sqrt B (should all be same)

Percentage Accurate: 54.7% → 100.0%

Time: 18.5s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

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(Float64(2.0 * x) * x))
end

MATLAB

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

Wolfram

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

Initial Program: 54.7% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * x) * x))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] * x), $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 50.7%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

   3. pow1/2 50.3%

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

   4. associate-*l* 50.3%

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

   5. *-commutative 50.3%

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

   6. unpow-prod-down 50.3%

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

   7. pow1/2 50.3%

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

   8. sqrt-unprod 50.9%

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

   9. add-sqr-sqrt 51.0%

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

   10. pow1/2 51.0%

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

4. Applied egg-rr 51.0%

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

   1. unpow2 51.0%

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

   2. add-sqr-sqrt 52.2%

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

   3. *-commutative 52.2%

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

   4. add-sqr-sqrt 52.1%

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

   5. associate-*l* 52.3%

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

   6. pow1/2 52.3%

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

   7. sqrt-pow1 52.3%

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

   8. metadata-eval 52.3%

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

   9. pow1/2 52.3%

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

   10. sqrt-pow1 52.3%

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

   11. metadata-eval 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in x around 0 52.2%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.2%

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

   4. rem-sqrt-square 50.5%

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

   5. unpow1/2 50.5%

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

   6. *-commutative 50.5%

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

   7. unpow1/2 50.5%

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

   8. *-commutative 50.5%

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

   9. swap-sqr 50.3%

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

   10. unpow1/2 50.3%

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

   11. unpow1/2 50.3%

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

   12. rem-square-sqrt 50.7%

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

   13. unpow2 50.7%

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

   14. count-2 50.7%

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

   15. unpow2 50.7%

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

   16. unpow2 50.7%

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

   17. hypot-define 100.0%

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

9. Simplified 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 6.9% 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 50.7%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

   3. pow1/2 50.3%

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

   4. associate-*l* 50.3%

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

   5. *-commutative 50.3%

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

   6. unpow-prod-down 50.3%

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

   7. pow1/2 50.3%

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

   8. sqrt-unprod 50.9%

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

   9. add-sqr-sqrt 51.0%

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

   10. pow1/2 51.0%

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

4. Applied egg-rr 51.0%

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

   1. unpow2 51.0%

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

   2. add-sqr-sqrt 52.2%

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

   3. *-commutative 52.2%

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

   4. add-sqr-sqrt 52.1%

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

   5. associate-*l* 52.3%

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

   6. pow1/2 52.3%

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

   7. sqrt-pow1 52.3%

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

   8. metadata-eval 52.3%

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

   9. pow1/2 52.3%

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

   10. sqrt-pow1 52.3%

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

   11. metadata-eval 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in x around 0 52.2%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.2%

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

   4. rem-sqrt-square 50.5%

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

   5. unpow1/2 50.5%

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

   6. *-commutative 50.5%

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

   7. unpow1/2 50.5%

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

   8. *-commutative 50.5%

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

   9. swap-sqr 50.3%

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

   10. unpow1/2 50.3%

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

   11. unpow1/2 50.3%

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

   12. rem-square-sqrt 50.7%

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

   13. unpow2 50.7%

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

   14. count-2 50.7%

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

   15. unpow2 50.7%

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

   16. unpow2 50.7%

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

   17. hypot-define 100.0%

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

9. Simplified 100.0%

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

    1. hypot-undefine 50.7%

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

    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-+ 6.8%

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

    14. sqrt-unprod 7.1%

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

    15. add-sqr-sqrt 7.1%

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

    16. distribute-lft-out 7.1%

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

    17. *-commutative 7.1%

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

    18. *-un-lft-identity 7.1%

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

    19. distribute-rgt-out 7.1%

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

    20. distribute-lft-out 7.1%

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

    21. metadata-eval 7.1%

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

11. Applied egg-rr 7.1%

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

12. Final simplification 7.1%

    \[\leadsto x \cdot \left(x \cdot 2\right) \]
13. 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 50.7%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

   3. pow1/2 50.3%

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

   4. associate-*l* 50.3%

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

   5. *-commutative 50.3%

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

   6. unpow-prod-down 50.3%

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

   7. pow1/2 50.3%

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

   8. sqrt-unprod 50.9%

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

   9. add-sqr-sqrt 51.0%

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

   10. pow1/2 51.0%

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

4. Applied egg-rr 51.0%

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

   1. unpow2 51.0%

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

   2. add-sqr-sqrt 52.2%

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

   3. *-commutative 52.2%

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

   4. add-sqr-sqrt 52.1%

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

   5. associate-*l* 52.3%

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

   6. pow1/2 52.3%

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

   7. sqrt-pow1 52.3%

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

   8. metadata-eval 52.3%

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

   9. pow1/2 52.3%

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

   10. sqrt-pow1 52.3%

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

   11. metadata-eval 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in x around 0 52.2%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.2%

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

   4. rem-sqrt-square 50.5%

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

   5. unpow1/2 50.5%

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

   6. *-commutative 50.5%

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

   7. unpow1/2 50.5%

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

   8. *-commutative 50.5%

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

   9. swap-sqr 50.3%

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

   10. unpow1/2 50.3%

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

   11. unpow1/2 50.3%

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

   12. rem-square-sqrt 50.7%

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

   13. unpow2 50.7%

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

   14. count-2 50.7%

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

   15. unpow2 50.7%

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

   16. unpow2 50.7%

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

   17. hypot-define 100.0%

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

9. Simplified 100.0%

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

11. Applied egg-rr 0.0%

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

13. Simplified 5.4%

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

14. Final simplification 5.4%

    \[\leadsto 32 \]
15. Add Preprocessing

Alternative 4: 5.4% accurate, 105.0× speedup

Math

\[\begin{array}{l} \\ 64 \end{array} \]

FPCore

(FPCore (x) :precision binary64 64.0)

C

double code(double x) {
	return 64.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 64.0d0
end function

Java

public static double code(double x) {
	return 64.0;
}

Python

def code(x):
	return 64.0

Julia

function code(x)
	return 64.0
end

MATLAB

function tmp = code(x)
	tmp = 64.0;
end

Wolfram

code[x_] := 64.0

Derivation

1. Initial program 50.7%

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

   1. add-sqr-sqrt 50.3%

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

   2. pow2 50.3%

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

   3. pow1/2 50.3%

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

   4. associate-*l* 50.3%

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

   5. *-commutative 50.3%

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

   6. unpow-prod-down 50.3%

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

   7. pow1/2 50.3%

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

   8. sqrt-unprod 50.9%

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

   9. add-sqr-sqrt 51.0%

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

   10. pow1/2 51.0%

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

4. Applied egg-rr 51.0%

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

   1. unpow2 51.0%

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

   2. add-sqr-sqrt 52.2%

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

   3. *-commutative 52.2%

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

   4. add-sqr-sqrt 52.1%

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

   5. associate-*l* 52.3%

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

   6. pow1/2 52.3%

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

   7. sqrt-pow1 52.3%

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

   8. metadata-eval 52.3%

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

   9. pow1/2 52.3%

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

   10. sqrt-pow1 52.3%

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

   11. metadata-eval 52.3%

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

6. Applied egg-rr 52.3%

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

7. Taylor expanded in x around 0 52.2%

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

   1. rem-square-sqrt 51.0%

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

   2. fabs-sqr 51.0%

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

   3. rem-square-sqrt 99.2%

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

   4. rem-sqrt-square 50.5%

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

   5. unpow1/2 50.5%

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

   6. *-commutative 50.5%

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

   7. unpow1/2 50.5%

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

   8. *-commutative 50.5%

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

   9. swap-sqr 50.3%

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

   10. unpow1/2 50.3%

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

   11. unpow1/2 50.3%

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

   12. rem-square-sqrt 50.7%

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

   13. unpow2 50.7%

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

   14. count-2 50.7%

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

   15. unpow2 50.7%

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

   16. unpow2 50.7%

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

   17. hypot-define 100.0%

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

9. Simplified 100.0%

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

11. Applied egg-rr 0.0%

    \[\leadsto \color{blue}{{\left(\frac{0}{0}\right)}^{6}} \]

13. Simplified 5.4%

    \[\leadsto \color{blue}{64} \]

14. Final simplification 5.4%

    \[\leadsto 64 \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))