sqrt B (should all be same)

Percentage Accurate: 54.9% → 99.3%

Time: 4.7s

Alternatives: 7

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: 54.9% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4e-310) (* (sqrt 2.0) (- x)) (* x (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = x * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = x * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(x * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 58.5%

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

   2. Taylor expanded in x around -inf 99.3%

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{2} \cdot x\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 99.3%

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

      2. distribute-rgt-neg-in 99.3%

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

   4. Simplified 99.3%

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

   if -3.999999999999988e-310 < x

   1. Initial program 58.4%

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

   2. Taylor expanded in x around 0 99.3%

      \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]

Alternative 2: 97.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left({\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{2}\right)}^{1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (pow (cbrt (* x (sqrt 2.0))) 2.0) 1.5))

C

double code(double x) {
	return pow(pow(cbrt((x * sqrt(2.0))), 2.0), 1.5);
}

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.cbrt((x * Math.sqrt(2.0))), 2.0), 1.5);
}

Julia

function code(x)
	return (cbrt(Float64(x * sqrt(2.0))) ^ 2.0) ^ 1.5
end

Wolfram

code[x_] := N[Power[N[Power[N[Power[N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision], 1.5], $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.7%

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

   2. sqrt-prod 57.8%

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

   3. pow2 57.8%

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

   4. associate-*l* 57.8%

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

   5. associate-*l* 57.8%

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

3. Applied egg-rr 57.8%

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

   1. unpow2 57.8%

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

   2. rem-sqrt-square 57.8%

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

   3. rem-square-sqrt 57.6%

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

   4. unpow1/2 57.6%

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

   5. unpow1/2 57.6%

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

   6. fabs-sqr 57.6%

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

   7. unpow1/2 57.6%

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

   8. pow-sqr 57.8%

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

   9. metadata-eval 57.8%

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

   10. unpow1 57.8%

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

   11. *-commutative 57.8%

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

   12. pow-plus 57.8%

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

   13. metadata-eval 57.8%

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

5. Simplified 57.8%

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

   1. add-sqr-sqrt 57.8%

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

   2. swap-sqr 57.8%

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

   3. cbrt-unprod 97.9%

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

   4. pow2 97.9%

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

   5. *-commutative 97.9%

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

7. Applied egg-rr 97.9%

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

8. Final simplification 97.9%

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

Alternative 3: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{2} \cdot {\left(\left|\sqrt[3]{x}\right|\right)}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (sqrt 2.0) (pow (fabs (cbrt x)) 3.0)))

C

double code(double x) {
	return sqrt(2.0) * pow(fabs(cbrt(x)), 3.0);
}

Java

public static double code(double x) {
	return Math.sqrt(2.0) * Math.pow(Math.abs(Math.cbrt(x)), 3.0);
}

Julia

function code(x)
	return Float64(sqrt(2.0) * (abs(cbrt(x)) ^ 3.0))
end

Wolfram

code[x_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Abs[N[Power[x, 1/3], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.4%

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

   2. pow3 57.5%

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

   3. associate-*l* 57.5%

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

   4. sqrt-prod 57.4%

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

   5. sqrt-unprod 46.9%

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

   6. add-sqr-sqrt 48.0%

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

3. Applied egg-rr 48.0%

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

   1. add-sqr-sqrt 46.9%

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

   2. associate-*r* 47.0%

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

   3. sqrt-prod 46.7%

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

   4. sqrt-unprod 57.5%

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

   5. associate-*r* 57.5%

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

   6. add-cube-cbrt 57.2%

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

   7. sqrt-prod 57.2%

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

   8. sqrt-unprod 57.2%

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

   9. add-cbrt-cube 57.6%

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

   10. cbrt-prod 57.4%

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

   11. sqrt-prod 57.6%

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

   12. cbrt-prod 97.7%

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

   13. pow2 97.7%

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

5. Applied egg-rr 97.7%

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

   1. unpow2 97.7%

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

   2. rem-sqrt-square 97.7%

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

7. Simplified 97.7%

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

8. Taylor expanded in x around 0 43.1%

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

10. Simplified 97.7%

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

11. Final simplification 97.7%

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

Alternative 4: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\left|\sqrt[3]{x \cdot \sqrt{2}}\right|\right)}^{3} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (fabs (cbrt (* x (sqrt 2.0)))) 3.0))

C

double code(double x) {
	return pow(fabs(cbrt((x * sqrt(2.0)))), 3.0);
}

Java

public static double code(double x) {
	return Math.pow(Math.abs(Math.cbrt((x * Math.sqrt(2.0)))), 3.0);
}

Julia

function code(x)
	return abs(cbrt(Float64(x * sqrt(2.0)))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Abs[N[Power[N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.4%

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

   2. pow3 57.5%

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

   3. associate-*l* 57.5%

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

   4. sqrt-prod 57.4%

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

   5. sqrt-unprod 46.9%

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

   6. add-sqr-sqrt 48.0%

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

3. Applied egg-rr 48.0%

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

   1. add-sqr-sqrt 46.9%

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

   2. associate-*r* 47.0%

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

   3. sqrt-prod 46.7%

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

   4. sqrt-unprod 57.5%

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

   5. associate-*r* 57.5%

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

   6. add-cube-cbrt 57.2%

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

   7. sqrt-prod 57.2%

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

   8. sqrt-unprod 57.2%

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

   9. add-cbrt-cube 57.6%

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

   10. add-sqr-sqrt 57.6%

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

   11. rem-sqrt-square 57.6%

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

   12. add-cbrt-cube 57.2%

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

5. Applied egg-rr 97.8%

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

6. Final simplification 97.8%

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

Alternative 5: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{x} \cdot \sqrt[3]{x + x}\right)}^{1.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow (* (cbrt x) (cbrt (+ x x))) 1.5))

C

double code(double x) {
	return pow((cbrt(x) * cbrt((x + x))), 1.5);
}

Java

public static double code(double x) {
	return Math.pow((Math.cbrt(x) * Math.cbrt((x + x))), 1.5);
}

Julia

function code(x)
	return Float64(cbrt(x) * cbrt(Float64(x + x))) ^ 1.5
end

Wolfram

code[x_] := N[Power[N[(N[Power[x, 1/3], $MachinePrecision] * N[Power[N[(x + x), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision]

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.7%

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

   2. sqrt-prod 57.8%

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

   3. pow2 57.8%

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

   4. associate-*l* 57.8%

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

   5. associate-*l* 57.8%

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

3. Applied egg-rr 57.8%

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

   1. unpow2 57.8%

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

   2. rem-sqrt-square 57.8%

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

   3. rem-square-sqrt 57.6%

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

   4. unpow1/2 57.6%

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

   5. unpow1/2 57.6%

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

   6. fabs-sqr 57.6%

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

   7. unpow1/2 57.6%

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

   8. pow-sqr 57.8%

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

   9. metadata-eval 57.8%

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

   10. unpow1 57.8%

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

   11. *-commutative 57.8%

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

   12. pow-plus 57.8%

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

   13. metadata-eval 57.8%

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

5. Simplified 57.8%

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

   1. associate-*r* 57.8%

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

   2. cbrt-prod 97.6%

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

   3. add-log-exp 7.2%

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

   4. *-commutative 7.2%

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

   5. exp-lft-sqr 7.1%

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

   6. log-prod 7.1%

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

   7. add-log-exp 13.5%

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

   8. add-log-exp 97.6%

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

7. Applied egg-rr 97.6%

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

8. Final simplification 97.6%

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

Alternative 6: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around 0 48.7%

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

3. Final simplification 48.7%

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

Alternative 7: 3.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return sqrt(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.7%

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

   2. sqrt-prod 57.8%

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

   3. pow2 57.8%

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

   4. associate-*l* 57.8%

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

   5. associate-*l* 57.8%

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

3. Applied egg-rr 57.8%

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

   1. unpow2 57.8%

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

   2. rem-sqrt-square 57.8%

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

   3. rem-square-sqrt 57.6%

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

   4. unpow1/2 57.6%

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

   5. unpow1/2 57.6%

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

   6. fabs-sqr 57.6%

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

   7. unpow1/2 57.6%

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

   8. pow-sqr 57.8%

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

   9. metadata-eval 57.8%

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

   10. unpow1 57.8%

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

   11. *-commutative 57.8%

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

   12. pow-plus 57.8%

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

   13. metadata-eval 57.8%

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

5. Simplified 57.8%

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

   1. add-sqr-sqrt 57.8%

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

   2. swap-sqr 57.8%

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

   3. cbrt-unprod 97.9%

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

   4. pow2 97.9%

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

   5. *-commutative 97.9%

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

7. Applied egg-rr 97.9%

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

   1. pow-pow 48.0%

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

   2. metadata-eval 48.0%

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

   3. rem-cube-cbrt 48.7%

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

   4. *-commutative 48.7%

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

   5. pow1/2 48.7%

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

   6. metadata-eval 48.7%

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

   7. pow-pow 48.7%

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

   8. pow1/3 48.1%

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

   9. add-sqr-sqrt 47.2%

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

   10. associate-*r* 47.1%

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

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{0}{0} \cdot \sqrt{x}} \]

11. Simplified 3.3%

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

12. Final simplification 3.3%

    \[\leadsto \sqrt{x} \]

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))