sqrt B (should all be same)

Percentage Accurate: 53.9% → 100.0%

Time: 3.5s

Alternatives: 1

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot x\right) \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))

C

double code(double x) {
	return sqrt(((2.0 * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((2.0d0 * x) * x))
end function

Java

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}

Python

def code(x):
	return math.sqrt(((2.0 * x) * x))

Julia

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((2.0 * x) * x));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(2 \cdot x\right) \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* (* 2.0 x) x)))

C

double code(double x) {
	return sqrt(((2.0 * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(((2.0d0 * x) * x))
end function

Java

public static double code(double x) {
	return Math.sqrt(((2.0 * x) * x));
}

Python

def code(x):
	return math.sqrt(((2.0 * x) * x))

Julia

function code(x)
	return sqrt(Float64(Float64(2.0 * x) * x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(((2.0 * x) * x));
end

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.3%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
2. Step-by-step derivation

   1. add-cube-cbrt 55.7%

      \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\left(2 \cdot x\right) \cdot x} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x}\right) \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x}}} \]

   2. sqrt-prod 55.7%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\left(2 \cdot x\right) \cdot x} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x}} \cdot \sqrt{\sqrt[3]{\left(2 \cdot x\right) \cdot x}}} \]

   3. pow2 55.7%

      \[\leadsto \sqrt{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot x\right) \cdot x}\right)}^{2}}} \cdot \sqrt{\sqrt[3]{\left(2 \cdot x\right) \cdot x}} \]

   4. associate-*l* 55.7%

      \[\leadsto \sqrt{{\left(\sqrt[3]{\color{blue}{2 \cdot \left(x \cdot x\right)}}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\left(2 \cdot x\right) \cdot x}} \]

   5. associate-*l* 55.7%

      \[\leadsto \sqrt{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\color{blue}{2 \cdot \left(x \cdot x\right)}}} \]

3. Applied egg-rr 55.7%

   \[\leadsto \color{blue}{\sqrt{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}}} \]
4. Step-by-step derivation

   1. unpow2 55.7%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{2 \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{2 \cdot \left(x \cdot x\right)}}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   2. rem-sqrt-square 55.7%

      \[\leadsto \color{blue}{\left|\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right|} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   3. unpow1 55.7%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{1}}\right| \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   4. metadata-eval 55.7%

      \[\leadsto \left|{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{\color{blue}{\left(2 \cdot 0.5\right)}}\right| \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   5. pow-sqr 55.5%

      \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5}}\right| \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   6. fabs-sqr 55.5%

      \[\leadsto \color{blue}{\left({\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5} \cdot {\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5}\right)} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   7. pow-sqr 55.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{\left(2 \cdot 0.5\right)}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   8. metadata-eval 55.7%

      \[\leadsto {\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{\color{blue}{1}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   9. unpow1 55.7%

      \[\leadsto \color{blue}{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt{\sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   10. unpow1/2 55.7%

       \[\leadsto \sqrt[3]{2 \cdot \left(x \cdot x\right)} \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5}} \]

   11. *-commutative 55.7%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{0.5} \cdot \sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]

   12. pow-plus 55.7%

       \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{\left(0.5 + 1\right)}} \]

   13. *-commutative 55.7%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(x \cdot x\right) \cdot 2}}\right)}^{\left(0.5 + 1\right)} \]

   14. associate-*l* 55.7%

       \[\leadsto {\left(\sqrt[3]{\color{blue}{x \cdot \left(x \cdot 2\right)}}\right)}^{\left(0.5 + 1\right)} \]

   15. metadata-eval 55.7%

       \[\leadsto {\left(\sqrt[3]{x \cdot \left(x \cdot 2\right)}\right)}^{\color{blue}{1.5}} \]

5. Simplified 55.7%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \left(x \cdot 2\right)}\right)}^{1.5}} \]
6. Step-by-step derivation

   1. pow1/3 51.8%

      \[\leadsto {\color{blue}{\left({\left(x \cdot \left(x \cdot 2\right)\right)}^{0.3333333333333333}\right)}}^{1.5} \]

   2. pow-pow 56.3%

      \[\leadsto \color{blue}{{\left(x \cdot \left(x \cdot 2\right)\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]

   3. metadata-eval 56.3%

      \[\leadsto {\left(x \cdot \left(x \cdot 2\right)\right)}^{\color{blue}{0.5}} \]

   4. pow1/2 56.3%

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

   5. associate-*r* 56.2%

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

   6. sqrt-prod 56.0%

      \[\leadsto \color{blue}{\sqrt{x \cdot x} \cdot \sqrt{2}} \]

   7. sqrt-unprod 47.1%

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

   8. add-sqr-sqrt 48.6%

      \[\leadsto \color{blue}{x} \cdot \sqrt{2} \]

   9. *-commutative 48.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   10. add-sqr-sqrt 48.4%

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

   11. associate-*l* 48.5%

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

   12. pow1/2 48.5%

       \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   13. sqrt-pow1 48.5%

       \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   14. metadata-eval 48.5%

       \[\leadsto {2}^{\color{blue}{0.25}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   15. pow1/2 48.5%

       \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   16. sqrt-pow1 48.5%

       \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   17. metadata-eval 48.5%

       \[\leadsto {2}^{0.25} \cdot \left({2}^{\color{blue}{0.25}} \cdot x\right) \]

7. Applied egg-rr 48.5%

   \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]

8. Taylor expanded in x around 0 48.6%

   \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
9. Step-by-step derivation

   1. *-commutative 48.6%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]

   2. unpow1/2 48.6%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot x \]

   3. metadata-eval 48.6%

      \[\leadsto {2}^{\color{blue}{\left(2 \cdot 0.25\right)}} \cdot x \]

   4. pow-sqr 48.4%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot x \]

   5. rem-square-sqrt 47.1%

      \[\leadsto \left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \]

   6. swap-sqr 47.1%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot \sqrt{x}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right)} \]

   7. exp-to-pow 47.1%

      \[\leadsto \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot \sqrt{x}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   8. *-commutative 47.1%

      \[\leadsto \left(e^{\color{blue}{0.25 \cdot \log 2}} \cdot \sqrt{x}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   9. rem-exp-log 43.9%

      \[\leadsto \left(e^{0.25 \cdot \log 2} \cdot \color{blue}{e^{\log \left(\sqrt{x}\right)}}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   10. unpow1/2 43.9%

       \[\leadsto \left(e^{0.25 \cdot \log 2} \cdot e^{\log \color{blue}{\left({x}^{0.5}\right)}}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   11. log-pow 43.9%

       \[\leadsto \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{0.5 \cdot \log x}}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   12. metadata-eval 43.9%

       \[\leadsto \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{\left(0.25 \cdot 2\right)} \cdot \log x}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   13. associate-*r* 43.9%

       \[\leadsto \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{0.25 \cdot \left(2 \cdot \log x\right)}}\right) \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   14. exp-sum 43.9%

       \[\leadsto \color{blue}{e^{0.25 \cdot \log 2 + 0.25 \cdot \left(2 \cdot \log x\right)}} \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   15. distribute-lft-in 43.9%

       \[\leadsto e^{\color{blue}{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)}} \cdot \left({2}^{0.25} \cdot \sqrt{x}\right) \]

   16. exp-to-pow 43.9%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(\color{blue}{e^{\log 2 \cdot 0.25}} \cdot \sqrt{x}\right) \]

   17. *-commutative 43.9%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{\color{blue}{0.25 \cdot \log 2}} \cdot \sqrt{x}\right) \]

   18. rem-exp-log 43.4%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{0.25 \cdot \log 2} \cdot \color{blue}{e^{\log \left(\sqrt{x}\right)}}\right) \]

   19. unpow1/2 43.4%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{0.25 \cdot \log 2} \cdot e^{\log \color{blue}{\left({x}^{0.5}\right)}}\right) \]

   20. log-pow 43.3%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{0.5 \cdot \log x}}\right) \]

   21. metadata-eval 43.3%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{\left(0.25 \cdot 2\right)} \cdot \log x}\right) \]

   22. associate-*r* 43.3%

       \[\leadsto e^{0.25 \cdot \left(\log 2 + 2 \cdot \log x\right)} \cdot \left(e^{0.25 \cdot \log 2} \cdot e^{\color{blue}{0.25 \cdot \left(2 \cdot \log x\right)}}\right) \]

10. Simplified 100.0%

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

11. Final simplification 100.0%

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

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))