sqrt C (should all be same)

Percentage Accurate: 54.0% → 100.0%

Time: 13.8s

Alternatives: 3

Speedup: 1.0×

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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return hypot(x, x)
end

MATLAB

function tmp = code(x)
	tmp = hypot(x, x);
end

Wolfram

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 57.6%

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

   1. rem-cube-cbrt 56.7%

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

   2. *-commutative 56.7%

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

   3. sqrt-prod 56.6%

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

   4. sqrt-prod 43.2%

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

   5. add-sqr-sqrt 44.3%

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

4. Applied egg-rr 44.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 44.9%

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

   2. *-commutative 44.9%

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

   3. add-sqr-sqrt 44.9%

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

   4. associate-*l* 45.0%

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

   5. pow1/2 45.0%

      \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   6. sqrt-pow1 45.0%

      \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   7. metadata-eval 45.0%

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

   8. pow1/2 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   9. sqrt-pow1 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   10. metadata-eval 45.0%

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

6. Applied egg-rr 45.0%

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

   1. associate-*r* 44.9%

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

   2. pow-prod-down 44.9%

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

   3. metadata-eval 44.9%

      \[\leadsto {\color{blue}{4}}^{0.25} \cdot x \]

   4. pow1 44.9%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{x}^{1}} \]

   5. metadata-eval 44.9%

      \[\leadsto {4}^{0.25} \cdot {x}^{\color{blue}{\left(4 \cdot 0.25\right)}} \]

   6. pow-pow 27.3%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{\left({x}^{4}\right)}^{0.25}} \]

   7. unpow-prod-down 27.4%

      \[\leadsto \color{blue}{{\left(4 \cdot {x}^{4}\right)}^{0.25}} \]

   8. *-commutative 27.4%

      \[\leadsto {\color{blue}{\left({x}^{4} \cdot 4\right)}}^{0.25} \]

   9. add-sqr-sqrt 27.2%

      \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}} \cdot \sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   10. sqrt-unprod 27.4%

       \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25} \cdot {\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   11. pow-prod-up 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left({x}^{4} \cdot 4\right)}^{\left(0.25 + 0.25\right)}}} \]

   12. metadata-eval 27.4%

       \[\leadsto \sqrt{{\left({x}^{4} \cdot 4\right)}^{\color{blue}{0.5}}} \]

   13. pow1/2 27.4%

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

   14. *-commutative 27.4%

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

   15. sqrt-prod 27.4%

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

   16. metadata-eval 27.4%

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

   17. sqrt-pow1 57.6%

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

   18. metadata-eval 57.6%

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

8. Applied egg-rr 57.6%

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

10. Simplified 100.0%

    \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

11. Final simplification 100.0%

    \[\leadsto \mathsf{hypot}\left(x, x\right) \]
12. Add Preprocessing

Alternative 2: 7.0% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot 2\right) \end{array} \]

FPCore

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

C

double code(double x) {
	return x * (x * 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x * 2.0);
}

Python

def code(x):
	return x * (x * 2.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.6%

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

   1. rem-cube-cbrt 56.7%

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

   2. *-commutative 56.7%

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

   3. sqrt-prod 56.6%

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

   4. sqrt-prod 43.2%

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

   5. add-sqr-sqrt 44.3%

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

4. Applied egg-rr 44.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 44.9%

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

   2. *-commutative 44.9%

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

   3. add-sqr-sqrt 44.9%

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

   4. associate-*l* 45.0%

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

   5. pow1/2 45.0%

      \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   6. sqrt-pow1 45.0%

      \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   7. metadata-eval 45.0%

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

   8. pow1/2 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   9. sqrt-pow1 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   10. metadata-eval 45.0%

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

6. Applied egg-rr 45.0%

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

   1. associate-*r* 44.9%

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

   2. pow-prod-down 44.9%

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

   3. metadata-eval 44.9%

      \[\leadsto {\color{blue}{4}}^{0.25} \cdot x \]

   4. pow1 44.9%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{x}^{1}} \]

   5. metadata-eval 44.9%

      \[\leadsto {4}^{0.25} \cdot {x}^{\color{blue}{\left(4 \cdot 0.25\right)}} \]

   6. pow-pow 27.3%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{\left({x}^{4}\right)}^{0.25}} \]

   7. unpow-prod-down 27.4%

      \[\leadsto \color{blue}{{\left(4 \cdot {x}^{4}\right)}^{0.25}} \]

   8. *-commutative 27.4%

      \[\leadsto {\color{blue}{\left({x}^{4} \cdot 4\right)}}^{0.25} \]

   9. add-sqr-sqrt 27.2%

      \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}} \cdot \sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   10. sqrt-unprod 27.4%

       \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25} \cdot {\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   11. pow-prod-up 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left({x}^{4} \cdot 4\right)}^{\left(0.25 + 0.25\right)}}} \]

   12. metadata-eval 27.4%

       \[\leadsto \sqrt{{\left({x}^{4} \cdot 4\right)}^{\color{blue}{0.5}}} \]

   13. pow1/2 27.4%

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

   14. *-commutative 27.4%

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

   15. sqrt-prod 27.4%

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

   16. metadata-eval 27.4%

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

   17. sqrt-pow1 57.6%

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

   18. metadata-eval 57.6%

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

8. Applied egg-rr 57.6%

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

10. Simplified 100.0%

    \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
11. Step-by-step derivation

    1. hypot-undefine 57.6%

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

    2. flip-+ 0.0%

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

    3. difference-of-squares 0.0%

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

    4. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \color{blue}{0}}{x \cdot x - x \cdot x}} \]

    5. metadata-eval 0.0%

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

    6. +-inverses 0.0%

       \[\leadsto \sqrt{\frac{\left(x \cdot x + x \cdot x\right) \cdot \log 1}{\color{blue}{0}}} \]

    7. metadata-eval 0.0%

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

    8. associate-*r/ 0.0%

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

    9. metadata-eval 0.0%

       \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\color{blue}{0}}{\log 1}} \]

    10. +-inverses 0.0%

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

    11. metadata-eval 0.0%

        \[\leadsto \sqrt{\left(x \cdot x + x \cdot x\right) \cdot \frac{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - \left(x \cdot x\right) \cdot \left(x \cdot x\right)}{\color{blue}{0}}} \]

    12. +-inverses 0.0%

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

    13. flip-+ 6.3%

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

    14. sqrt-unprod 6.6%

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

    15. add-sqr-sqrt 6.6%

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

    16. distribute-lft-out 6.6%

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

    17. *-commutative 6.6%

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

    18. *-un-lft-identity 6.6%

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

    19. distribute-rgt-out 6.6%

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

    20. distribute-lft-out 6.6%

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

    21. metadata-eval 6.6%

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

12. Applied egg-rr 6.6%

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

13. Final simplification 6.6%

    \[\leadsto x \cdot \left(x \cdot 2\right) \]
14. Add Preprocessing

Alternative 3: 5.4% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 32 \end{array} \]

FPCore

(FPCore (x) :precision binary64 32.0)

C

double code(double x) {
	return 32.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 32.0d0
end function

Java

public static double code(double x) {
	return 32.0;
}

Python

def code(x):
	return 32.0

Julia

function code(x)
	return 32.0
end

MATLAB

function tmp = code(x)
	tmp = 32.0;
end

Wolfram

code[x_] := 32.0

Derivation

1. Initial program 57.6%

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

   1. rem-cube-cbrt 56.7%

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

   2. *-commutative 56.7%

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

   3. sqrt-prod 56.6%

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

   4. sqrt-prod 43.2%

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

   5. add-sqr-sqrt 44.3%

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

4. Applied egg-rr 44.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{x \cdot \sqrt{2}}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 44.9%

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

   2. *-commutative 44.9%

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

   3. add-sqr-sqrt 44.9%

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

   4. associate-*l* 45.0%

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

   5. pow1/2 45.0%

      \[\leadsto \sqrt{\color{blue}{{2}^{0.5}}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   6. sqrt-pow1 45.0%

      \[\leadsto \color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot \left(\sqrt{\sqrt{2}} \cdot x\right) \]

   7. metadata-eval 45.0%

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

   8. pow1/2 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   9. sqrt-pow1 45.0%

      \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   10. metadata-eval 45.0%

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

6. Applied egg-rr 45.0%

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

   1. associate-*r* 44.9%

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

   2. pow-prod-down 44.9%

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

   3. metadata-eval 44.9%

      \[\leadsto {\color{blue}{4}}^{0.25} \cdot x \]

   4. pow1 44.9%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{x}^{1}} \]

   5. metadata-eval 44.9%

      \[\leadsto {4}^{0.25} \cdot {x}^{\color{blue}{\left(4 \cdot 0.25\right)}} \]

   6. pow-pow 27.3%

      \[\leadsto {4}^{0.25} \cdot \color{blue}{{\left({x}^{4}\right)}^{0.25}} \]

   7. unpow-prod-down 27.4%

      \[\leadsto \color{blue}{{\left(4 \cdot {x}^{4}\right)}^{0.25}} \]

   8. *-commutative 27.4%

      \[\leadsto {\color{blue}{\left({x}^{4} \cdot 4\right)}}^{0.25} \]

   9. add-sqr-sqrt 27.2%

      \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}} \cdot \sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   10. sqrt-unprod 27.4%

       \[\leadsto \color{blue}{\sqrt{{\left({x}^{4} \cdot 4\right)}^{0.25} \cdot {\left({x}^{4} \cdot 4\right)}^{0.25}}} \]

   11. pow-prod-up 27.4%

       \[\leadsto \sqrt{\color{blue}{{\left({x}^{4} \cdot 4\right)}^{\left(0.25 + 0.25\right)}}} \]

   12. metadata-eval 27.4%

       \[\leadsto \sqrt{{\left({x}^{4} \cdot 4\right)}^{\color{blue}{0.5}}} \]

   13. pow1/2 27.4%

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

   14. *-commutative 27.4%

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

   15. sqrt-prod 27.4%

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

   16. metadata-eval 27.4%

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

   17. sqrt-pow1 57.6%

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

   18. metadata-eval 57.6%

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

8. Applied egg-rr 57.6%

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

10. Simplified 100.0%

    \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

12. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{{\left(\frac{0}{0}\right)}^{5}} \]

14. Simplified 5.2%

    \[\leadsto \color{blue}{32} \]

15. Final simplification 5.2%

    \[\leadsto 32 \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))