sqrt C (should all be same)

Percentage Accurate: 54.9% → 99.3%

Time: 4.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

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(2.0 * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Initial Program: 54.9% accurate, 1.0× speedup

Math

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

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(2.0 * Float64(x * x)))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= -4e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = sqrt(2.0) * x;
	}
	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 = sqrt(2.0d0) * x
    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 = Math.sqrt(2.0) * x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -3.999999999999988e-310

   1. Initial program 58.5%

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

   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%

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

   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%

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

Alternative 2: 97.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around -inf 52.6%

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

   1. mul-1-neg 52.6%

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

   2. distribute-rgt-neg-in 52.6%

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

4. Simplified 52.6%

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

   1. add-sqr-sqrt 51.4%

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

   2. sqrt-unprod 58.1%

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

   3. sqr-neg 58.1%

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

   4. sqrt-prod 58.4%

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

   5. add-cube-cbrt 57.7%

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

   6. sqrt-prod 57.8%

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

   7. sqrt-unprod 57.6%

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

   8. add-sqr-sqrt 57.8%

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

   9. pow1 57.8%

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

   10. pow1/2 57.8%

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

   11. pow-prod-up 57.8%

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

   12. metadata-eval 57.8%

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

6. Applied egg-rr 57.8%

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

   1. *-commutative 57.8%

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

   2. cbrt-prod 57.4%

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

   3. cbrt-prod 97.5%

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

   4. add-sqr-sqrt 97.5%

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

   5. cbrt-unprod 97.8%

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

   6. swap-sqr 97.7%

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

   7. cbrt-prod 98.2%

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

   8. cbrt-prod 97.9%

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

   9. pow2 97.9%

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

   10. *-commutative 97.9%

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

8. Applied egg-rr 97.9%

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

9. Final simplification 97.9%

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

Alternative 3: 97.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. add-cube-cbrt 57.4%

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

   2. pow3 57.5%

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

   3. *-commutative 57.5%

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

   4. sqrt-prod 57.4%

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

   5. sqrt-prod 46.9%

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

   6. add-sqr-sqrt 48.0%

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

3. Applied egg-rr 48.0%

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

   1. add-sqr-sqrt 46.8%

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

   2. sqrt-unprod 97.8%

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

   3. cbrt-prod 98.0%

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

   4. cbrt-prod 97.7%

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

   5. swap-sqr 97.6%

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

   6. cbrt-prod 57.6%

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

   7. cbrt-unprod 57.4%

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

   8. add-sqr-sqrt 57.4%

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

   9. cbrt-prod 57.6%

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

   10. *-commutative 57.6%

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

   11. add-sqr-sqrt 57.6%

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

   12. rem-sqrt-square 57.6%

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

   13. *-commutative 57.6%

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

   14. cbrt-prod 57.4%

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

   15. cbrt-prod 97.5%

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

   16. add-sqr-sqrt 97.5%

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

   17. cbrt-unprod 97.6%

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

5. Applied egg-rr 97.8%

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

6. Final simplification 97.8%

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

Alternative 4: 97.9% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around -inf 52.6%

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

   1. mul-1-neg 52.6%

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

   2. distribute-rgt-neg-in 52.6%

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

4. Simplified 52.6%

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

   1. add-sqr-sqrt 51.4%

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

   2. sqrt-unprod 58.1%

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

   3. sqr-neg 58.1%

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

   4. sqrt-prod 58.4%

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

   5. add-cube-cbrt 57.7%

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

   6. sqrt-prod 57.8%

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

   7. sqrt-unprod 57.6%

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

   8. add-sqr-sqrt 57.8%

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

   9. pow1 57.8%

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

   10. pow1/2 57.8%

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

   11. pow-prod-up 57.8%

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

   12. metadata-eval 57.8%

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

6. Applied egg-rr 57.8%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{2 \cdot \left(x \cdot x\right)}\right)}^{1.5}} \]
7. 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%

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

8. Applied egg-rr 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 5.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.2e-206) (sqrt (* x 0.0)) (sqrt x)))

C

double code(double x) {
	double tmp;
	if (x <= 3.2e-206) {
		tmp = sqrt((x * 0.0));
	} else {
		tmp = sqrt(x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.2d-206) then
        tmp = sqrt((x * 0.0d0))
    else
        tmp = sqrt(x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.2e-206) {
		tmp = Math.sqrt((x * 0.0));
	} else {
		tmp = Math.sqrt(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.2e-206:
		tmp = math.sqrt((x * 0.0))
	else:
		tmp = math.sqrt(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.2e-206)
		tmp = sqrt(Float64(x * 0.0));
	else
		tmp = sqrt(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.2e-206)
		tmp = sqrt((x * 0.0));
	else
		tmp = sqrt(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.2e-206], N[Sqrt[N[(x * 0.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.19999999999999976e-206

   1. Initial program 53.4%

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

      1. add-cube-cbrt 52.6%

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

      2. pow3 52.6%

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

      3. *-commutative 52.6%

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

      4. sqrt-prod 52.6%

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

      5. sqrt-prod 9.9%

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

      6. add-sqr-sqrt 11.7%

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

   3. Applied egg-rr 11.7%

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

      1. rem-cube-cbrt 11.8%

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

      2. *-commutative 11.8%

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

      3. add-sqr-sqrt 10.0%

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

      4. sqrt-prod 53.2%

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

      5. sqr-neg 53.2%

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

      6. sqrt-unprod 88.9%

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

      7. add-sqr-sqrt 88.8%

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

      8. add-sqr-sqrt 89.7%

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

      9. associate-*l* 89.9%

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

      10. pow1/2 89.9%

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

      11. sqrt-pow1 89.9%

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

      12. metadata-eval 89.9%

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

      13. pow1/2 89.9%

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

      14. sqrt-pow1 89.9%

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

      15. metadata-eval 89.9%

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

      16. add-sqr-sqrt 88.8%

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

      17. sqrt-unprod 53.1%

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

      18. sqr-neg 53.1%

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

      19. sqrt-prod 10.0%

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

      20. add-sqr-sqrt 11.8%

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

   5. Applied egg-rr 11.8%

      \[\leadsto \color{blue}{{2}^{0.25} \cdot \left({2}^{0.25} \cdot x\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 11.8%

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

      2. pow-prod-up 11.8%

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

      3. metadata-eval 11.8%

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

      4. pow1/2 11.8%

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

      5. add-sqr-sqrt 10.0%

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

      6. sqrt-prod 53.2%

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

      7. sqrt-prod 53.4%

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

      8. pow1/2 53.4%

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

   7. Applied egg-rr 53.4%

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

   9. Simplified 4.0%

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

   if 3.19999999999999976e-206 < x

   1. Initial program 65.3%

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

   2. Taylor expanded in x around -inf 1.3%

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

      1. mul-1-neg 1.3%

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

      2. distribute-rgt-neg-in 1.3%

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

   4. Simplified 1.3%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 64.9%

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

      3. sqr-neg 64.9%

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

      4. sqrt-prod 65.3%

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

      5. add-cube-cbrt 64.4%

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

      6. sqrt-prod 64.5%

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

      7. sqrt-unprod 64.3%

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

      8. add-sqr-sqrt 64.5%

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

      9. pow1 64.5%

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

      10. pow1/2 64.5%

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

      11. pow-prod-up 64.5%

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

      12. metadata-eval 64.5%

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

   6. Applied egg-rr 64.5%

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

      1. pow1/3 59.8%

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

      2. pow-pow 65.3%

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

      3. metadata-eval 65.3%

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

      4. pow1/2 65.3%

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

      5. associate-*r* 65.3%

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

      6. count-2 65.3%

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

      7. sqrt-prod 99.0%

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

      8. flip-+ 0.0%

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

      9. difference-of-squares 0.0%

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

      10. +-inverses 0.0%

          \[\leadsto \sqrt{\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}} \cdot \sqrt{x} \]

      11. +-inverses 0.0%

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

      12. +-inverses 0.0%

          \[\leadsto \sqrt{\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}} \cdot \sqrt{x} \]

      13. +-inverses 0.0%

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

      14. associate-*r/ 0.0%

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

      15. +-inverses 0.0%

          \[\leadsto \sqrt{\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \cdot \sqrt{x} \]

      16. +-inverses 0.0%

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

      17. flip-+ 7.9%

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

      18. sqrt-unprod 8.1%

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

      19. add-sqr-sqrt 8.1%

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

   8. Applied egg-rr 8.1%

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

   10. Simplified 7.2%

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

4. Final simplification 5.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-206}:\\ \;\;\;\;\sqrt{x \cdot 0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x}\\ \end{array} \]

Alternative 6: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

2. Taylor expanded in x around 0 48.7%

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

3. Final simplification 48.7%

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

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%

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

2. Taylor expanded in x around -inf 52.6%

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

   1. mul-1-neg 52.6%

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

   2. distribute-rgt-neg-in 52.6%

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

4. Simplified 52.6%

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

   1. add-sqr-sqrt 51.4%

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

   2. sqrt-unprod 58.1%

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

   3. sqr-neg 58.1%

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

   4. sqrt-prod 58.4%

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

   5. add-cube-cbrt 57.7%

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

   6. sqrt-prod 57.8%

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

   7. sqrt-unprod 57.6%

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

   8. add-sqr-sqrt 57.8%

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

   9. pow1 57.8%

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

   10. pow1/2 57.8%

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

   11. pow-prod-up 57.8%

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

   12. metadata-eval 57.8%

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

6. Applied egg-rr 57.8%

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

   1. pow1/3 53.5%

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

   2. pow-pow 58.4%

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

   3. metadata-eval 58.4%

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

   4. pow1/2 58.4%

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

   5. associate-*r* 58.4%

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

   6. count-2 58.4%

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

   7. sqrt-prod 47.6%

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

   8. flip-+ 0.0%

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

   9. difference-of-squares 0.0%

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

   10. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x + x\right) \cdot \color{blue}{0}}{x - x}} \cdot \sqrt{x} \]

   11. +-inverses 0.0%

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

   12. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x + x\right) \cdot \left(x \cdot x - x \cdot x\right)}{\color{blue}{0}}} \cdot \sqrt{x} \]

   13. +-inverses 0.0%

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

   14. associate-*r/ 0.0%

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

   15. +-inverses 0.0%

       \[\leadsto \sqrt{\left(x + x\right) \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \cdot \sqrt{x} \]

   16. +-inverses 0.0%

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

   17. flip-+ 3.9%

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

   18. sqrt-unprod 3.9%

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

   19. add-sqr-sqrt 3.9%

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

8. Applied egg-rr 3.9%

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

10. Simplified 3.3%

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

11. Final simplification 3.3%

    \[\leadsto \sqrt{x} \]

Reproduce

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