sqrt B (should all be same)

Percentage Accurate: 53.4% → 99.4%

Time: 5.3s

Alternatives: 2

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: 53.4% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. add-cbrt-cube 37.8%

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

   2. pow3 37.8%

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

   3. pow1/2 37.8%

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

   4. associate-*l* 37.8%

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

   5. *-commutative 37.8%

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

   6. unpow-prod-down 37.8%

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

   7. pow1/2 37.8%

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

   8. sqrt-unprod 18.5%

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

   9. add-sqr-sqrt 20.1%

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

   10. pow1/2 20.1%

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

4. Applied egg-rr 20.1%

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

   1. rem-cbrt-cube 49.4%

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

   2. *-commutative 49.4%

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

   3. add-sqr-sqrt 47.9%

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

   4. add-sqr-sqrt 47.8%

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

   5. associate-*r* 47.8%

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

   6. sqrt-prod 32.8%

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

   7. pow1 32.8%

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

   8. pow1/2 32.8%

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

   9. pow-prod-up 32.8%

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

   10. metadata-eval 32.8%

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

   11. associate-*l* 32.8%

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

   12. sqrt-prod 32.9%

       \[\leadsto \color{blue}{\sqrt{2 \cdot {x}^{1.5}}} \cdot \sqrt{\sqrt{x}} \]

   13. *-commutative 32.9%

       \[\leadsto \color{blue}{\sqrt{\sqrt{x}} \cdot \sqrt{2 \cdot {x}^{1.5}}} \]

   14. sqrt-prod 32.8%

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

   15. associate-*r* 32.9%

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

   16. pow1/2 32.9%

       \[\leadsto \left(\color{blue}{{\left(\sqrt{x}\right)}^{0.5}} \cdot \sqrt{2}\right) \cdot \sqrt{{x}^{1.5}} \]

   17. sqrt-pow2 32.9%

       \[\leadsto \left(\color{blue}{{x}^{\left(\frac{0.5}{2}\right)}} \cdot \sqrt{2}\right) \cdot \sqrt{{x}^{1.5}} \]

   18. metadata-eval 32.9%

       \[\leadsto \left({x}^{\color{blue}{0.25}} \cdot \sqrt{2}\right) \cdot \sqrt{{x}^{1.5}} \]

   19. sqrt-pow1 48.0%

       \[\leadsto \left({x}^{0.25} \cdot \sqrt{2}\right) \cdot \color{blue}{{x}^{\left(\frac{1.5}{2}\right)}} \]

   20. metadata-eval 48.0%

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

6. Applied egg-rr 48.0%

   \[\leadsto \color{blue}{\left({x}^{0.25} \cdot \sqrt{2}\right) \cdot {x}^{0.75}} \]
7. Step-by-step derivation

   1. add-log-exp 6.9%

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

   2. *-un-lft-identity 6.9%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{{x}^{0.25} \cdot \sqrt{2}}\right)} \cdot {x}^{0.75} \]

   3. log-prod 6.9%

      \[\leadsto \color{blue}{\left(\log 1 + \log \left(e^{{x}^{0.25} \cdot \sqrt{2}}\right)\right)} \cdot {x}^{0.75} \]

   4. metadata-eval 6.9%

      \[\leadsto \left(\color{blue}{0} + \log \left(e^{{x}^{0.25} \cdot \sqrt{2}}\right)\right) \cdot {x}^{0.75} \]

   5. add-log-exp 48.0%

      \[\leadsto \left(0 + \color{blue}{{x}^{0.25} \cdot \sqrt{2}}\right) \cdot {x}^{0.75} \]

   6. add-sqr-sqrt 48.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{{x}^{0.25} \cdot \sqrt{2}} \cdot \sqrt{{x}^{0.25} \cdot \sqrt{2}}}\right) \cdot {x}^{0.75} \]

   7. sqrt-unprod 48.0%

      \[\leadsto \left(0 + \color{blue}{\sqrt{\left({x}^{0.25} \cdot \sqrt{2}\right) \cdot \left({x}^{0.25} \cdot \sqrt{2}\right)}}\right) \cdot {x}^{0.75} \]

   8. pow1/2 48.0%

      \[\leadsto \left(0 + \sqrt{\left({x}^{0.25} \cdot \color{blue}{{2}^{0.5}}\right) \cdot \left({x}^{0.25} \cdot \sqrt{2}\right)}\right) \cdot {x}^{0.75} \]

   9. metadata-eval 48.0%

      \[\leadsto \left(0 + \sqrt{\left({x}^{0.25} \cdot {2}^{\color{blue}{\left(0.25 \cdot 2\right)}}\right) \cdot \left({x}^{0.25} \cdot \sqrt{2}\right)}\right) \cdot {x}^{0.75} \]

   10. pow-pow 48.0%

       \[\leadsto \left(0 + \sqrt{\left({x}^{0.25} \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}\right) \cdot \left({x}^{0.25} \cdot \sqrt{2}\right)}\right) \cdot {x}^{0.75} \]

   11. *-commutative 48.0%

       \[\leadsto \left(0 + \sqrt{\color{blue}{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right)} \cdot \left({x}^{0.25} \cdot \sqrt{2}\right)}\right) \cdot {x}^{0.75} \]

   12. pow1/2 48.0%

       \[\leadsto \left(0 + \sqrt{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right) \cdot \left({x}^{0.25} \cdot \color{blue}{{2}^{0.5}}\right)}\right) \cdot {x}^{0.75} \]

   13. metadata-eval 48.0%

       \[\leadsto \left(0 + \sqrt{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right) \cdot \left({x}^{0.25} \cdot {2}^{\color{blue}{\left(0.25 \cdot 2\right)}}\right)}\right) \cdot {x}^{0.75} \]

   14. pow-pow 48.0%

       \[\leadsto \left(0 + \sqrt{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right) \cdot \left({x}^{0.25} \cdot \color{blue}{{\left({2}^{0.25}\right)}^{2}}\right)}\right) \cdot {x}^{0.75} \]

   15. *-commutative 48.0%

       \[\leadsto \left(0 + \sqrt{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right) \cdot \color{blue}{\left({\left({2}^{0.25}\right)}^{2} \cdot {x}^{0.25}\right)}}\right) \cdot {x}^{0.75} \]

   16. swap-sqr 47.9%

       \[\leadsto \left(0 + \sqrt{\color{blue}{\left({\left({2}^{0.25}\right)}^{2} \cdot {\left({2}^{0.25}\right)}^{2}\right) \cdot \left({x}^{0.25} \cdot {x}^{0.25}\right)}}\right) \cdot {x}^{0.75} \]

8. Applied egg-rr 48.1%

   \[\leadsto \color{blue}{\left(0 + \sqrt{2 \cdot \sqrt{x}}\right)} \cdot {x}^{0.75} \]
9. Step-by-step derivation

   1. +-lft-identity 48.1%

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

10. Simplified 48.1%

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

11. Final simplification 48.1%

    \[\leadsto \sqrt{2 \cdot \sqrt{x}} \cdot {x}^{0.75} \]
12. 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 54.2%

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

   1. associate-*l* 54.2%

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

   2. sqrt-prod 54.0%

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

   3. sqrt-unprod 47.9%

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

   4. add-sqr-sqrt 49.4%

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

4. Applied egg-rr 49.4%

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

5. Final simplification 49.4%

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

Reproduce

herbie shell --seed 2024047 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))