sqrt B (should all be same)

Percentage Accurate: 54.2% → 99.6%

Time: 5.2s

Alternatives: 4

Speedup: 1.3×

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: 54.2% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot {64}^{0.0625}\right) \cdot {4}^{0.0625} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (* x_m (pow 64.0 0.0625)) (pow 4.0 0.0625)))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * pow(64.0, 0.0625)) * pow(4.0, 0.0625);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * (64.0d0 ** 0.0625d0)) * (4.0d0 ** 0.0625d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * Math.pow(64.0, 0.0625)) * Math.pow(4.0, 0.0625);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * math.pow(64.0, 0.0625)) * math.pow(4.0, 0.0625)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * (64.0 ^ 0.0625)) * (4.0 ^ 0.0625))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * (64.0 ^ 0.0625)) * (4.0 ^ 0.0625);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * N[Power[64.0, 0.0625], $MachinePrecision]), $MachinePrecision] * N[Power[4.0, 0.0625], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow-prod-down N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow-prod-down N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. sqr-pow N/A

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

   14. *-commutative N/A

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

   15. sqr-pow N/A

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{{4}^{0.0625} \cdot \left({4}^{0.0625} \cdot \left({4}^{0.125} \cdot x\right)\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto {4}^{\frac{1}{16}} \cdot \color{blue}{\left({64}^{\frac{1}{16}} \cdot x\right)} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto {4}^{\frac{1}{16}} \cdot \color{blue}{\left({64}^{\frac{1}{16}} \cdot x\right)} \]

   2. lower-pow.f64 52.5

      \[\leadsto {4}^{0.0625} \cdot \left(\color{blue}{{64}^{0.0625}} \cdot x\right) \]

7. Applied rewrites 52.5%

   \[\leadsto {4}^{0.0625} \cdot \color{blue}{\left({64}^{0.0625} \cdot x\right)} \]

8. Final simplification 52.5%

   \[\leadsto \left(x \cdot {64}^{0.0625}\right) \cdot {4}^{0.0625} \]
9. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt{2 \cdot \sqrt{x\_m}} \cdot {x\_m}^{0.75} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt (* 2.0 (sqrt x_m))) (pow x_m 0.75)))

C

x_m = fabs(x);
double code(double x_m) {
	return sqrt((2.0 * sqrt(x_m))) * pow(x_m, 0.75);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = sqrt((2.0d0 * sqrt(x_m))) * (x_m ** 0.75d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.sqrt((2.0 * Math.sqrt(x_m))) * Math.pow(x_m, 0.75);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.sqrt((2.0 * math.sqrt(x_m))) * math.pow(x_m, 0.75)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(Float64(2.0 * sqrt(x_m))) * (x_m ^ 0.75))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = sqrt((2.0 * sqrt(x_m))) * (x_m ^ 0.75);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[N[(2.0 * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$95$m, 0.75], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. pow1/2 N/A

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

   8. pow1/2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. sqrt-prod N/A

       \[\leadsto \color{blue}{\sqrt{{x}^{\frac{1}{2}} \cdot x} \cdot \sqrt{{x}^{\frac{1}{2}} \cdot 2}} \]

   13. rem-square-sqrt N/A

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

   14. sqrt-prod N/A

       \[\leadsto \sqrt{{x}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{x \cdot x}}} \cdot \sqrt{{x}^{\frac{1}{2}} \cdot 2} \]

   15. pow1/2 N/A

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

   16. pow-prod-down N/A

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

   17. sqrt-pow1 N/A

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

   18. pow-prod-down N/A

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

   19. unpow-prod-down N/A

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

   20. sqr-pow N/A

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

   21. *-commutative N/A

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

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{{x}^{0.75} \cdot \sqrt{\sqrt{x} \cdot 2}} \]

5. Final simplification 51.3%

   \[\leadsto \sqrt{2 \cdot \sqrt{x}} \cdot {x}^{0.75} \]
6. Add Preprocessing

Alternative 3: 99.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \sqrt{2} \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt 2.0) x_m))

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(2.0) * x_m)
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[2.0], $MachinePrecision] * x$95$m), $MachinePrecision]

Derivation

1. Initial program 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. unpow-prod-down N/A

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

   7. pow1/2 N/A

      \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{x \cdot x}} \]

   8. sqrt-prod N/A

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

   9. rem-square-sqrt N/A

      \[\leadsto {2}^{\frac{1}{2}} \cdot \color{blue}{x} \]

   10. lower-*.f64 N/A

       \[\leadsto \color{blue}{{2}^{\frac{1}{2}} \cdot x} \]

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 52.4

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

4. Applied rewrites 52.4%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
5. Add Preprocessing

Alternative 4: 20.3% accurate, 21.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 x_m)

C

x_m = fabs(x);
double code(double x_m) {
	return x_m;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m

Julia

x_m = abs(x)
function code(x_m)
	return x_m
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := x$95$m

Derivation

1. Initial program 49.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. rem-square-sqrt N/A

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

   7. pow1/2 N/A

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

   8. pow1/2 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*l* N/A

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

   12. sqrt-prod N/A

       \[\leadsto \color{blue}{\sqrt{{x}^{\frac{1}{2}} \cdot x} \cdot \sqrt{{x}^{\frac{1}{2}} \cdot 2}} \]

   13. rem-square-sqrt N/A

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

   14. sqrt-prod N/A

       \[\leadsto \sqrt{{x}^{\frac{1}{2}} \cdot \color{blue}{\sqrt{x \cdot x}}} \cdot \sqrt{{x}^{\frac{1}{2}} \cdot 2} \]

   15. pow1/2 N/A

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

   16. pow-prod-down N/A

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

   17. sqrt-pow1 N/A

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

   18. pow-prod-down N/A

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

   19. unpow-prod-down N/A

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

   20. sqr-pow N/A

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

   21. *-commutative N/A

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

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{{x}^{0.75} \cdot \sqrt{\sqrt{x} \cdot 2}} \]

6. Applied rewrites 11.6%

   \[\leadsto \color{blue}{x \cdot 1} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot 1} \]

   2. *-rgt-identity 11.6

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

8. Applied rewrites 11.6%

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

Reproduce

herbie shell --seed 2024273 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))