sqrt B (should all be same)

Percentage Accurate: 54.6% → 99.6%

Time: 13.4s

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: 54.6% 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 53.1%

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

   1. add-sqr-sqrt 52.8%

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

   2. pow2 52.8%

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

   3. pow1/2 52.8%

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

   4. associate-*l* 52.7%

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

   5. *-commutative 52.7%

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

   6. unpow-prod-down 52.6%

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

   7. pow1/2 52.6%

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

   8. sqrt-unprod 50.5%

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

   9. add-sqr-sqrt 50.6%

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

   10. pow1/2 50.6%

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

4. Applied egg-rr 50.6%

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

   1. sqrt-prod 50.6%

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

   2. *-commutative 50.6%

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

   3. pow1/2 50.6%

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

   4. sqrt-pow1 50.6%

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

   5. metadata-eval 50.6%

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

6. Applied egg-rr 50.6%

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

   1. unpow2 50.6%

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

   2. swap-sqr 50.7%

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

   3. add-sqr-sqrt 51.9%

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

   4. associate-*r* 52.0%

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

   5. sqr-pow 52.0%

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

   6. associate-*l* 52.0%

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

   7. metadata-eval 52.0%

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

   8. metadata-eval 52.0%

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

8. Applied egg-rr 52.0%

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

9. Taylor expanded in x around 0 52.1%

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

10. Final simplification 52.1%

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

Alternative 2: 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 53.1%

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

   1. sqrt-prod 50.9%

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

4. Applied egg-rr 50.9%

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

   1. *-commutative 50.9%

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

6. Simplified 50.9%

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

7. Final simplification 50.9%

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

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 53.1%

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

   1. associate-*l* 53.0%

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

   2. sqrt-prod 52.8%

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

   3. sqrt-unprod 50.6%

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

   4. add-sqr-sqrt 51.9%

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

4. Applied egg-rr 51.9%

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

5. Final simplification 51.9%

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

Reproduce

herbie shell --seed 2024079 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))