sqrt D (should all be same)

Average Accuracy: 54.9% → 100.0%

Time: 4.1s

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 54.0%

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

2. Simplified 54.0%

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

   [Start] 54.0	\[ \sqrt{2 \cdot {x}^{2}} \]
   unpow2 [=>] 54.0	\[ \sqrt{2 \cdot \color{blue}{\left(x \cdot x\right)}} \]

3. Taylor expanded in x around -inf 50.4%

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

4. Simplified 50.4%

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

   [Start] 50.4	\[ -1 \cdot \left(\sqrt{2} \cdot x\right) \]
   mul-1-neg [=>] 50.4	\[ \color{blue}{-\sqrt{2} \cdot x} \]
   distribute-rgt-neg-in [=>] 50.4	\[ \color{blue}{\sqrt{2} \cdot \left(-x\right)} \]

5. Applied egg-rr 30.2%

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

   [Start] 50.4	\[ \sqrt{2} \cdot \left(-x\right) \]
   expm1-log1p-u [=>] 48.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \left(-x\right)\right)\right)} \]
   expm1-udef [=>] 25.2	\[ \color{blue}{e^{\mathsf{log1p}\left(\sqrt{2} \cdot \left(-x\right)\right)} - 1} \]
   log1p-udef [=>] 25.2	\[ e^{\color{blue}{\log \left(1 + \sqrt{2} \cdot \left(-x\right)\right)}} - 1 \]
   add-exp-log [<=] 27.4	\[ \color{blue}{\left(1 + \sqrt{2} \cdot \left(-x\right)\right)} - 1 \]
   add-sqr-sqrt [=>] 25.8	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\left(\sqrt{-x} \cdot \sqrt{-x}\right)}\right) - 1 \]
   sqrt-unprod [=>] 31.9	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}\right) - 1 \]
   sqr-neg [=>] 31.9	\[ \left(1 + \sqrt{2} \cdot \sqrt{\color{blue}{x \cdot x}}\right) - 1 \]
   sqrt-unprod [<=] 28.7	\[ \left(1 + \sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}\right) - 1 \]
   add-sqr-sqrt [<=] 30.2	\[ \left(1 + \sqrt{2} \cdot \color{blue}{x}\right) - 1 \]

6. Simplified 100.0%

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

   [Start] 30.2	\[ \left(1 + \sqrt{2} \cdot x\right) - 1 \]
   +-commutative [=>] 30.2	\[ \color{blue}{\left(\sqrt{2} \cdot x + 1\right)} - 1 \]
   associate--l+ [=>] 51.1	\[ \color{blue}{\sqrt{2} \cdot x + \left(1 - 1\right)} \]
   metadata-eval [=>] 51.1	\[ \sqrt{2} \cdot x + \color{blue}{0} \]
   +-rgt-identity [=>] 51.1	\[ \color{blue}{\sqrt{2} \cdot x} \]
   unpow1 [<=] 51.1	\[ \color{blue}{{\left(\sqrt{2} \cdot x\right)}^{1}} \]
   sqr-pow [=>] 49.8	\[ \color{blue}{{\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)}} \]
   fabs-sqr [<=] 49.8	\[ \color{blue}{\left|{\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\sqrt{2} \cdot x\right)}^{\left(\frac{1}{2}\right)}\right|} \]
   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 [<=] 53.8	\[ \color{blue}{\sqrt{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}} \]
   *-commutative [=>] 53.8	\[ \sqrt{\color{blue}{\left(x \cdot \sqrt{2}\right)} \cdot \left(\sqrt{2} \cdot x\right)} \]
   associate-*l* [=>] 53.7	\[ \sqrt{\color{blue}{x \cdot \left(\sqrt{2} \cdot \left(\sqrt{2} \cdot x\right)\right)}} \]
   *-commutative [=>] 53.7	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(x \cdot \sqrt{2}\right)}\right)} \]
   rem-log-exp [<=] 53.7	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \left(x \cdot \color{blue}{\log \left(e^{\sqrt{2}}\right)}\right)\right)} \]
   log-pow [<=] 7.5	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\log \left({\left(e^{\sqrt{2}}\right)}^{x}\right)}\right)} \]
   sqr-pow [=>] 7.4	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \log \color{blue}{\left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)}\right)} \]
   log-prod [=>] 7.8	\[ \sqrt{x \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right) + \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)\right)}\right)} \]
   distribute-lft-in [=>] 7.8	\[ \sqrt{x \cdot \color{blue}{\left(\sqrt{2} \cdot \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right) + \sqrt{2} \cdot \log \left({\left(e^{\sqrt{2}}\right)}^{\left(\frac{x}{2}\right)}\right)\right)}} \]

7. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023160 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))