sqrt B (should all be same)

Percentage Accurate: 54.2% → 99.5%

Time: 3.9s

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: 54.2% 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: 99.5% 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 59.8%

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

   1. pow1/2 N/A

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

   2. unpow-prod-down N/A

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

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot x\right)}^{\frac{1}{2}}\right), \color{blue}{\left({x}^{\frac{1}{2}}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot x\right), \frac{1}{2}\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \left(\sqrt{x}\right)\right) \]

   7. sqrt-lowering-sqrt.f64 47.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, x\right), \frac{1}{2}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

4. Applied egg-rr 47.8%

   \[\leadsto \color{blue}{{\left(2 \cdot x\right)}^{0.5} \cdot \sqrt{x}} \]
5. Step-by-step derivation

   1. unpow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot x}\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot x\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{x}\right)\right) \]

   3. *-lowering-*.f64 47.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

6. Applied egg-rr 47.8%

   \[\leadsto \color{blue}{\sqrt{2 \cdot x}} \cdot \sqrt{x} \]
7. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x\_m}{{4}^{-0.25}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ x_m (pow 4.0 -0.25)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m / pow(4.0, -0.25);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m / (4.0d0 ** (-0.25d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m / Math.pow(4.0, -0.25);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m / math.pow(4.0, -0.25)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m / (4.0 ^ -0.25))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m / (4.0 ^ -0.25);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m / N[Power[4.0, -0.25], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.8%

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

   1. pow1/2 N/A

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

   2. associate-*l* N/A

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{\frac{1}{2}} \]

   3. unpow-prod-down N/A

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

   4. *-commutative N/A

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

   5. unpow-prod-down N/A

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

   6. sqr-pow N/A

      \[\leadsto \left({x}^{\frac{1}{2}} \cdot \left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right) \cdot {2}^{\frac{1}{2}} \]

   7. associate-*r* N/A

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

   8. associate-*l* N/A

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

   9. sqr-pow N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-down N/A

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

   12. sqr-pow N/A

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

   13. pow-prod-down N/A

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

   14. pow-prod-down N/A

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

   15. pow-prod-down N/A

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

   16. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot \left(2 \cdot 2\right)\right)\right), \color{blue}{\left(\frac{\frac{1}{2}}{2}\right)}\right) \]

4. Applied egg-rr 30.6%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{4}^{\frac{1}{4}} \cdot x} \]
6. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \frac{{4}^{\frac{1}{4}} \cdot x}{\color{blue}{1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x \cdot {4}^{\frac{1}{4}}}{1} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{x}{1} \cdot \color{blue}{{4}^{\frac{1}{4}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{4}^{\frac{1}{4}}}}} \]

   5. exp-to-pow N/A

      \[\leadsto \frac{x}{\frac{1}{e^{\log 4 \cdot \frac{1}{4}}}} \]

   6. exp-neg N/A

      \[\leadsto \frac{x}{e^{\mathsf{neg}\left(\log 4 \cdot \frac{1}{4}\right)}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{\mathsf{neg}\left(\log 4 \cdot \frac{1}{4}\right)}\right)}\right) \]

   8. distribute-rgt-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \frac{-1}{4}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\log 4 \cdot \left(\frac{1}{4} \cdot -1\right)}\right)\right) \]

   11. exp-to-pow N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left({4}^{\color{blue}{\left(\frac{1}{4} \cdot -1\right)}}\right)\right) \]

   12. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{pow.f64}\left(4, \color{blue}{\left(\frac{1}{4} \cdot -1\right)}\right)\right) \]

   13. metadata-eval 48.9%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{pow.f64}\left(4, \frac{-1}{4}\right)\right) \]

7. Simplified 48.9%

   \[\leadsto \color{blue}{\frac{x}{{4}^{-0.25}}} \]
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 59.8%

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

   1. pow1/2 N/A

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

   2. associate-*l* N/A

      \[\leadsto {\left(2 \cdot \left(x \cdot x\right)\right)}^{\frac{1}{2}} \]

   3. unpow-prod-down N/A

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

   4. pow1/2 N/A

      \[\leadsto {2}^{\frac{1}{2}} \cdot \sqrt{x \cdot x} \]

   5. sqrt-prod N/A

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

   6. rem-square-sqrt N/A

      \[\leadsto {2}^{\frac{1}{2}} \cdot x \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}}\right), \color{blue}{x}\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), x\right) \]

   9. sqrt-lowering-sqrt.f64 48.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), x\right) \]

4. Applied egg-rr 48.9%

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

5. Final simplification 48.9%

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

Reproduce

herbie shell --seed 2024192 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))