sqrt D (should all be same)

Percentage Accurate: 54.0% → 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: 54.0% 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 53.4%

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

   1. unpow2 53.4%

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

3. Simplified 53.4%

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

   1. add-cube-cbrt 52.4%

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

   2. pow3 52.5%

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

   3. *-commutative 52.5%

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

   4. sqrt-prod 52.5%

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

   5. sqrt-prod 49.7%

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

   6. add-sqr-sqrt 51.1%

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

5. Applied egg-rr 51.1%

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

   1. rem-cube-cbrt 51.7%

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

   2. *-commutative 51.7%

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

   3. add-sqr-sqrt 51.6%

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

   4. associate-*l* 51.7%

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

   5. pow1/2 51.7%

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

   6. sqrt-pow1 51.7%

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

   7. metadata-eval 51.7%

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

   8. pow1/2 51.7%

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

   9. sqrt-pow1 51.7%

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

   10. metadata-eval 51.7%

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

7. Applied egg-rr 51.7%

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

8. Taylor expanded in x around 0 51.7%

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

   1. *-commutative 51.7%

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

   2. rem-exp-log 46.4%

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

   3. *-rgt-identity 46.4%

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

   4. metadata-eval 46.4%

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

   5. distribute-rgt-out 46.4%

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

   6. distribute-lft-out 46.4%

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

   7. log-prod 50.0%

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

   8. log-pow 50.0%

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

   9. swap-sqr 50.0%

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

   10. rem-square-sqrt 50.0%

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

   11. *-commutative 50.0%

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

   12. metadata-eval 50.0%

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

   13. pow-sqr 50.0%

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

   14. exp-to-pow 50.0%

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

   15. exp-to-pow 50.0%

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

   16. prod-exp 50.0%

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

   17. rem-log-exp 50.0%

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

   18. distribute-lft-out 50.0%

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

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 2023222 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))