sqrt B (should all be same)

Average Accuracy: 52.7% → 100.0%

Time: 4.3s

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 52.7%

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

2. Applied egg-rr 49.8%

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

   [Start] 52.7	\[ \sqrt{\left(2 \cdot x\right) \cdot x} \]
   add-sqr-sqrt [=>] 52.4	\[ \color{blue}{\sqrt{\sqrt{\left(2 \cdot x\right) \cdot x}} \cdot \sqrt{\sqrt{\left(2 \cdot x\right) \cdot x}}} \]
   pow2 [=>] 52.4	\[ \color{blue}{{\left(\sqrt{\sqrt{\left(2 \cdot x\right) \cdot x}}\right)}^{2}} \]
   associate-*l* [=>] 52.4	\[ {\left(\sqrt{\sqrt{\color{blue}{2 \cdot \left(x \cdot x\right)}}}\right)}^{2} \]
   sqrt-prod [=>] 52.3	\[ {\left(\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{x \cdot x}}}\right)}^{2} \]
   sqrt-unprod [<=] 49.7	\[ {\left(\sqrt{\sqrt{2} \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)}}\right)}^{2} \]
   add-sqr-sqrt [<=] 49.8	\[ {\left(\sqrt{\sqrt{2} \cdot \color{blue}{x}}\right)}^{2} \]

3. Applied egg-rr 52.4%

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

   [Start] 49.8	\[ {\left(\sqrt{\sqrt{2} \cdot x}\right)}^{2} \]
   add-sqr-sqrt [=>] 49.8	\[ {\left(\sqrt{\color{blue}{\sqrt{\sqrt{2} \cdot x} \cdot \sqrt{\sqrt{2} \cdot x}}}\right)}^{2} \]
   sqrt-unprod [=>] 52.3	\[ {\left(\sqrt{\color{blue}{\sqrt{\left(\sqrt{2} \cdot x\right) \cdot \left(\sqrt{2} \cdot x\right)}}}\right)}^{2} \]
   swap-sqr [=>] 52.2	\[ {\left(\sqrt{\sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \left(x \cdot x\right)}}}\right)}^{2} \]
   add-sqr-sqrt [<=] 52.4	\[ {\left(\sqrt{\sqrt{\color{blue}{2} \cdot \left(x \cdot x\right)}}\right)}^{2} \]

4. Taylor expanded in x around 0 49.8%

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

5. Simplified 99.2%

   \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(x, x\right)}}\right)}^{2} \]
   Proof

   [Start] 49.8	\[ {\left(\sqrt{\sqrt{2} \cdot x}\right)}^{2} \]
   *-commutative [<=] 49.8	\[ {\left(\sqrt{\color{blue}{x \cdot \sqrt{2}}}\right)}^{2} \]
   rem-square-sqrt [<=] 49.8	\[ {\left(\sqrt{\color{blue}{\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}}}\right)}^{2} \]
   fabs-sqr [<=] 49.8	\[ {\left(\sqrt{\color{blue}{\left|\sqrt{x \cdot \sqrt{2}} \cdot \sqrt{x \cdot \sqrt{2}}\right|}}\right)}^{2} \]
   rem-square-sqrt [=>] 99.0	\[ {\left(\sqrt{\left|\color{blue}{x \cdot \sqrt{2}}\right|}\right)}^{2} \]
   rem-sqrt-square [<=] 52.3	\[ {\left(\sqrt{\color{blue}{\sqrt{\left(x \cdot \sqrt{2}\right) \cdot \left(x \cdot \sqrt{2}\right)}}}\right)}^{2} \]
   swap-sqr [=>] 52.2	\[ {\left(\sqrt{\sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}}}\right)}^{2} \]
   rem-square-sqrt [=>] 52.4	\[ {\left(\sqrt{\sqrt{\left(x \cdot x\right) \cdot \color{blue}{2}}}\right)}^{2} \]
   *-commutative [<=] 52.4	\[ {\left(\sqrt{\sqrt{\color{blue}{2 \cdot \left(x \cdot x\right)}}}\right)}^{2} \]
   unpow2 [<=] 52.4	\[ {\left(\sqrt{\sqrt{2 \cdot \color{blue}{{x}^{2}}}}\right)}^{2} \]
   count-2 [<=] 52.4	\[ {\left(\sqrt{\sqrt{\color{blue}{{x}^{2} + {x}^{2}}}}\right)}^{2} \]
   unpow2 [=>] 52.4	\[ {\left(\sqrt{\sqrt{\color{blue}{x \cdot x} + {x}^{2}}}\right)}^{2} \]
   unpow2 [=>] 52.4	\[ {\left(\sqrt{\sqrt{x \cdot x + \color{blue}{x \cdot x}}}\right)}^{2} \]
   hypot-def [=>] 99.2	\[ {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(x, x\right)}}\right)}^{2} \]

6. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{\left(1 + \mathsf{hypot}\left(x, x\right)\right) - 1} \]
   Proof

   [Start] 99.2	\[ {\left(\sqrt{\mathsf{hypot}\left(x, x\right)}\right)}^{2} \]
   unpow2 [=>] 99.2	\[ \color{blue}{\sqrt{\mathsf{hypot}\left(x, x\right)} \cdot \sqrt{\mathsf{hypot}\left(x, x\right)}} \]
   add-sqr-sqrt [<=] 100.0	\[ \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
   expm1-log1p-u [=>] 95.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(x, x\right)\right)\right)} \]
   expm1-udef [=>] 50.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\mathsf{hypot}\left(x, x\right)\right)} - 1} \]
   log1p-udef [=>] 50.0	\[ e^{\color{blue}{\log \left(1 + \mathsf{hypot}\left(x, x\right)\right)}} - 1 \]
   add-exp-log [<=] 54.4	\[ \color{blue}{\left(1 + \mathsf{hypot}\left(x, x\right)\right)} - 1 \]

7. Simplified 100.0%

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

   [Start] 54.4	\[ \left(1 + \mathsf{hypot}\left(x, x\right)\right) - 1 \]
   +-commutative [=>] 54.4	\[ \color{blue}{\left(\mathsf{hypot}\left(x, x\right) + 1\right)} - 1 \]
   associate--l+ [=>] 100.0	\[ \color{blue}{\mathsf{hypot}\left(x, x\right) + \left(1 - 1\right)} \]
   metadata-eval [=>] 100.0	\[ \mathsf{hypot}\left(x, x\right) + \color{blue}{0} \]
   +-rgt-identity [=>] 100.0	\[ \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

8. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023135 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))