sqrt A (should all be same)

Percentage Accurate: 53.9% → 100.0%

Time: 4.6s

Alternatives: 5

Speedup: 0.2×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))

C

double code(double x) {
	return sqrt(((x * x) + (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt(((x * x) + (x * x)));
}

Python

def code(x):
	return math.sqrt(((x * x) + (x * x)))

Julia

function code(x)
	return sqrt(Float64(Float64(x * x) + Float64(x * x)))
end

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (+ (* x x) (* x x))))

C

double code(double x) {
	return sqrt(((x * x) + (x * x)));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt(((x * x) + (x * x)));
}

Python

def code(x):
	return math.sqrt(((x * x) + (x * x)))

Julia

function code(x)
	return sqrt(Float64(Float64(x * x) + Float64(x * x)))
end

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.2× 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 52.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. accelerator-lowering-hypot.f64 100.0

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

4. Applied egg-rr 100.0%

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

Alternative 2: 49.4% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (sqrt x) (sqrt (+ x x))))

C

double code(double x) {
	return sqrt(x) * sqrt((x + x));
}

Fortran

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

Java

public static double code(double x) {
	return Math.sqrt(x) * Math.sqrt((x + x));
}

Python

def code(x):
	return math.sqrt(x) * math.sqrt((x + x))

Julia

function code(x)
	return Float64(sqrt(x) * sqrt(Float64(x + x)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. sqrt-prod N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 45.4

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

4. Applied egg-rr 45.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x + x}} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\sqrt{2} \cdot 0.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (* (sqrt 2.0) 0.5)))

C

double code(double x) {
	return x / (sqrt(2.0) * 0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (sqrt(2.0d0) * 0.5d0)
end function

Java

public static double code(double x) {
	return x / (Math.sqrt(2.0) * 0.5);
}

Python

def code(x):
	return x / (math.sqrt(2.0) * 0.5)

Julia

function code(x)
	return Float64(x / Float64(sqrt(2.0) * 0.5))
end

MATLAB

function tmp = code(x)
	tmp = x / (sqrt(2.0) * 0.5);
end

Wolfram

code[x_] := N[(x / N[(N[Sqrt[2.0], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. sqrt-prod N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 45.4

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

4. Applied egg-rr 45.4%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \sqrt{2}\right)\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right)\right) \cdot \sqrt{2}\right)\right) \]

   4. distribute-rgt-neg-out N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)} \cdot \sqrt{2}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\sqrt{-1} \cdot \left(\left(\mathsf{neg}\left(\sqrt{-1}\right)\right) \cdot \sqrt{2}\right)\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \left(\sqrt{x} \cdot \sqrt{-1}\right)\right)} \]

   9. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{x}\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{x}\right)} \]

   11. +-lft-identity N/A

       \[\leadsto \sqrt{x} \cdot \left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(0 + \sqrt{x}\right)}\right) \]

   12. +-commutative N/A

       \[\leadsto \sqrt{x} \cdot \left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{x} + 0\right)}\right) \]

   13. distribute-rgt-in N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) + 0 \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right)\right)} \]

   14. mul0-lft N/A

       \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) + \color{blue}{0}\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{x}, \left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}, 0\right)} \]

7. Simplified 45.2%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, 0\right)} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. rem-square-sqrt N/A

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

   4. *-commutative N/A

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

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

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

   6. sqrt-lowering-sqrt.f64 46.4

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

9. Applied egg-rr 46.4%

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

    1. pow1/2 N/A

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

    2. metadata-eval N/A

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

    3. metadata-eval N/A

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

    4. pow-plus N/A

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

    5. pow-flip N/A

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

    6. pow1/2 N/A

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

    7. associate-*r* N/A

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

    8. *-commutative N/A

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

    9. associate-/r/ N/A

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

    10. clear-num N/A

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

    11. metadata-eval N/A

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

    12. div-inv N/A

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

    13. associate-/l/ N/A

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

    14. metadata-eval N/A

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

    15. div-inv N/A

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

    16. clear-num N/A

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

    17. div-inv N/A

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

    18. *-commutative N/A

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

    19. pow1/2 N/A

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

    20. pow-flip N/A

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

    21. pow-plus N/A

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

    22. metadata-eval N/A

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

    23. metadata-eval N/A

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

    24. pow1/2 N/A

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

    25. /-lowering-/.f64 N/A

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

11. Applied egg-rr 46.4%

    \[\leadsto \color{blue}{\frac{x}{\sqrt{2} \cdot 0.5}} \]
12. Add Preprocessing

Alternative 4: 50.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. sqrt-prod N/A

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

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

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

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

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

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

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

   6. +-lowering-+.f64 45.4

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

4. Applied egg-rr 45.4%

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

5. Taylor expanded in x around 0

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

   1. *-lft-identity N/A

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

   2. metadata-eval N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot \sqrt{2}\right)\right) \]

   3. rem-square-sqrt N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\left(\mathsf{neg}\left(\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}\right)\right) \cdot \sqrt{2}\right)\right) \]

   4. distribute-rgt-neg-out N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\color{blue}{\left(\sqrt{-1} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)} \cdot \sqrt{2}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \color{blue}{\left(\sqrt{-1} \cdot \left(\left(\mathsf{neg}\left(\sqrt{-1}\right)\right) \cdot \sqrt{2}\right)\right)}\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\sqrt{-1} \cdot \color{blue}{\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)}\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{-1}\right) \cdot \left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right)\right)} \]

   8. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \left(\sqrt{x} \cdot \sqrt{-1}\right)\right)} \]

   9. *-commutative N/A

      \[\leadsto \sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{x}\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \sqrt{x}\right)} \]

   11. +-lft-identity N/A

       \[\leadsto \sqrt{x} \cdot \left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(0 + \sqrt{x}\right)}\right) \]

   12. +-commutative N/A

       \[\leadsto \sqrt{x} \cdot \left(\left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) \cdot \color{blue}{\left(\sqrt{x} + 0\right)}\right) \]

   13. distribute-rgt-in N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) + 0 \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right)\right)} \]

   14. mul0-lft N/A

       \[\leadsto \sqrt{x} \cdot \left(\sqrt{x} \cdot \left(\left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}\right) + \color{blue}{0}\right) \]

   15. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{x}, \left(\sqrt{2} \cdot \left(\mathsf{neg}\left(\sqrt{-1}\right)\right)\right) \cdot \sqrt{-1}, 0\right)} \]

7. Simplified 45.2%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{2}, 0\right)} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. rem-square-sqrt N/A

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

   4. *-commutative N/A

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

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

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

   6. sqrt-lowering-sqrt.f64 46.4

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

9. Applied egg-rr 46.4%

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

10. Final simplification 46.4%

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

Alternative 5: 3.9% accurate, 24.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 52.5%

   \[\sqrt{x \cdot x + x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

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

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

   4. +-lowering-+.f64 52.6

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

4. Applied egg-rr 52.6%

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

   1. pow1/2 N/A

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

   2. *-commutative N/A

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

   3. distribute-lft-out N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

      \[\leadsto {\left(\frac{\color{blue}{0}}{x \cdot x - x \cdot x}\right)}^{\frac{1}{2}} \]

   6. +-inverses N/A

      \[\leadsto {\left(\frac{0}{\color{blue}{0}}\right)}^{\frac{1}{2}} \]

   7. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 - 0}}{0}\right)}^{\frac{1}{2}} \]

   8. metadata-eval N/A

      \[\leadsto {\left(\frac{\color{blue}{0 \cdot 0} - 0}{0}\right)}^{\frac{1}{2}} \]

   9. metadata-eval N/A

      \[\leadsto {\left(\frac{0 \cdot 0 - \color{blue}{0 \cdot 0}}{0}\right)}^{\frac{1}{2}} \]

   10. metadata-eval N/A

       \[\leadsto {\left(\frac{0 \cdot 0 - 0 \cdot 0}{\color{blue}{0 + 0}}\right)}^{\frac{1}{2}} \]

   11. flip-- N/A

       \[\leadsto {\color{blue}{\left(0 - 0\right)}}^{\frac{1}{2}} \]

   12. metadata-eval N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   13. metadata-eval 3.6

       \[\leadsto \color{blue}{0} \]

6. Applied egg-rr 3.6%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024198 
(FPCore (x)
  :name "sqrt A (should all be same)"
  :precision binary64
  (sqrt (+ (* x x) (* x x))))