sqrt B (should all be same)

Percentage Accurate: 54.0% → 99.5%

Time: 8.5s

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: 54.0% 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

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

FPCore

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

C

x = abs(x);
double code(double x) {
	return sqrt((x * 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 = sqrt((x * 2.0d0)) * sqrt(x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. sqrt-prod 49.0%

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

3. Applied egg-rr 49.0%

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

   1. *-commutative 49.0%

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

5. Simplified 49.0%

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

6. Final simplification 49.0%

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

Alternative 2: 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 55.6%

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

   1. associate-*l* 55.6%

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

   2. sqrt-prod 55.3%

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

   3. sqrt-unprod 48.7%

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

   4. add-sqr-sqrt 50.1%

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

3. Applied egg-rr 50.1%

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

4. Final simplification 50.1%

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

Alternative 3: 5.4% accurate, 105.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 1 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 1.0)

C

x = abs(x);
double code(double x) {
	return 1.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 1.0;
}

Python

x = abs(x)
def code(x):
	return 1.0

Julia

x = abs(x)
function code(x)
	return 1.0
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 1.0;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 1.0

Derivation

1. Initial program 55.6%

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

   1. rem-cube-cbrt 54.7%

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

   2. associate-*l* 54.7%

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

   3. sqrt-prod 54.6%

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

   4. sqrt-unprod 48.2%

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

   5. add-sqr-sqrt 49.5%

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

   6. *-commutative 49.5%

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

3. Applied egg-rr 49.5%

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

   1. rem-cube-cbrt 50.1%

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

   2. *-commutative 50.1%

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

   3. add-sqr-sqrt 50.0%

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

   4. associate-*l* 50.1%

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

   5. pow1/2 50.1%

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

   6. sqrt-pow1 50.1%

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

   7. metadata-eval 50.1%

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

   8. pow1/2 50.1%

      \[\leadsto {2}^{0.25} \cdot \left(\sqrt{\color{blue}{{2}^{0.5}}} \cdot x\right) \]

   9. sqrt-pow1 50.1%

      \[\leadsto {2}^{0.25} \cdot \left(\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}} \cdot x\right) \]

   10. metadata-eval 50.1%

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

5. Applied egg-rr 50.1%

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

   1. associate-*r* 50.0%

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

   2. *-commutative 50.0%

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

   3. pow-sqr 50.1%

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

   4. metadata-eval 50.1%

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

   5. pow1/2 50.1%

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

   6. +-rgt-identity 50.1%

      \[\leadsto \color{blue}{x \cdot \sqrt{2} + 0} \]

   7. metadata-eval 50.1%

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

   8. associate--l+ 29.1%

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

   9. flip-- 18.8%

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

   10. clear-num 18.8%

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

   11. *-un-lft-identity 18.8%

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

   12. associate-/l* 18.8%

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

   13. flip-- 29.0%

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

   14. associate--l+ 50.1%

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

   15. metadata-eval 50.1%

       \[\leadsto \frac{1}{\frac{1}{x \cdot \sqrt{2} + \color{blue}{0}}} \]

   16. +-rgt-identity 50.1%

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

7. Applied egg-rr 50.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{x \cdot \sqrt{2}}}} \]
8. Step-by-step derivation

   1. /-rgt-identity 50.1%

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

   2. add-sqr-sqrt 48.6%

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

   3. associate-/l* 48.7%

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

   4. metadata-eval 48.7%

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

   5. sqrt-div 48.8%

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

   6. add-sqr-sqrt 48.8%

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

   7. sqrt-unprod 28.9%

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

   8. sqr-neg 28.9%

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

   9. sqrt-unprod 0.0%

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

   10. add-sqr-sqrt 0.0%

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

   11. add-exp-log 0.0%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 27.3%

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

9. Applied egg-rr 2.7%

   \[\leadsto \frac{1}{\frac{1}{\color{blue}{\frac{\sqrt{x \cdot \sqrt{2}}}{\sqrt{x \cdot \sqrt{2}}}}}} \]
10. Step-by-step derivation

    1. *-inverses 5.4%

       \[\leadsto \frac{1}{\frac{1}{\color{blue}{1}}} \]

11. Simplified 5.4%

    \[\leadsto \frac{1}{\frac{1}{\color{blue}{1}}} \]

12. Final simplification 5.4%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023301 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))