sqrt C (should all be same)

Percentage Accurate: 53.5% → 99.6%

Time: 16.5s

Alternatives: 4

Speedup: 1.3×

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: 53.5% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {64}^{0.0625} \cdot \left({16}^{0.03125} \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (pow 64.0 0.0625) (* (pow 16.0 0.03125) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(64.0, 0.0625) * (pow(16.0, 0.03125) * x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (64.0d0 ** 0.0625d0) * ((16.0d0 ** 0.03125d0) * x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(64.0, 0.0625) * (Math.pow(16.0, 0.03125) * x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(64.0, 0.0625) * (math.pow(16.0, 0.03125) * x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((64.0 ^ 0.0625) * Float64((16.0 ^ 0.03125) * x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (64.0 ^ 0.0625) * ((16.0 ^ 0.03125) * x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[64.0, 0.0625], $MachinePrecision] * N[(N[Power[16.0, 0.03125], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. unpow-prod-down N/A

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

   7. lower-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f64 50.2

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

4. Applied rewrites 50.2%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqrt-prod N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-sqrt.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. rem-square-sqrt N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow1/2 N/A

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

   14. sqr-pow N/A

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

   15. metadata-eval N/A

       \[\leadsto x \cdot \left({2}^{\color{blue}{\frac{1}{4}}} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto x \cdot \left({2}^{\color{blue}{\left(\frac{1}{8} + \frac{1}{8}\right)}} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

   17. metadata-eval N/A

       \[\leadsto x \cdot \left({2}^{\left(\frac{1}{8} + \frac{1}{8}\right)} \cdot {2}^{\color{blue}{\frac{1}{4}}}\right) \]

   18. metadata-eval N/A

       \[\leadsto x \cdot \left({2}^{\left(\frac{1}{8} + \frac{1}{8}\right)} \cdot {2}^{\color{blue}{\left(\frac{1}{8} + \frac{1}{8}\right)}}\right) \]

   19. pow-prod-down N/A

       \[\leadsto x \cdot \color{blue}{{\left(2 \cdot 2\right)}^{\left(\frac{1}{8} + \frac{1}{8}\right)}} \]

   20. metadata-eval N/A

       \[\leadsto x \cdot {\color{blue}{4}}^{\left(\frac{1}{8} + \frac{1}{8}\right)} \]

   21. pow-prod-up N/A

       \[\leadsto x \cdot \color{blue}{\left({4}^{\frac{1}{8}} \cdot {4}^{\frac{1}{8}}\right)} \]

   22. sqr-pow N/A

       \[\leadsto x \cdot \left(\color{blue}{\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right)} \cdot {4}^{\frac{1}{8}}\right) \]

   23. lift-pow.f64 N/A

       \[\leadsto x \cdot \left(\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot {4}^{\left(\frac{\frac{1}{8}}{2}\right)}\right) \cdot \color{blue}{{4}^{\frac{1}{8}}}\right) \]

   24. associate-*l* N/A

       \[\leadsto x \cdot \color{blue}{\left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot \left({4}^{\left(\frac{\frac{1}{8}}{2}\right)} \cdot {4}^{\frac{1}{8}}\right)\right)} \]

6. Applied rewrites 51.4%

   \[\leadsto \color{blue}{{64}^{0.0625} \cdot \left({16}^{0.03125} \cdot x\right)} \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. unpow-prod-down N/A

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

   7. lower-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. pow1/2 N/A

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

   13. lower-sqrt.f64 50.2

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

4. Applied rewrites 50.2%

   \[\leadsto \color{blue}{\sqrt{x \cdot 2} \cdot \sqrt{x}} \]
5. 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 53.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 51.3

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

4. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
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 53.2%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow2 N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. pow-prod-up N/A

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

   8. metadata-eval N/A

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

   9. pow-plus N/A

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

   10. associate-*r* N/A

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

   11. associate-*r* N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. *-commutative N/A

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

   15. pow-plus N/A

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

   16. metadata-eval N/A

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

   17. lower-pow.f64 N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 N/A

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

   22. pow1/2 N/A

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

   23. lower-sqrt.f64 27.4

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

4. Applied rewrites 27.4%

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

   1. lift-pow.f64 N/A

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

   2. metadata-eval N/A

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

   3. pow-plus N/A

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

   4. pow1/2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lower-*.f64 27.4

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

6. Applied rewrites 27.4%

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

8. Applied rewrites 11.5%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 11.5

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

10. Applied rewrites 11.5%

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

Reproduce

herbie shell --seed 2024320 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))