sqrt C (should all be same)

Percentage Accurate: 53.9% → 99.4%

Time: 4.6s

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: 53.9% 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, 0.3× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (pow x 0.75) (sqrt (* 2.0 (sqrt x)))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 0.75d0) * sqrt((2.0d0 * sqrt(x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Power[x, 0.75], $MachinePrecision] * N[Sqrt[N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. associate-*r* 55.9%

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

   2. sqrt-prod 53.6%

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

3. Applied egg-rr 53.6%

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

   1. pow1/2 53.6%

      \[\leadsto \sqrt{2 \cdot x} \cdot \color{blue}{{x}^{0.5}} \]

   2. metadata-eval 53.6%

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

   3. pow-prod-up 53.4%

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

   4. pow-prod-up 53.6%

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

   5. metadata-eval 53.6%

      \[\leadsto \sqrt{2 \cdot x} \cdot {x}^{\color{blue}{0.5}} \]

   6. pow1/2 53.6%

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

   7. add-cbrt-cube 33.1%

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

   8. add-sqr-sqrt 33.1%

      \[\leadsto \sqrt{2 \cdot x} \cdot \sqrt[3]{\color{blue}{x} \cdot \sqrt{x}} \]

   9. cbrt-prod 53.1%

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

   10. add-cbrt-cube 33.0%

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

   11. cbrt-prod 33.1%

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

   12. pow1 33.1%

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

   13. pow1/2 33.1%

       \[\leadsto \sqrt[3]{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right) \cdot \sqrt{2 \cdot x}} \cdot \sqrt[3]{{x}^{1} \cdot \color{blue}{{x}^{0.5}}} \]

   14. pow-prod-up 33.0%

       \[\leadsto \sqrt[3]{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right) \cdot \sqrt{2 \cdot x}} \cdot \sqrt[3]{\color{blue}{{x}^{\left(1 + 0.5\right)}}} \]

   15. metadata-eval 33.0%

       \[\leadsto \sqrt[3]{\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right) \cdot \sqrt{2 \cdot x}} \cdot \sqrt[3]{{x}^{\color{blue}{1.5}}} \]

   16. cbrt-unprod 18.1%

       \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{2 \cdot x} \cdot \sqrt{2 \cdot x}\right) \cdot \sqrt{2 \cdot x}\right) \cdot {x}^{1.5}}} \]

5. Applied egg-rr 18.1%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot x\right)}^{1.5} \cdot {x}^{1.5}}} \]
6. Step-by-step derivation

   1. *-commutative 18.1%

      \[\leadsto \sqrt[3]{{\color{blue}{\left(x \cdot 2\right)}}^{1.5} \cdot {x}^{1.5}} \]

7. Simplified 18.1%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 18.0%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{{\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}}} \cdot \sqrt{\sqrt[3]{{\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}}}} \]

   2. sqrt-unprod 18.1%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{{\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}} \cdot \sqrt[3]{{\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}}}} \]

   3. cbrt-unprod 9.9%

      \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left({\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}\right) \cdot \left({\left(x \cdot 2\right)}^{1.5} \cdot {x}^{1.5}\right)}}} \]

   4. swap-sqr 9.9%

      \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left({\left(x \cdot 2\right)}^{1.5} \cdot {\left(x \cdot 2\right)}^{1.5}\right) \cdot \left({x}^{1.5} \cdot {x}^{1.5}\right)}}} \]

   5. pow-prod-up 9.9%

      \[\leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(x \cdot 2\right)}^{\left(1.5 + 1.5\right)}} \cdot \left({x}^{1.5} \cdot {x}^{1.5}\right)}} \]

   6. metadata-eval 9.9%

      \[\leadsto \sqrt{\sqrt[3]{{\left(x \cdot 2\right)}^{\color{blue}{3}} \cdot \left({x}^{1.5} \cdot {x}^{1.5}\right)}} \]

   7. pow3 9.9%

      \[\leadsto \sqrt{\sqrt[3]{\color{blue}{\left(\left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right) \cdot \left(x \cdot 2\right)\right)} \cdot \left({x}^{1.5} \cdot {x}^{1.5}\right)}} \]

   8. pow-prod-up 22.0%

      \[\leadsto \sqrt{\sqrt[3]{\left(\left(\left(x \cdot 2\right) \cdot \left(x \cdot 2\right)\right) \cdot \left(x \cdot 2\right)\right) \cdot \color{blue}{{x}^{\left(1.5 + 1.5\right)}}}} \]

   9. metadata-eval 22.0%

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

   10. pow3 22.0%

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

   11. cbrt-unprod 37.6%

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

   12. add-cbrt-cube 37.7%

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

   13. add-cbrt-cube 55.9%

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

   14. *-commutative 55.9%

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

   15. associate-*l* 55.9%

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

9. Applied egg-rr 53.6%

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

10. Final simplification 53.6%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \sqrt{x} \cdot \sqrt{x \cdot 2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* (sqrt x) (sqrt (* x 2.0))))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(x) * sqrt((x * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[Sqrt[x], $MachinePrecision] * N[Sqrt[N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. associate-*r* 55.9%

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

   2. sqrt-prod 53.6%

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

3. Applied egg-rr 53.6%

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

4. Final simplification 53.6%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \cdot \sqrt{2} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* x (sqrt 2.0)))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x * sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.9%

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

   1. sqrt-prod 55.6%

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

   2. sqrt-prod 53.4%

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

   3. add-sqr-sqrt 54.5%

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

3. Applied egg-rr 54.5%

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

4. Final simplification 54.5%

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

Reproduce

herbie shell --seed 2023310 
(FPCore (x)
  :name "sqrt C (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (* x x))))