrt{\color{blue}{\left(2 \cdot x\right) \cdot x}} \]
   count-2 [<=] 30.4	\[ \sqrt{\color{blue}{\left(x + x\right)} \cdot x} \]
   *-commutative [<=] 30.4	\[ \sqrt{\color{blue}{x \cdot \left(x + x\right)}} \]
   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 2023011 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))