sqrt D (should all be same)

Percentage Accurate: 54.6% → 99.6%

Time: 4.9s

Alternatives: 3

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: 54.6% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ {2}^{0.125} \cdot \left({8}^{0.125} \cdot x\right) \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (pow 2.0 0.125) (* (pow 8.0 0.125) x)))

C

x = abs(x);
double code(double x) {
	return pow(2.0, 0.125) * (pow(8.0, 0.125) * x);
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 ** 0.125d0) * ((8.0d0 ** 0.125d0) * x)
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return Math.pow(2.0, 0.125) * (Math.pow(8.0, 0.125) * x);
}

Python

x = abs(x)
def code(x):
	return math.pow(2.0, 0.125) * (math.pow(8.0, 0.125) * x)

Julia

x = abs(x)
function code(x)
	return Float64((2.0 ^ 0.125) * Float64((8.0 ^ 0.125) * x))
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = (2.0 ^ 0.125) * ((8.0 ^ 0.125) * x);
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(N[Power[8.0, 0.125], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. pow1/2 53.6%

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

   2. sqr-pow 53.4%

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

   3. pow-prod-down 32.4%

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

   4. *-commutative 32.4%

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

   5. *-commutative 32.4%

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

   6. swap-sqr 32.4%

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

   7. pow-prod-up 32.4%

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

   8. metadata-eval 32.4%

      \[\leadsto {\left({x}^{\color{blue}{4}} \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{0.5}{2}\right)} \]

   9. metadata-eval 32.4%

      \[\leadsto {\left({x}^{4} \cdot \color{blue}{4}\right)}^{\left(\frac{0.5}{2}\right)} \]

   10. metadata-eval 32.4%

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

3. Applied egg-rr 32.4%

   \[\leadsto \color{blue}{{\left({x}^{4} \cdot 4\right)}^{0.25}} \]
4. Step-by-step derivation

   1. *-commutative 32.4%

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

   2. unpow-prod-down 32.2%

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

   3. metadata-eval 32.2%

      \[\leadsto {\color{blue}{\left(2 \cdot 2\right)}}^{0.25} \cdot {\left({x}^{4}\right)}^{0.25} \]

   4. pow-prod-down 32.2%

      \[\leadsto \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)} \cdot {\left({x}^{4}\right)}^{0.25} \]

   5. pow-pow 49.5%

      \[\leadsto \left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot \color{blue}{{x}^{\left(4 \cdot 0.25\right)}} \]

   6. metadata-eval 49.5%

      \[\leadsto \left({2}^{0.25} \cdot {2}^{0.25}\right) \cdot {x}^{\color{blue}{1}} \]

   7. pow1 49.5%

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

   8. associate-*r* 49.6%

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

   9. add-sqr-sqrt 49.6%

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

   10. associate-*l* 49.6%

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

   11. sqrt-pow1 49.6%

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

   12. metadata-eval 49.6%

       \[\leadsto {2}^{\color{blue}{0.125}} \cdot \left(\sqrt{{2}^{0.25}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   13. sqrt-pow1 49.6%

       \[\leadsto {2}^{0.125} \cdot \left(\color{blue}{{2}^{\left(\frac{0.25}{2}\right)}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

   14. metadata-eval 49.6%

       \[\leadsto {2}^{0.125} \cdot \left({2}^{\color{blue}{0.125}} \cdot \left({2}^{0.25} \cdot x\right)\right) \]

5. Applied egg-rr 49.6%

   \[\leadsto \color{blue}{{2}^{0.125} \cdot \left({2}^{0.125} \cdot \left({2}^{0.25} \cdot x\right)\right)} \]

6. Taylor expanded in x around 0 49.7%

   \[\leadsto {2}^{0.125} \cdot \color{blue}{\left({8}^{0.125} \cdot x\right)} \]

7. Final simplification 49.7%

   \[\leadsto {2}^{0.125} \cdot \left({8}^{0.125} \cdot x\right) \]

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (sqrt (* 2.0 x)) (sqrt x)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt((2.0d0 * x)) * sqrt(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Sqrt[N[(2.0 * x), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. unpow2 53.6%

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

   2. associate-*r* 53.6%

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

   3. sqrt-prod 48.6%

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

3. Applied egg-rr 48.6%

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

4. Final simplification 48.6%

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

Alternative 3: 99.3% accurate, 2.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* x (sqrt 2.0)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x * sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.6%

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

   1. sqrt-prod 53.3%

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

   2. unpow2 53.3%

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

   3. sqrt-prod 48.3%

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

   4. add-sqr-sqrt 49.5%

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

3. Applied egg-rr 49.5%

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

4. Final simplification 49.5%

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

Reproduce

herbie shell --seed 2023306 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))