sqrt D (should all be same)

Percentage Accurate: 53.5% → 100.0%

Time: 4.6s

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: 53.5% 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: 100.0% accurate, 2.0× 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 54.3%

   \[\sqrt{2 \cdot {x}^{2}} \]

2. Taylor expanded in x around 0 49.2%

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

   1. rem-square-sqrt 47.9%

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

   2. fabs-sqr 47.9%

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

   3. rem-square-sqrt 99.3%

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

   4. rem-sqrt-square 54.0%

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

   5. swap-sqr 53.7%

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

   6. unpow2 53.7%

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

   7. rem-square-sqrt 54.3%

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

   8. *-commutative 54.3%

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

   9. count-2 54.3%

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

   10. unpow2 54.3%

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

   11. unpow2 54.3%

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

   12. hypot-def 100.0%

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

4. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 6.9% accurate, 40.8× 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 54.3%

   \[\sqrt{2 \cdot {x}^{2}} \]

2. Taylor expanded in x around 0 49.2%

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

   1. rem-square-sqrt 47.9%

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

   2. fabs-sqr 47.9%

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

   3. rem-square-sqrt 99.3%

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

   4. rem-sqrt-square 54.0%

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

   5. swap-sqr 53.7%

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

   6. unpow2 53.7%

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

   7. rem-square-sqrt 54.3%

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

   8. *-commutative 54.3%

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

   9. count-2 54.3%

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

   10. unpow2 54.3%

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

   11. unpow2 54.3%

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

   12. hypot-def 100.0%

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

4. Simplified 100.0%

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

   1. hypot-udef 54.3%

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

   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. +-inverses 0.0%

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

   4. metadata-eval 0.0%

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

   5. +-inverses 0.0%

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

   6. +-inverses 0.0%

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

   7. +-inverses 0.0%

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

   8. metadata-eval 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. frac-times 0.0%

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

   12. flip-+ 0.0%

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

   13. flip-+ 6.4%

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

   14. sqrt-unprod 6.8%

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

   15. add-sqr-sqrt 6.8%

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

   16. count-2 6.8%

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

6. Applied egg-rr 6.8%

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

7. Final simplification 6.8%

   \[\leadsto x \cdot \left(x \cdot 2\right) \]

Alternative 3: 11.3% accurate, 68.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return x * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

   1. sqrt-prod 54.0%

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

   2. unpow2 54.0%

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

   3. sqrt-prod 48.0%

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

   4. add-sqr-sqrt 49.2%

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

3. Applied egg-rr 49.2%

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

4. Taylor expanded in x around 0 49.2%

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

6. Simplified 11.5%

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

7. Final simplification 11.5%

   \[\leadsto x \cdot -2 \]

Alternative 4: 1.7% accurate, 204.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -2.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0

Julia

function code(x)
	return -2.0
end

MATLAB

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

Wolfram

code[x_] := -2.0

Derivation

1. Initial program 54.3%

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

   1. sqrt-prod 54.0%

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

   2. unpow2 54.0%

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

   3. sqrt-prod 48.0%

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

   4. add-sqr-sqrt 49.2%

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

3. Applied egg-rr 49.2%

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

   1. add-sqr-sqrt 48.0%

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

   2. sqrt-unprod 54.0%

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

   3. sqrt-prod 54.3%

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

   4. count-2 54.3%

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

   5. flip-+ 0.0%

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

   6. +-inverses 0.0%

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

   7. metadata-eval 0.0%

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

   8. +-inverses 0.0%

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

   9. +-inverses 0.0%

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

   10. +-inverses 0.0%

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

   11. metadata-eval 0.0%

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

   12. +-inverses 0.0%

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

   13. +-inverses 0.0%

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

   14. frac-times 0.0%

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

   15. flip-+ 0.0%

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

   16. flip-+ 6.4%

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

5. Applied egg-rr 6.8%

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

7. Simplified 1.6%

   \[\leadsto \color{blue}{-2} \]

8. Final simplification 1.6%

   \[\leadsto -2 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))