sqrt B

Average Error: 30.6 → 0.0

Time: 2.3s

Precision: binary64

Cost: 6528

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 30.6

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

2. Taylor expanded in x around -inf 31.7

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

3. Simplified 31.7

   \[\leadsto \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]
   Proof
   (*.f64 (sqrt.f64 2) (neg.f64 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (sqrt.f64 2) x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (*.f64 (sqrt.f64 2) x))): 0 points increase in error, 0 points decrease in error

4. Applied egg-rr 46.0

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

5. 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)))): 144 points increase in error, 1 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 x 1))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (/.f64 x (Rewrite<= metadata-eval (/.f64 2 2)))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) 2))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (/.f64 (*.f64 x (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 2) (sqrt.f64 2)))) 2)) (*.f64 x x))): 34 points increase in error, 11 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (sqrt.f64 2)) (sqrt.f64 2))) 2)) (*.f64 x x))): 3 points increase in error, 8 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) x)) (sqrt.f64 2)) 2)) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) x) 2) (sqrt.f64 2)))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x (sqrt.f64 2))) 2) (sqrt.f64 2))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x 2) (sqrt.f64 2))) (sqrt.f64 2))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (*.f64 (/.f64 x 2) (Rewrite<= rem-log-exp_binary64 (log.f64 (exp.f64 (sqrt.f64 2))))) (sqrt.f64 2))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))) (sqrt.f64 2))) (*.f64 x x))): 131 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))))) (*.f64 x x))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (Rewrite<= /-rgt-identity_binary64 (/.f64 x 1))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (/.f64 x (Rewrite<= metadata-eval (/.f64 2 2)))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x 2) 2))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (/.f64 (*.f64 x (Rewrite<= rem-square-sqrt_binary64 (*.f64 (sqrt.f64 2) (sqrt.f64 2)))) 2)))): 4 points increase in error, 40 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (sqrt.f64 2)) (sqrt.f64 2))) 2)))): 12 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (/.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) x)) (sqrt.f64 2)) 2)))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (*.f64 (sqrt.f64 2) x) 2) (sqrt.f64 2)))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (*.f64 (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x (sqrt.f64 2))) 2) (sqrt.f64 2))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (*.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x 2) (sqrt.f64 2))) (sqrt.f64 2))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (*.f64 (*.f64 (/.f64 x 2) (Rewrite<= rem-log-exp_binary64 (log.f64 (exp.f64 (sqrt.f64 2))))) (sqrt.f64 2))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (*.f64 (Rewrite<= log-pow_binary64 (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))) (sqrt.f64 2))))): 66 points increase in error, 0 points decrease in error
   (sqrt.f64 (+.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))))) (*.f64 x (Rewrite=> *-commutative_binary64 (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (Rewrite<= distribute-lft-in_binary64 (*.f64 x (+.f64 (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))) (*.f64 (sqrt.f64 2) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (*.f64 x (Rewrite<= distribute-lft-in_binary64 (*.f64 (sqrt.f64 2) (+.f64 (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2))) (log.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (*.f64 x (*.f64 (sqrt.f64 2) (Rewrite<= log-prod_binary64 (log.f64 (*.f64 (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)) (pow.f64 (exp.f64 (sqrt.f64 2)) (/.f64 x 2)))))))): 4 points increase in error, 4 points decrease in error
   (sqrt.f64 (*.f64 x (*.f64 (sqrt.f64 2) (log.f64 (Rewrite<= sqr-pow_binary64 (pow.f64 (exp.f64 (sqrt.f64 2)) x)))))): 0 points increase in error, 3 points decrease in error
   (sqrt.f64 (*.f64 x (*.f64 (sqrt.f64 2) (Rewrite=> log-pow_binary64 (*.f64 x (log.f64 (exp.f64 (sqrt.f64 2)))))))): 0 points increase in error, 131 points decrease in error
   (sqrt.f64 (*.f64 x (*.f64 (sqrt.f64 2) (*.f64 x (Rewrite=> rem-log-exp_binary64 (sqrt.f64 2)))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (*.f64 x (*.f64 (sqrt.f64 2) (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) x))))): 0 points increase in error, 0 points decrease in error
   (sqrt.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x (sqrt.f64 2)) (*.f64 (sqrt.f64 2) x)))): 2 points increase in error, 5 points decrease in error
   (sqrt.f64 (*.f64 (Rewrite<= *-commutative_binary64 (*.f64 (sqrt.f64 2) x)) (*.f64 (sqrt.f64 2) x))): 0 points increase in error, 0 points decrease in error
   (Rewrite=> rem-sqrt-square_binary64 (fabs.f64 (*.f64 (sqrt.f64 2) x))): 0 points increase in error, 126 points decrease in error
   (fabs.f64 (Rewrite<= unpow1_binary64 (pow.f64 (*.f64 (sqrt.f64 2) x) 1))): 0 points increase in error, 0 points decrease in error
   (fabs.f64 (Rewrite=> sqr-pow_binary64 (*.f64 (pow.f64 (*.f64 (sqrt.f64 2) x) (/.f64 1 2)) (pow.f64 (*.f64 (sqrt.f64 2) x) (/.f64 1 2))))): 179 points increase in error, 15 points decrease in error
   (Rewrite=> fabs-sqr_binary64 (*.f64 (pow.f64 (*.f64 (sqrt.f64 2) x) (/.f64 1 2)) (pow.f64 (*.f64 (sqrt.f64 2) x) (/.f64 1 2)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sqr-pow_binary64 (pow.f64 (*.f64 (sqrt.f64 2) x) 1)): 15 points increase in error, 179 points decrease in error
   (Rewrite=> unpow1_binary64 (*.f64 (sqrt.f64 2) x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-rgt-identity_binary64 (+.f64 (*.f64 (sqrt.f64 2) x) 0)): 0 points increase in error, 0 points decrease in error
   (+.f64 (*.f64 (sqrt.f64 2) x) (Rewrite<= metadata-eval (-.f64 1 1))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (*.f64 (sqrt.f64 2) x) 1) 1)): 64 points increase in error, 64 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 1 (*.f64 (sqrt.f64 2) x))) 1): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.0

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

Alternatives

Alternative 1
Error	55.8
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;x + x\\ \end{array} \]

Alternative 2
Error	56.7
Cost	192

\[x + x \]

Alternative 3
Error	62.9
Cost	64

\[-2 \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "sqrt B"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))