sqrt B (should all be same)

Percentage Accurate: 54.7% → 99.6%

Time: 6.7s

Alternatives: 4

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: 54.7% 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

\[\begin{array}{l} x_m = \left|x\right| \\ {2}^{0.375} \cdot \left(x_m \cdot {2}^{0.125}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow 2.0 0.375) (* x_m (pow 2.0 0.125))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(2.0, 0.375) * (x_m * pow(2.0, 0.125));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (2.0d0 ** 0.375d0) * (x_m * (2.0d0 ** 0.125d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(2.0, 0.375) * (x_m * Math.pow(2.0, 0.125));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(2.0, 0.375) * (x_m * math.pow(2.0, 0.125))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((2.0 ^ 0.375) * Float64(x_m * (2.0 ^ 0.125)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (2.0 ^ 0.375) * (x_m * (2.0 ^ 0.125));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[2.0, 0.375], $MachinePrecision] * N[(x$95$m * N[Power[2.0, 0.125], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.8%

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

   1. add-sqr-sqrt 51.5%

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

   2. pow2 51.5%

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

   3. *-commutative 51.5%

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

   4. *-commutative 51.5%

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

   5. associate-*r* 51.4%

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

   6. sqrt-prod 51.3%

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

   7. sqrt-unprod 51.0%

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

   8. add-sqr-sqrt 51.1%

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

4. Applied egg-rr 51.1%

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

   1. sqrt-unprod 51.0%

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

   2. pow1/2 51.0%

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

   3. sqrt-pow1 51.0%

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

   4. metadata-eval 51.0%

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

   5. *-commutative 51.0%

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

6. Applied egg-rr 51.0%

   \[\leadsto {\color{blue}{\left({2}^{0.25} \cdot \sqrt{x}\right)}}^{2} \]
7. Step-by-step derivation

   1. unpow2 51.0%

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

   2. swap-sqr 51.0%

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

   3. add-sqr-sqrt 52.3%

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

   4. associate-*r* 52.3%

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

   5. metadata-eval 52.3%

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

   6. pow-prod-up 52.3%

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

   7. associate-*r* 52.4%

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

   8. *-commutative 52.4%

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

   9. associate-*r* 52.5%

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

   10. associate-*l* 52.5%

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

   11. pow-prod-up 52.5%

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

   12. metadata-eval 52.5%

       \[\leadsto {2}^{\color{blue}{0.375}} \cdot \left(x \cdot {2}^{0.125}\right) \]

8. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{{2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right)} \]

9. Final simplification 52.5%

   \[\leadsto {2}^{0.375} \cdot \left(x \cdot {2}^{0.125}\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.8%

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

   1. pow1/2 51.8%

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

   2. sqr-pow 51.5%

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

   3. pow-prod-down 28.2%

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

   4. pow2 28.2%

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

   5. *-commutative 28.2%

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

   6. *-commutative 28.2%

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

   7. associate-*r* 28.2%

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

   8. unpow-prod-down 28.2%

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

   9. pow-prod-down 28.2%

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

   10. pow-prod-up 28.2%

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

   11. metadata-eval 28.2%

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

   12. metadata-eval 28.2%

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

   13. metadata-eval 28.2%

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

4. Applied egg-rr 28.2%

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

   1. *-commutative 28.2%

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

   2. unpow-prod-down 28.1%

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

   3. metadata-eval 28.1%

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

   4. pow-prod-down 28.0%

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

   5. pow-pow 52.3%

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

   6. metadata-eval 52.3%

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

   7. pow1 52.3%

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

   8. associate-*r* 52.3%

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

   9. add-sqr-sqrt 52.3%

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

   10. associate-*l* 52.4%

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

   11. sqrt-pow1 52.4%

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

   12. metadata-eval 52.4%

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

   13. sqrt-pow1 52.4%

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

   14. metadata-eval 52.4%

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

6. Applied egg-rr 52.4%

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

7. Taylor expanded in x around 0 52.5%

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

8. Final simplification 52.5%

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

Alternative 3: 99.4% 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 51.8%

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

   1. sqrt-prod 51.3%

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

4. Applied egg-rr 51.3%

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

   1. *-commutative 51.3%

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

6. Simplified 51.3%

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

7. Final simplification 51.3%

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

Alternative 4: 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 51.8%

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

   1. associate-*l* 51.7%

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

   2. sqrt-prod 51.5%

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

   3. sqrt-unprod 51.1%

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

   4. add-sqr-sqrt 52.4%

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

4. Applied egg-rr 52.4%

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

5. Final simplification 52.4%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))