sqrt B (should all be same)

Average Accuracy: 51.8% → 100.0%

Time: 2.4s

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 51.8%

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

2. Taylor expanded in x around -inf 50.7%

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

3. Simplified 50.7%

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

   [Start] 50.7	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
   mul-1-neg [=>] 50.7	\[ \color{blue}{-\sqrt{2} \cdot x} \]
   distribute-rgt-neg-in [=>] 50.7	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

4. Applied egg-rr 28.6%

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

   [Start] 50.7	\[ \sqrt{2} \cdot \left(-x\right) \]
   expm1-log1p-u [=>] 48.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(-x\right)\right)\right)} \]
   expm1-udef [=>] 26.4	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(-x\right)\right)} - 1} \]
   log1p-udef [=>] 26.4	\[ e^{\color{blue}{\log \left(1 + \sqrt{2} \cdot \left(-x\right)\right)}} - 1 \]
   add-exp-log [<=] 28.7	\[ \color{blue}{\left(1 + \sqrt{2} \cdot \left(-x\right)\right)} - 1 \]
   add-sqr-sqrt [=>] 27.1	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) - 1 \]
   sqrt-unprod [=>] 29.1	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) - 1 \]
   sqr-neg [=>] 29.1	\[ \left(1 + \sqrt{2} \cdot \sqrt{\color{blue}{x \cdot x}}\right) - 1 \]
   sqrt-unprod [<=] 27.1	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) - 1 \]
   add-sqr-sqrt [<=] 28.6	\[ \left(1 + \sqrt{2} \cdot \color{blue}{x}\right) - 1 \]

5. Simplified 100.0%

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

   [Start] 28.6	\[ \left(1 + \sqrt{2} \cdot x\right) - 1 \]
   +-commutative [=>] 28.6	\[ \color{blue}{\left(\sqrt{2} \cdot x + 1\right)} - 1 \]
   associate--l+ [=>] 50.9	\[ \color{blue}{\sqrt{2} \cdot x + \left(1 - 1\right)} \]
   metadata-eval [=>] 50.9	\[ \sqrt{2} \cdot x + \color{blue}{0} \]
   +-rgt-identity [=>] 50.9	\[ \color{blue}{\sqrt{2} \cdot x} \]
   unpow1 [<=] 50.9	\[ \color{blue}{{\left(\sqrt{2} \cdot x\right)}^{1}} \]
   sqr-pow [=>] 49.6	\[ \color{blue}{{\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)}} \]
   fabs-sqr [<=] 49.6	\[ \color{blue}{\left|{\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)}\right|} \]
   sqr-pow [<=] 99.3	\[ \left|\color{blue}{{\left(\sqrt{2} \cdot x\right)}^{1}}\right| \]
   unpow1 [=>] 99.3	\[ \left|\color{blue}{\sqrt{2} \cdot x}\right| \]
   rem-sqrt-square [<=] 51.5	\[ \color{blue}{\sqrt{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]
   *-commutative [=>] 51.5	\[ \sqrt{\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \left(\sqrt{2} \cdot x\right)} \]
   associate-*l* [=>] 51.5	\[ \sqrt{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\sqrt{2} \cdot x\right)\right)}} \]
   *-commutative [=>] 51.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)}\right)} \]
   rem-log-exp [<=] 51.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\log \left(e^{\sqrt{2}}\right)}\right)\right)} \]
   log-pow [<=] 4.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\log \left({\left(e^{\sqrt{2}}\right)}^{x}\right)}\right)} \]
   sqr-pow [=>] 4.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \log \color{blue}{\left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)}\right)} \]
   log-prod [=>] 4.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right) + \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)\right)}\right)} \]
   distribute-lft-in [=>] 4.5	\[ \sqrt{x \cdot \color{blue}{\left(\sqrt{2} \cdot \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right) + \sqrt{2} \cdot \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)\right)}} \]

6. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023152 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))