sqrt A (should all be same)

Average Error: 30.3 → 0.0

Time: 1.5s

Precision: binary64

Cost: 6528

Math

\[\sqrt{x \cdot x + x \cdot x} \]

↓

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

FPCore

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

↓

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

C

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 30.3

   \[\sqrt{x \cdot x + x \cdot x} \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
   Proof
   (hypot.f64 x x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 x x) (*.f64 x x)))): 1 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	60.7
Cost	320

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

Alternative 2
Error	62.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022342 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))