sqrt D (should all be same)

Percentage Accurate: 54.1% → 100.0%

Time: 4.3s

Alternatives: 1

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.0× 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 60.4%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

   1. unpow2 60.4%

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

   2. sqr-neg 60.4%

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

   3. unpow2 60.4%

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

   4. unpow2 60.4%

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

   5. sqr-neg 60.4%

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

3. Simplified 60.4%

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

   1. add-cube-cbrt 59.3%

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

   2. pow3 59.3%

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

   3. *-commutative 59.3%

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

   4. sqrt-prod 59.3%

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

   5. sqrt-prod 52.0%

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

   6. add-sqr-sqrt 53.0%

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

5. Applied egg-rr 53.0%

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

   1. rem-cube-cbrt 53.8%

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

   2. *-commutative 53.8%

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

   3. add-sqr-sqrt 53.7%

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

   4. associate-*l* 53.8%

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

   5. pow1/2 53.8%

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

   6. sqrt-pow1 53.8%

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

   7. metadata-eval 53.8%

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

   8. pow1/2 53.8%

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

   9. sqrt-pow1 53.8%

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

   10. metadata-eval 53.8%

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

7. Applied egg-rr 53.8%

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

8. Taylor expanded in x around 0 53.8%

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

   1. unpow1/2 53.8%

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

   2. metadata-eval 53.8%

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

   3. pow-sqr 53.7%

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

   4. rem-square-sqrt 52.5%

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

   5. swap-sqr 52.5%

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

   6. unpow1/2 52.5%

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

   7. exp-to-pow 49.3%

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

   8. metadata-eval 49.3%

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

   9. associate-*l* 49.3%

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

   10. *-commutative 49.3%

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

   11. *-commutative 49.3%

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

   12. exp-to-pow 49.3%

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

   13. *-commutative 49.3%

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

   14. exp-sum 48.8%

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

   15. distribute-lft-in 48.8%

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

   16. +-commutative 48.8%

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

   17. *-commutative 48.8%

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

   18. exp-prod 30.6%

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

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 2023293 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))