sqrt B (should all be same)

Percentage Accurate: 54.1% → 99.6%

Time: 4.3s

Alternatives: 3

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: 54.1% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.125) (* (pow 8.0 0.125) x_m)))

C

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

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.125d0) * ((8.0d0 ** 0.125d0) * x_m)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.125], $MachinePrecision] * N[(N[Power[8.0, 0.125], $MachinePrecision] * x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. pow1/2 50.3%

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

   2. sqr-pow 50.1%

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

   3. pow-prod-down 29.9%

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

   4. pow2 29.9%

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

   5. *-commutative 29.9%

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

   6. *-commutative 29.9%

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

   7. associate-*r* 29.9%

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

   8. unpow-prod-down 29.9%

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

   9. pow-prod-down 29.9%

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

   10. pow-prod-up 29.9%

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

   11. metadata-eval 29.9%

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

   12. metadata-eval 29.9%

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

   13. metadata-eval 29.9%

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

3. Applied egg-rr 29.9%

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

   1. *-commutative 29.9%

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

   2. unpow-prod-down 29.8%

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

   3. metadata-eval 29.8%

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

   4. pow-prod-down 29.7%

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

   5. pow-pow 48.5%

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

   6. metadata-eval 48.5%

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

   7. pow1 48.5%

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

   8. associate-*r* 48.5%

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

   9. add-sqr-sqrt 48.5%

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

   10. associate-*l* 48.6%

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

   11. sqrt-pow1 48.6%

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

   12. metadata-eval 48.6%

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

   13. sqrt-pow1 48.6%

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

   14. metadata-eval 48.6%

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

5. Applied egg-rr 48.6%

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

6. Taylor expanded in x around 0 48.7%

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

7. Final simplification 48.7%

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

Alternative 2: 99.5% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (sqrt (* 2.0 x_m)) (sqrt x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Sqrt[N[(2.0 * x$95$m), $MachinePrecision]], $MachinePrecision] * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. sqrt-prod 47.4%

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

3. Applied egg-rr 47.4%

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

   1. *-commutative 47.4%

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

5. Simplified 47.4%

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

6. Final simplification 47.4%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (sqrt 2.0)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.3%

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

   1. associate-*l* 50.3%

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

   2. sqrt-prod 50.1%

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

   3. sqrt-unprod 47.2%

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

   4. add-sqr-sqrt 48.6%

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

3. Applied egg-rr 48.6%

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

4. Final simplification 48.6%

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

Reproduce

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