sqrt B (should all be same)

Percentage Accurate: 55.1% → 99.4%

Time: 7.5s

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: 55.1% 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: 99.4% 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 52.2%

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

   1. add-sqr-sqrt 51.9%

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

   2. pow2 51.9%

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

   3. *-commutative 51.9%

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

   4. *-commutative 51.9%

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

   5. associate-*r* 51.9%

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

   6. sqrt-prod 51.8%

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

   7. sqrt-unprod 47.0%

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

   8. add-sqr-sqrt 47.1%

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

4. Applied egg-rr 47.1%

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

   1. sqrt-unprod 47.2%

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

   2. pow1/2 47.2%

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

   3. sqrt-pow1 47.2%

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

   4. metadata-eval 47.2%

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

   5. *-commutative 47.2%

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

6. Applied egg-rr 47.2%

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

   1. unpow2 47.2%

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

   2. swap-sqr 47.1%

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

   3. pow-prod-up 47.1%

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

   4. metadata-eval 47.1%

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

   5. pow1/2 47.1%

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

   6. add-sqr-sqrt 48.4%

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

   7. *-commutative 48.4%

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

   8. pow1 48.4%

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

   9. metadata-eval 48.4%

      \[\leadsto {x}^{\color{blue}{\left(0.25 + 0.75\right)}} \cdot \sqrt{2} \]

   10. pow-prod-up 47.2%

       \[\leadsto \color{blue}{\left({x}^{0.25} \cdot {x}^{0.75}\right)} \cdot \sqrt{2} \]

   11. associate-*r* 47.2%

       \[\leadsto \color{blue}{{x}^{0.25} \cdot \left({x}^{0.75} \cdot \sqrt{2}\right)} \]

   12. *-commutative 47.2%

       \[\leadsto {x}^{0.25} \cdot \color{blue}{\left(\sqrt{2} \cdot {x}^{0.75}\right)} \]

   13. associate-*r* 47.2%

       \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right) \cdot {x}^{0.75}} \]

   14. pow1/2 47.2%

       \[\leadsto \left({x}^{0.25} \cdot \color{blue}{{2}^{0.5}}\right) \cdot {x}^{0.75} \]

   15. metadata-eval 47.2%

       \[\leadsto \left({x}^{0.25} \cdot {2}^{\color{blue}{\left(0.25 + 0.25\right)}}\right) \cdot {x}^{0.75} \]

   16. pow-prod-up 47.2%

       \[\leadsto \left({x}^{0.25} \cdot \color{blue}{\left({2}^{0.25} \cdot {2}^{0.25}\right)}\right) \cdot {x}^{0.75} \]

   17. pow-prod-down 47.2%

       \[\leadsto \left({x}^{0.25} \cdot \color{blue}{{\left(2 \cdot 2\right)}^{0.25}}\right) \cdot {x}^{0.75} \]

   18. metadata-eval 47.2%

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

   19. metadata-eval 47.2%

       \[\leadsto \left({x}^{0.25} \cdot {\color{blue}{\left(2 + 2\right)}}^{0.25}\right) \cdot {x}^{0.75} \]

   20. pow-prod-down 47.4%

       \[\leadsto \color{blue}{{\left(x \cdot \left(2 + 2\right)\right)}^{0.25}} \cdot {x}^{0.75} \]

   21. metadata-eval 47.4%

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

8. Applied egg-rr 47.4%

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

9. Final simplification 47.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. sqrt-prod 47.4%

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

4. Applied egg-rr 47.4%

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

   1. *-commutative 47.4%

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

6. Simplified 47.4%

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

7. Final simplification 47.4%

   \[\leadsto \sqrt{x \cdot 2} \cdot \sqrt{x} \]
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 52.2%

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

   1. associate-*l* 52.2%

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

   2. sqrt-prod 51.9%

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

   3. sqrt-unprod 47.1%

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

   4. add-sqr-sqrt 48.4%

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

4. Applied egg-rr 48.4%

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

5. Final simplification 48.4%

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

Reproduce

herbie shell --seed 2024011 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))