sqrt B (should all be same)

Percentage Accurate: 54.4% → 100.0%

Time: 2.3s

Alternatives: 4

Speedup: 0.2×

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: 54.4% 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, 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 52.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. count-2-rev N/A

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

   6. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 50.3% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 52.2

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

5. Applied rewrites 52.2%

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

Alternative 3: 3.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

      \[\leadsto \sqrt{x \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   7. +-inverses N/A

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

   8. associate-*r/ N/A

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

   9. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot \color{blue}{0}}{x \cdot x - x \cdot x}} \]

   10. +-inverses N/A

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

   11. distribute-lft-out-- N/A

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

   12. +-inverses N/A

       \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   13. +-inverses N/A

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

   14. flip-+ N/A

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

   15. lower-+.f64 3.6

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

4. Applied rewrites 3.6%

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

Alternative 4: 3.8% accurate, 21.0× 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 52.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. count-2-rev N/A

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

   5. flip-+ N/A

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

   6. +-inverses N/A

      \[\leadsto \sqrt{x \cdot \frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   7. +-inverses N/A

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

   8. associate-*r/ N/A

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

   9. +-inverses N/A

      \[\leadsto \sqrt{\frac{x \cdot \color{blue}{0}}{x \cdot x - x \cdot x}} \]

   10. +-inverses N/A

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

   11. distribute-lft-out-- N/A

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

   12. +-inverses N/A

       \[\leadsto \sqrt{\frac{x \cdot x - x \cdot x}{\color{blue}{0}}} \]

   13. +-inverses N/A

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

   14. flip-+ N/A

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

   15. lower-+.f64 3.6

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

4. Applied rewrites 3.6%

   \[\leadsto \sqrt{\color{blue}{x + x}} \]
5. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-+.f64 N/A

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

   4. flip-+ N/A

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

   5. +-inverses N/A

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

   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

      \[\leadsto {\left(\frac{\color{blue}{{0}^{3}} - 0}{0}\right)}^{\frac{1}{2}} \]

   9. metadata-eval N/A

      \[\leadsto {\left(\frac{{0}^{3} - \color{blue}{{0}^{3}}}{0}\right)}^{\frac{1}{2}} \]

   10. mul0-rgt N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\color{blue}{\left(\left(x \cdot x\right) \cdot 0\right)}}^{3}}{0}\right)}^{\frac{1}{2}} \]

   11. metadata-eval N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{\color{blue}{0 + 0}}\right)}^{\frac{1}{2}} \]

   12. metadata-eval N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{\color{blue}{0 \cdot 0} + 0}\right)}^{\frac{1}{2}} \]

   13. metadata-eval N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \color{blue}{\left(0 + 0\right)}}\right)}^{\frac{1}{2}} \]

   14. metadata-eval N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{0 \cdot 0} + 0\right)}\right)}^{\frac{1}{2}} \]

   15. mul0-rgt N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \left(0 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0\right)} + 0\right)}\right)}^{\frac{1}{2}} \]

   16. mul0-rgt N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \left(\color{blue}{\left(\left(x \cdot x\right) \cdot 0\right)} \cdot \left(\left(x \cdot x\right) \cdot 0\right) + 0\right)}\right)}^{\frac{1}{2}} \]

   17. metadata-eval N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \left(\left(\left(x \cdot x\right) \cdot 0\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right) + \color{blue}{0 \cdot 0}\right)}\right)}^{\frac{1}{2}} \]

   18. mul0-rgt N/A

       \[\leadsto {\left(\frac{{0}^{3} - {\left(\left(x \cdot x\right) \cdot 0\right)}^{3}}{0 \cdot 0 + \left(\left(\left(x \cdot x\right) \cdot 0\right) \cdot \left(\left(x \cdot x\right) \cdot 0\right) + 0 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0\right)}\right)}\right)}^{\frac{1}{2}} \]

   19. flip3-- N/A

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

   20. mul0-rgt N/A

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

   21. metadata-eval N/A

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

   22. metadata-eval 3.8

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

6. Applied rewrites 3.8%

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

Reproduce

herbie shell --seed 2024338 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))