sqrt C (should all be same)

Percentage Accurate: 53.7% → 99.4%

Time: 6.5s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

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(2.0 * Float64(x * x)))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(x \cdot x\right)} \end{array} \]

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(2.0 * Float64(x * x)))
end

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\left(x\_m \cdot 4\right)}^{0.25} \cdot {x\_m}^{0.75} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow (* x_m 4.0) 0.25) (pow x_m 0.75)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow((x_m * 4.0), 0.25) * pow(x_m, 0.75);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = ((x_m * 4.0d0) ** 0.25d0) * (x_m ** 0.75d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow((x_m * 4.0), 0.25) * Math.pow(x_m, 0.75);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow((x_m * 4.0), 0.25) * math.pow(x_m, 0.75)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((Float64(x_m * 4.0) ^ 0.25) * (x_m ^ 0.75))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = ((x_m * 4.0) ^ 0.25) * (x_m ^ 0.75);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[N[(x$95$m * 4.0), $MachinePrecision], 0.25], $MachinePrecision] * N[Power[x$95$m, 0.75], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. pow1/2 N/A

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

   2. rem-square-sqrt N/A

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

   3. pow1/2 N/A

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

   4. pow1/2 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. unpow-prod-down N/A

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

   8. unpow-prod-down N/A

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

   9. pow1/2 N/A

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

   10. sqrt-pow2 N/A

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

   11. *-commutative N/A

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

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

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

4. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{0.25} \cdot {x}^{0.75}} \]
5. Add Preprocessing

Alternative 2: 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 53.4%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

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

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

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

   6. pow1/2 N/A

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

   7. sqrt-lowering-sqrt.f64 53.4%

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

4. Applied egg-rr 53.4%

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

5. Final simplification 53.4%

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

Alternative 3: 6.0% accurate, 11.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot x\_m\right) \cdot \left(4 \cdot \left(x\_m \cdot x\_m\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (* x_m x_m) (* 4.0 (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) * (4.0 * (x_m * x_m));
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) * (4.0 * (x_m * x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) * (4.0 * (x_m * x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) * Float64(4.0 * Float64(x_m * x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) * (4.0 * (x_m * x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] * N[(4.0 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. pow1/2 N/A

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

   2. rem-square-sqrt N/A

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

   3. pow1/2 N/A

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

   4. pow1/2 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. unpow-prod-down N/A

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

   8. unpow-prod-down N/A

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

   9. pow1/2 N/A

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

   10. sqrt-pow2 N/A

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

   11. *-commutative N/A

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

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

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

4. Applied egg-rr 52.5%

   \[\leadsto \color{blue}{{\left(x \cdot 4\right)}^{0.25} \cdot {x}^{0.75}} \]

6. Applied egg-rr 5.9%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(4 \cdot \left(x \cdot x\right)\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))