sqrt C (should all be same)

Percentage Accurate: 54.4% → 99.4%

Time: 26.7s

Alternatives: 6

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: 54.4% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(4 \cdot x\_m\right)}^{0.25} \cdot {x\_m}^{0.75} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow (* 4.0 x_m) 0.25) (pow x_m 0.75)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((4.0 * x_m), 0.25) * pow(x_m, 0.75);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((4.0d0 * x_m) ** 0.25d0) * (x_m ** 0.75d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((4.0 * x_m), 0.25) * Math.pow(x_m, 0.75);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((4.0 * x_m), 0.25) * math.pow(x_m, 0.75)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((Float64(4.0 * x_m) ^ 0.25) * (x_m ^ 0.75))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((4.0 * x_m) ^ 0.25) * (x_m ^ 0.75);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[(4.0 * x$95$m), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[x$95$m, 0.75], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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. pow2 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. unpow-prod-down N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 54.8%

   \[\leadsto \color{blue}{{\left(4 \cdot x\right)}^{0.25} \cdot {x}^{0.75}} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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. pow2 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. metadata-eval N/A

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

   11. pow-plus N/A

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

   12. associate-*r* N/A

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

   13. associate-*r* N/A

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

   14. unpow-prod-down N/A

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

   15. lower-*.f64 N/A

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

4. Applied rewrites 54.8%

   \[\leadsto \color{blue}{{\left(4 \cdot x\right)}^{0.25} \cdot {x}^{0.75}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(4 \cdot x\right)}^{\frac{1}{4}}} \cdot {x}^{\frac{3}{4}} \]

   2. lift-*.f64 N/A

      \[\leadsto {\color{blue}{\left(4 \cdot x\right)}}^{\frac{1}{4}} \cdot {x}^{\frac{3}{4}} \]

   3. *-commutative N/A

      \[\leadsto {\color{blue}{\left(x \cdot 4\right)}}^{\frac{1}{4}} \cdot {x}^{\frac{3}{4}} \]

   4. pow-prod-down N/A

      \[\leadsto \color{blue}{\left({x}^{\frac{1}{4}} \cdot {4}^{\frac{1}{4}}\right)} \cdot {x}^{\frac{3}{4}} \]

   5. metadata-eval N/A

      \[\leadsto \left({x}^{\color{blue}{\left(4 \cdot \frac{1}{16}\right)}} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   6. metadata-eval N/A

      \[\leadsto \left({x}^{\left(4 \cdot \color{blue}{\frac{\frac{1}{8}}{2}}\right)} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   7. metadata-eval N/A

      \[\leadsto \left({x}^{\left(4 \cdot \frac{\color{blue}{\frac{\frac{1}{4}}{2}}}{2}\right)} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   8. pow-pow N/A

      \[\leadsto \left(\color{blue}{{\left({x}^{4}\right)}^{\left(\frac{\frac{\frac{1}{4}}{2}}{2}\right)}} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   9. lift-pow.f64 N/A

      \[\leadsto \left({\color{blue}{\left({x}^{4}\right)}}^{\left(\frac{\frac{\frac{1}{4}}{2}}{2}\right)} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   10. sqrt-pow1 N/A

       \[\leadsto \left(\color{blue}{\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}}} \cdot {4}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   11. metadata-eval N/A

       \[\leadsto \left(\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot {\color{blue}{\left(2 \cdot 2\right)}}^{\frac{1}{4}}\right) \cdot {x}^{\frac{3}{4}} \]

   12. pow-prod-down N/A

       \[\leadsto \left(\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot \color{blue}{\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{4}}\right)}\right) \cdot {x}^{\frac{3}{4}} \]

   13. pow-sqr N/A

       \[\leadsto \left(\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot \color{blue}{{2}^{\left(2 \cdot \frac{1}{4}\right)}}\right) \cdot {x}^{\frac{3}{4}} \]

   14. metadata-eval N/A

       \[\leadsto \left(\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot {2}^{\color{blue}{\frac{1}{2}}}\right) \cdot {x}^{\frac{3}{4}} \]

   15. pow1/2 N/A

       \[\leadsto \left(\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)}} \cdot \color{blue}{\sqrt{2}}\right) \cdot {x}^{\frac{3}{4}} \]

   16. sqrt-unprod N/A

       \[\leadsto \color{blue}{\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot 2}} \cdot {x}^{\frac{3}{4}} \]

   17. lower-sqrt.f64 N/A

       \[\leadsto \color{blue}{\sqrt{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot 2}} \cdot {x}^{\frac{3}{4}} \]

   18. lower-*.f64 N/A

       \[\leadsto \sqrt{\color{blue}{{\left({x}^{4}\right)}^{\left(\frac{\frac{1}{4}}{2}\right)} \cdot 2}} \cdot {x}^{\frac{3}{4}} \]

   19. sqrt-pow1 N/A

       \[\leadsto \sqrt{\color{blue}{\sqrt{{\left({x}^{4}\right)}^{\frac{1}{4}}}} \cdot 2} \cdot {x}^{\frac{3}{4}} \]

   20. lift-pow.f64 N/A

       \[\leadsto \sqrt{\sqrt{{\color{blue}{\left({x}^{4}\right)}}^{\frac{1}{4}}} \cdot 2} \cdot {x}^{\frac{3}{4}} \]

   21. pow-pow N/A

       \[\leadsto \sqrt{\sqrt{\color{blue}{{x}^{\left(4 \cdot \frac{1}{4}\right)}}} \cdot 2} \cdot {x}^{\frac{3}{4}} \]

   22. metadata-eval N/A

       \[\leadsto \sqrt{\sqrt{{x}^{\color{blue}{1}}} \cdot 2} \cdot {x}^{\frac{3}{4}} \]

   23. unpow1 N/A

       \[\leadsto \sqrt{\sqrt{\color{blue}{x}} \cdot 2} \cdot {x}^{\frac{3}{4}} \]

   24. lower-sqrt.f64 54.8

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

6. Applied rewrites 54.8%

   \[\leadsto \color{blue}{\sqrt{\sqrt{x} \cdot 2}} \cdot {x}^{0.75} \]
7. Add Preprocessing

Alternative 3: 99.5% 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 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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 54.8

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

4. Applied rewrites 54.8%

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

Alternative 4: 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 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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 55.7

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

4. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\sqrt{2} \cdot x} \]
5. 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 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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 55.7

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

4. Applied rewrites 55.7%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. sqr-pow N/A

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

   6. pow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto {\left({\color{blue}{16}}^{\left(\frac{\frac{\frac{1}{4}}{2}}{2}\right)}\right)}^{2} \cdot x \]

   13. metadata-eval N/A

       \[\leadsto {\left({16}^{\left(\frac{\color{blue}{\frac{1}{8}}}{2}\right)}\right)}^{2} \cdot x \]

   14. metadata-eval 55.6

       \[\leadsto {\left({16}^{\color{blue}{0.0625}}\right)}^{2} \cdot x \]

6. Applied rewrites 55.6%

   \[\leadsto \color{blue}{{\left({16}^{0.0625}\right)}^{2}} \cdot x \]

8. Applied rewrites 12.2%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 12.2

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

10. Applied rewrites 12.2%

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

Alternative 6: 5.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 1.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 1.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 1.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 1.0

Julia

x_m = abs(x)
function code(x_m)
	return 1.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 1.0

Derivation

1. Initial program 52.4%

   \[\sqrt{2 \cdot \left(x \cdot x\right)} \]
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 55.7

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

4. Applied rewrites 55.7%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

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

   4. pow-pow N/A

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

   5. sqr-pow N/A

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

   6. pow-prod-down N/A

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

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto {\left({\color{blue}{16}}^{\left(\frac{\frac{\frac{1}{4}}{2}}{2}\right)}\right)}^{2} \cdot x \]

   13. metadata-eval N/A

       \[\leadsto {\left({16}^{\left(\frac{\color{blue}{\frac{1}{8}}}{2}\right)}\right)}^{2} \cdot x \]

   14. metadata-eval 55.6

       \[\leadsto {\left({16}^{\color{blue}{0.0625}}\right)}^{2} \cdot x \]

6. Applied rewrites 55.6%

   \[\leadsto \color{blue}{{\left({16}^{0.0625}\right)}^{2}} \cdot x \]

8. Applied rewrites 12.2%

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

10. Applied rewrites 5.6%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))