sqrt D (should all be same)

Percentage Accurate: 53.9% → 100.0%

Time: 5.6s

Alternatives: 2

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: 53.9% 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 54.1%

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

   1. unpow2 54.1%

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

3. Simplified 54.1%

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

   1. rem-3cbrt-rft 53.2%

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

   2. pow1/3 51.2%

      \[\leadsto \color{blue}{{\left(\sqrt{2 \cdot \left(x \cdot x\right)}\right)}^{0.3333333333333333}} \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}}\right) \]

   3. pow1/2 51.2%

      \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(x \cdot x\right)\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}}\right) \]

   4. pow-pow 51.2%

      \[\leadsto \color{blue}{{\left(2 \cdot \left(x \cdot x\right)\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}}\right) \]

   5. metadata-eval 51.2%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \left(\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}} \cdot \sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)}}\right) \]

   6. cbrt-prod 51.2%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\sqrt[3]{\sqrt{2 \cdot \left(x \cdot x\right)} \cdot \sqrt{2 \cdot \left(x \cdot x\right)}}} \]

   7. add-sqr-sqrt 51.2%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{2 \cdot \left(x \cdot x\right)}} \]

6. Applied egg-rr 51.2%

   \[\leadsto \color{blue}{{\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{2 \cdot \left(x \cdot x\right)}} \]
7. Step-by-step derivation

   1. unpow1/3 50.0%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(2 \cdot \left(x \cdot x\right)\right)}^{0.3333333333333333}} \]

   2. metadata-eval 50.0%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot {\left(2 \cdot \left(x \cdot x\right)\right)}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}} \]

   3. pow-sqr 50.0%

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\left({\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666} \cdot {\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666}\right)} \]

   4. cube-mult 50.0%

      \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(x \cdot x\right)\right)}^{0.16666666666666666}\right)}^{3}} \]

   5. exp-to-pow 50.1%

      \[\leadsto {\color{blue}{\left(e^{\log \left(2 \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666}\right)}}^{3} \]

   6. *-commutative 50.1%

      \[\leadsto {\left(e^{\color{blue}{0.16666666666666666 \cdot \log \left(2 \cdot \left(x \cdot x\right)\right)}}\right)}^{3} \]

   7. metadata-eval 50.1%

      \[\leadsto {\left(e^{\color{blue}{\left(0.3333333333333333 \cdot 0.5\right)} \cdot \log \left(2 \cdot \left(x \cdot x\right)\right)}\right)}^{3} \]

   8. associate-*r* 50.1%

      \[\leadsto {\left(e^{\color{blue}{0.3333333333333333 \cdot \left(0.5 \cdot \log \left(2 \cdot \left(x \cdot x\right)\right)\right)}}\right)}^{3} \]

   9. *-commutative 50.1%

      \[\leadsto {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(2 \cdot \left(x \cdot x\right)\right)\right) \cdot 0.3333333333333333}}\right)}^{3} \]

   10. exp-prod 50.2%

       \[\leadsto {\color{blue}{\left({\left(e^{0.5 \cdot \log \left(2 \cdot \left(x \cdot x\right)\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   11. unpow1/3 50.6%

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

   12. rem-cube-cbrt 50.6%

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

   13. *-commutative 50.6%

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

   14. exp-to-pow 54.1%

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

   15. unpow1/2 54.1%

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

   16. count-2 54.1%

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

   17. hypot-define 100.0%

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

8. Simplified 100.0%

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

Alternative 2: 11.1% accurate, 68.0× speedup

Math

\[\begin{array}{l} \\ x + x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x x))

C

double code(double x) {
	return x + x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x + x
end function

Java

public static double code(double x) {
	return x + x;
}

Python

def code(x):
	return x + x

Julia

function code(x)
	return Float64(x + x)
end

MATLAB

function tmp = code(x)
	tmp = x + x;
end

Wolfram

code[x_] := N[(x + x), $MachinePrecision]

Derivation

1. Initial program 54.1%

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

   1. unpow2 54.1%

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

3. Simplified 54.1%

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

5. Taylor expanded in x around 0 49.7%

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

7. Applied egg-rr 11.1%

   \[\leadsto \color{blue}{x + x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))