sqrt D (should all be same)

Percentage Accurate: 54.2% → 99.6%

Time: 13.8s

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: 54.2% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.7%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 52.7%

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

3. Simplified 52.7%

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

   1. pow1/2 N/A

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

   2. sqr-pow N/A

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

   3. associate-*r* N/A

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

   4. unpow-prod-down N/A

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

   5. unpow-prod-down N/A

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

   6. associate-*r* N/A

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

   7. sqr-pow N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. sqr-pow N/A

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

   12. associate-*l* N/A

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

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

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

6. Applied egg-rr 48.9%

   \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left(x \cdot {2}^{0.25}\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

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

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{4}} \cdot {2}^{\frac{1}{8}}\right), \color{blue}{\left({2}^{\frac{1}{8}} \cdot x\right)}\right) \]

   6. pow-prod-up N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\left(\frac{1}{4} + \frac{1}{8}\right)}\right), \left(\color{blue}{{2}^{\frac{1}{8}}} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \left(\frac{1}{4} + \frac{1}{8}\right)\right), \left(\color{blue}{{2}^{\frac{1}{8}}} \cdot x\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{3}{8}\right), \left({2}^{\color{blue}{\frac{1}{8}}} \cdot x\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{3}{8}\right), \mathsf{*.f64}\left(\left({2}^{\frac{1}{8}}\right), \color{blue}{x}\right)\right) \]

   10. pow-lowering-pow.f64 49.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{3}{8}\right), \mathsf{*.f64}\left(\mathsf{pow.f64}\left(2, \frac{1}{8}\right), x\right)\right) \]

8. Applied egg-rr 49.0%

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

Alternative 2: 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 52.7%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 52.7%

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

3. Simplified 52.7%

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

   1. pow1/2 N/A

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

   2. associate-*r* N/A

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

   3. unpow-prod-down N/A

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

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

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

   5. pow1/2 N/A

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

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

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

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

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

   8. pow1/2 N/A

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

   9. sqrt-lowering-sqrt.f64 47.8%

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

6. Applied egg-rr 47.8%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x} \cdot \sqrt{x}} \]
7. 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 52.7%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 52.7%

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

3. Simplified 52.7%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

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

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

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

   6. pow1/2 N/A

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

   7. sqrt-lowering-sqrt.f64 48.9%

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

6. Applied egg-rr 48.9%

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

7. Final simplification 48.9%

   \[\leadsto x \cdot \sqrt{2} \]
8. 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 52.7%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

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

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

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

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 52.7%

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

3. Simplified 52.7%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

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

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

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

   6. pow1/2 N/A

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

   7. sqrt-lowering-sqrt.f64 48.9%

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

6. Applied egg-rr 48.9%

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

   1. pow1/2 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\left({4}^{\left(2 \cdot \frac{1}{8}\right)}\right), x\right) \]

   7. pow-sqr N/A

      \[\leadsto \mathsf{*.f64}\left(\left({4}^{\frac{1}{8}} \cdot {4}^{\frac{1}{8}}\right), x\right) \]

   8. pow-prod-down N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(4 \cdot 4\right)}^{\frac{1}{8}}\right), x\right) \]

   9. pow-to-exp N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left(4 \cdot 4\right) \cdot \frac{1}{8}}\right), x\right) \]

   10. exp-lowering-exp.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\log \left(4 \cdot 4\right) \cdot \frac{1}{8}\right)\right), x\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\log \left(4 \cdot 4\right), \frac{1}{8}\right)\right), x\right) \]

   12. log-lowering-log.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(4 \cdot 4\right)\right), \frac{1}{8}\right)\right), x\right) \]

   13. metadata-eval 48.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(16\right), \frac{1}{8}\right)\right), x\right) \]

8. Applied egg-rr 48.7%

   \[\leadsto \color{blue}{e^{\log 16 \cdot 0.125}} \cdot x \]

10. Applied egg-rr 10.8%

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

    1. *-rgt-identity 10.8%

       \[\leadsto x \]

12. Applied egg-rr 10.8%

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

Reproduce

herbie shell --seed 2024148 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))