sqrt C (should all be same)

Percentage Accurate: 54.0% → 99.4%

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: 54.0% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.125} \cdot \left({2}^{0.125} \cdot \left(x\_m \cdot {2}^{0.25}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (pow 2.0 0.125) (* (pow 2.0 0.125) (* x_m (pow 2.0 0.25)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.125) * (pow(2.0, 0.125) * (x_m * pow(2.0, 0.25)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.125d0) * ((2.0d0 ** 0.125d0) * (x_m * (2.0d0 ** 0.25d0)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.125) * (Math.pow(2.0, 0.125) * (x_m * Math.pow(2.0, 0.25)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.125) * (math.pow(2.0, 0.125) * (x_m * math.pow(2.0, 0.25)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.125) * Float64((2.0 ^ 0.125) * Float64(x_m * (2.0 ^ 0.25))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.125) * ((2.0 ^ 0.125) * (x_m * (2.0 ^ 0.25)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(x$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow1/2 N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. unpow-prod-down N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lift-*.f64 N/A

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

   12. unpow-prod-down N/A

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

   13. sqr-pow N/A

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

   14. associate-*l* N/A

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

   15. lower-*.f64 N/A

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

4. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left(x \cdot {2}^{0.25}\right)\right)} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.25} \cdot \left(x\_m \cdot {2}^{0.25}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.25) (* x_m (pow 2.0 0.25))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.25) * (x_m * pow(2.0, 0.25));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.25d0) * (x_m * (2.0d0 ** 0.25d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.25) * (x_m * Math.pow(2.0, 0.25));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.25) * (x_m * math.pow(2.0, 0.25))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.25) * Float64(x_m * (2.0 ^ 0.25)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.25) * (x_m * (2.0 ^ 0.25));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.25], $MachinePrecision] * N[(x$95$m * N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.4%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow1/2 N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. unpow-prod-down N/A

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

   9. pow-prod-down N/A

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

   10. lift-*.f64 N/A

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

   11. unpow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. lower-*.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. associate-*r* N/A

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

4. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{{2}^{0.25} \cdot \left(x \cdot {2}^{0.25}\right)} \]
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 45.4%

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

   1. lift-*.f64 N/A

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

   2. sqrt-prod N/A

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

   3. pow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

   7. lower-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 52.7

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

4. Applied egg-rr 52.7%

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

5. Final simplification 52.7%

   \[\leadsto x \cdot \sqrt{2} \]
6. 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 45.4%

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

   1. lift-*.f64 N/A

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

   2. sqrt-prod N/A

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

   3. pow1/2 N/A

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

   4. lift-*.f64 N/A

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

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

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

   7. lower-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 52.7

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

4. Applied egg-rr 52.7%

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

   1. pow1/2 N/A

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

   2. metadata-eval N/A

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

   3. pow-sqr N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. pow-prod-up N/A

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

   7. pow-prod-down N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. pow-to-exp N/A

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

   11. lower-exp.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto e^{\log \left(16 \cdot \left(\color{blue}{4} \cdot \left(2 \cdot 2\right)\right)\right) \cdot \frac{\frac{1}{8}}{2}} \cdot x \]

   18. metadata-eval N/A

       \[\leadsto e^{\log \left(16 \cdot \left(4 \cdot \color{blue}{4}\right)\right) \cdot \frac{\frac{1}{8}}{2}} \cdot x \]

   19. metadata-eval N/A

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

   20. metadata-eval N/A

       \[\leadsto e^{\log \color{blue}{256} \cdot \frac{\frac{1}{8}}{2}} \cdot x \]

   21. metadata-eval 52.7

       \[\leadsto e^{\log 256 \cdot \color{blue}{0.0625}} \cdot x \]

6. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{e^{\log 256 \cdot 0.0625}} \cdot x \]
7. Step-by-step derivation

   1. exp-to-pow N/A

      \[\leadsto \color{blue}{{256}^{\frac{1}{16}}} \cdot x \]

   2. sqr-pow N/A

      \[\leadsto \color{blue}{\left({256}^{\left(\frac{\frac{1}{16}}{2}\right)} \cdot {256}^{\left(\frac{\frac{1}{16}}{2}\right)}\right)} \cdot x \]

   3. pow2 N/A

      \[\leadsto \color{blue}{{\left({256}^{\left(\frac{\frac{1}{16}}{2}\right)}\right)}^{2}} \cdot x \]

   4. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\left({256}^{\left(\frac{\frac{1}{16}}{2}\right)}\right)}^{2}} \cdot x \]

   5. lower-pow.f64 N/A

      \[\leadsto {\color{blue}{\left({256}^{\left(\frac{\frac{1}{16}}{2}\right)}\right)}}^{2} \cdot x \]

   6. metadata-eval 52.7

      \[\leadsto {\left({256}^{\color{blue}{0.03125}}\right)}^{2} \cdot x \]

8. Applied egg-rr 52.7%

   \[\leadsto \color{blue}{{\left({256}^{0.03125}\right)}^{2}} \cdot x \]

10. Applied egg-rr 11.7%

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

11. Final simplification 11.7%

    \[\leadsto x \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024208 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))