sqrt D (should all be same)

Percentage Accurate: 54.0% → 99.5%

Time: 4.5s

Alternatives: 4

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 54.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot {x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

double code(double x) {
	return sqrt((2.0 * pow(x, 2.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return math.sqrt((2.0 * math.pow(x, 2.0)))

Julia

function code(x)
	return sqrt(Float64(2.0 * (x ^ 2.0)))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.5% accurate, 1.0× 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 59.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot {x}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left({x}^{2} \cdot 2\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(x \cdot x\right) \cdot 2\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(x \cdot \left(x \cdot 2\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot 2\right)\right)\right) \]

   5. *-lowering-*.f64 59.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 2\right)\right)\right) \]

5. Simplified 59.1%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. unpow-prod-down N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot 2\right)}^{\frac{1}{2}}\right), \color{blue}{\left({x}^{\frac{1}{2}}\right)}\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot 2\right), \frac{1}{2}\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 54.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

7. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{{\left(2 \cdot x\right)}^{0.5} \cdot \sqrt{x}} \]
8. Step-by-step derivation

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot x}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot x\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 54.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

9. Applied egg-rr 54.4%

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

Alternative 2: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\sqrt{2}}{\frac{1}{x\_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (sqrt 2.0) (/ 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return sqrt(2.0) / (1.0 / x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = sqrt(2.0d0) / (1.0d0 / x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.sqrt(2.0) / (1.0 / x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.sqrt(2.0) / (1.0 / x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(sqrt(2.0) / Float64(1.0 / x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = sqrt(2.0) / (1.0 / x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[2.0], $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot {x}^{2}\right)}\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left({x}^{2} \cdot 2\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(x \cdot x\right) \cdot 2\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(x \cdot \left(x \cdot 2\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \left(x \cdot 2\right)\right)\right) \]

   5. *-lowering-*.f64 59.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(x, \mathsf{*.f64}\left(x, 2\right)\right)\right) \]

5. Simplified 59.1%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. unpow-prod-down N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot 2\right)}^{\frac{1}{2}}\right), \color{blue}{\left({x}^{\frac{1}{2}}\right)}\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot 2\right), \frac{1}{2}\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 54.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

7. Applied egg-rr 54.4%

   \[\leadsto \color{blue}{{\left(2 \cdot x\right)}^{0.5} \cdot \sqrt{x}} \]
8. Step-by-step derivation

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot x}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot x\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 54.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

9. Applied egg-rr 54.4%

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

    1. sqrt-prod N/A

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

    2. pow1/2 N/A

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

    3. metadata-eval N/A

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

    4. metadata-eval N/A

       \[\leadsto \left({2}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)\right)} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

    5. pow-sqr N/A

       \[\leadsto \left(\left({2}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right) \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

    6. pow-prod-down N/A

       \[\leadsto \left({\left(2 \cdot 2\right)}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

    7. metadata-eval N/A

       \[\leadsto \left({4}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot \sqrt{x}\right) \cdot \sqrt{x} \]

    8. pow-flip N/A

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

    9. associate-*r* N/A

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

    10. rem-square-sqrt N/A

        \[\leadsto \frac{1}{{4}^{\frac{-1}{4}}} \cdot x \]

    11. associate-/r/ N/A

        \[\leadsto \frac{1}{\color{blue}{\frac{{4}^{\frac{-1}{4}}}{x}}} \]

    12. div-inv N/A

        \[\leadsto \frac{1}{{4}^{\frac{-1}{4}} \cdot \color{blue}{\frac{1}{x}}} \]

    13. associate-/r* N/A

        \[\leadsto \frac{\frac{1}{{4}^{\frac{-1}{4}}}}{\color{blue}{\frac{1}{x}}} \]

    14. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{4}^{\frac{-1}{4}}}\right), \color{blue}{\left(\frac{1}{x}\right)}\right) \]

    15. pow-flip N/A

        \[\leadsto \mathsf{/.f64}\left(\left({4}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    16. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot 2\right)}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right), \left(\frac{1}{x}\right)\right) \]

    17. pow-prod-down N/A

        \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)} \cdot {2}^{\left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)}\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    18. pow-sqr N/A

        \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(2 \cdot \left(\mathsf{neg}\left(\frac{-1}{4}\right)\right)\right)}\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    19. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left({2}^{\left(2 \cdot \frac{1}{4}\right)}\right), \left(\frac{1}{x}\right)\right) \]

    20. metadata-eval N/A

        \[\leadsto \mathsf{/.f64}\left(\left({2}^{\frac{1}{2}}\right), \left(\frac{1}{x}\right)\right) \]

    21. pow1/2 N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2}\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    22. sqrt-lowering-sqrt.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    23. /-lowering-/.f64 55.6%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

11. Applied egg-rr 55.6%

    \[\leadsto \color{blue}{\frac{\sqrt{2}}{\frac{1}{x}}} \]
12. Add Preprocessing

Alternative 3: 99.3% accurate, 2.0× 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 59.1%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 55.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 55.6%

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

Alternative 4: 20.3% accurate, 204.0× 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 59.1%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 55.6%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 55.6%

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

   1. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({2}^{\color{blue}{\frac{1}{2}}}\right)\right) \]

   2. sqr-pow N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \color{blue}{{2}^{\left(\frac{\frac{1}{2}}{2}\right)}}\right)\right) \]

   3. pow2 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left({\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)}^{\color{blue}{2}}\right)\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\left({2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \color{blue}{2}\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{\frac{1}{2}}{2}\right)\right), 2\right)\right) \]

   6. metadata-eval 55.4%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{pow.f64}\left(\mathsf{pow.f64}\left(2, \frac{1}{4}\right), 2\right)\right) \]

7. Applied egg-rr 55.4%

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

9. Applied egg-rr 12.3%

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

    1. *-rgt-identity 12.3%

       \[\leadsto x \]

11. Applied egg-rr 12.3%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))