sqrt C (should all be same)

Percentage Accurate: 53.8% → 99.4%

Time: 13.5s

Alternatives: 4

Speedup: 1.3×

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: 53.8% 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, 0.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{0.75} \cdot \sqrt{2 \cdot \sqrt{x\_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* (pow x_m 0.75) (sqrt (* 2.0 (sqrt x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 0.75) * sqrt((2.0 * sqrt(x_m)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 0.75d0) * sqrt((2.0d0 * sqrt(x_m)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 0.75) * Math.sqrt((2.0 * Math.sqrt(x_m)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 0.75) * math.sqrt((2.0 * math.sqrt(x_m)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 0.75) * sqrt(Float64(2.0 * sqrt(x_m))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 0.75) * sqrt((2.0 * sqrt(x_m)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 0.75], $MachinePrecision] * N[Sqrt[N[(2.0 * N[Sqrt[x$95$m], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \sqrt{2 \cdot {x}^{\color{blue}{\left(\frac{3}{2} + \frac{1}{2}\right)}}} \]

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. pow-prod-up N/A

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

   9. metadata-eval N/A

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

   10. pow-plus N/A

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

   11. associate-*r* N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. sqrt-prod N/A

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

   15. pow1/2 N/A

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

   16. pow1/2 N/A

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

   17. lower-*.f64 N/A

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

4. Applied rewrites 33.2%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-plus N/A

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

   8. pow-pow N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval N/A

       \[\leadsto {x}^{\left(\color{blue}{\frac{3}{2}} \cdot \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \sqrt{x}} \]

   11. metadata-eval 45.5

       \[\leadsto {x}^{\color{blue}{0.75}} \cdot \sqrt{2 \cdot \sqrt{x}} \]

6. Applied rewrites 45.5%

   \[\leadsto \color{blue}{{x}^{0.75}} \cdot \sqrt{2 \cdot \sqrt{x}} \]
7. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. unpow-prod-down N/A

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

   7. lower-*.f64 N/A

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

   8. pow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. pow1/2 N/A

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

   12. lower-sqrt.f64 45.4

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

4. Applied rewrites 45.4%

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

5. Final simplification 45.4%

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

Alternative 3: 99.3% accurate, 1.3× 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 61.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 46.7

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

4. Applied rewrites 46.7%

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

5. Final simplification 46.7%

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

Alternative 4: 20.3% accurate, 21.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

x_m = abs(x)
function code(x_m)
	return x_m
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := x$95$m

Derivation

1. Initial program 61.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-prod N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. pow1/2 N/A

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

   10. lower-sqrt.f64 46.7

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

4. Applied rewrites 46.7%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. metadata-eval N/A

      \[\leadsto {2}^{\color{blue}{\left(\frac{1}{4} \cdot 2\right)}} \cdot x \]

   4. pow-pow N/A

      \[\leadsto \color{blue}{{\left({2}^{\frac{1}{4}}\right)}^{2}} \cdot x \]

   5. lift-pow.f64 N/A

      \[\leadsto {\color{blue}{\left({2}^{\frac{1}{4}}\right)}}^{2} \cdot x \]

   6. lower-pow.f64 46.6

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

6. Applied rewrites 46.6%

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

8. Applied rewrites 10.6%

   \[\leadsto \color{blue}{x \cdot 1} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot 1} \]

   2. *-rgt-identity 10.6

      \[\leadsto \color{blue}{x} \]

10. Applied rewrites 10.6%

    \[\leadsto \color{blue}{x} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024237 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))