sqrt B (should all be same)

Percentage Accurate: 53.7% → 99.6%

Time: 8.1s

Alternatives: 5

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: 53.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.375} \cdot \left(x\_m \cdot {2}^{0.125}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.375) (* x_m (pow 2.0 0.125))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.375) * (x_m * pow(2.0, 0.125));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.375d0) * (x_m * (2.0d0 ** 0.125d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.375) * (x_m * Math.pow(2.0, 0.125));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.375) * (x_m * math.pow(2.0, 0.125))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.375) * Float64(x_m * (2.0 ^ 0.125)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.375) * (x_m * (2.0 ^ 0.125));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.375], $MachinePrecision] * N[(x$95$m * N[Power[2.0, 0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
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. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 47.0

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

4. Applied rewrites 47.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. metadata-eval N/A

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

   11. pow-sqr N/A

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

   12. sqr-pow N/A

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

   13. lift-pow.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{{2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right)} \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
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. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 47.0

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

4. Applied rewrites 47.0%

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

Alternative 3: 99.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
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 48.2

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

4. Applied rewrites 48.2%

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

5. Final simplification 48.2%

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

Alternative 4: 20.3% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
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. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 47.0

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

4. Applied rewrites 47.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. metadata-eval N/A

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

   11. pow-sqr N/A

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

   12. sqr-pow N/A

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

   13. lift-pow.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{{2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right)} \]

8. Applied rewrites 10.9%

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

9. Final simplification 10.9%

   \[\leadsto 2 \cdot x \]
10. Add Preprocessing

Alternative 5: 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 54.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
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. unpow-prod-down N/A

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

   5. lower-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 47.0

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

4. Applied rewrites 47.0%

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

   1. lift-*.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. associate-*l* N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. metadata-eval N/A

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

   11. pow-sqr N/A

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

   12. sqr-pow N/A

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

   13. lift-pow.f64 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-sqrt.f64 N/A

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

   17. lift-sqrt.f64 N/A

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

   18. rem-square-sqrt N/A

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

   19. associate-*r* N/A

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

   20. lower-*.f64 N/A

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

6. Applied rewrites 48.4%

   \[\leadsto \color{blue}{{2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right)} \]

8. Applied rewrites 10.9%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 10.9

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

10. Applied rewrites 10.9%

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

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))