sqrt B (should all be same)

Percentage Accurate: 52.8% → 100.0%

Time: 9.0s

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: 52.8% 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: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{hypot}\left(x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (hypot x x))

C

double code(double x) {
	return hypot(x, x);
}

Java

public static double code(double x) {
	return Math.hypot(x, x);
}

Python

def code(x):
	return math.hypot(x, x)

Julia

function code(x)
	return hypot(x, x)
end

MATLAB

function tmp = code(x)
	tmp = hypot(x, x);
end

Wolfram

code[x_] := N[Sqrt[x ^ 2 + x ^ 2], $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. add-cube-cbrt 55.1%

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

   2. cbrt-prod 55.5%

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

   3. add-sqr-sqrt 55.4%

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

   4. *-commutative 55.4%

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

   5. pow1/3 52.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\left(2 \cdot x\right) \cdot x}\right)}^{0.3333333333333333}} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x} \]

   6. pow1/2 52.9%

      \[\leadsto {\color{blue}{\left({\left(\left(2 \cdot x\right) \cdot x\right)}^{0.5}\right)}}^{0.3333333333333333} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x} \]

   7. pow-pow 52.9%

      \[\leadsto \color{blue}{{\left(\left(2 \cdot x\right) \cdot x\right)}^{\left(0.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x} \]

   8. *-commutative 52.9%

      \[\leadsto {\color{blue}{\left(x \cdot \left(2 \cdot x\right)\right)}}^{\left(0.5 \cdot 0.3333333333333333\right)} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x} \]

   9. metadata-eval 52.9%

      \[\leadsto {\left(x \cdot \left(2 \cdot x\right)\right)}^{\color{blue}{0.16666666666666666}} \cdot \sqrt[3]{\left(2 \cdot x\right) \cdot x} \]

   10. *-commutative 52.9%

       \[\leadsto {\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{\color{blue}{x \cdot \left(2 \cdot x\right)}} \]

4. Applied egg-rr 52.9%

   \[\leadsto \color{blue}{{\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot \sqrt[3]{x \cdot \left(2 \cdot x\right)}} \]
5. Step-by-step derivation

   1. unpow1/3 51.7%

      \[\leadsto {\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot \color{blue}{{\left(x \cdot \left(2 \cdot x\right)\right)}^{0.3333333333333333}} \]

   2. metadata-eval 51.7%

      \[\leadsto {\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot {\left(x \cdot \left(2 \cdot x\right)\right)}^{\color{blue}{\left(2 \cdot 0.16666666666666666\right)}} \]

   3. pow-sqr 51.6%

      \[\leadsto {\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot \color{blue}{\left({\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666} \cdot {\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666}\right)} \]

   4. cube-mult 51.6%

      \[\leadsto \color{blue}{{\left({\left(x \cdot \left(2 \cdot x\right)\right)}^{0.16666666666666666}\right)}^{3}} \]

   5. exp-to-pow 51.8%

      \[\leadsto {\color{blue}{\left(e^{\log \left(x \cdot \left(2 \cdot x\right)\right) \cdot 0.16666666666666666}\right)}}^{3} \]

   6. metadata-eval 51.8%

      \[\leadsto {\left(e^{\log \left(x \cdot \left(2 \cdot x\right)\right) \cdot \color{blue}{\left(0.5 \cdot 0.3333333333333333\right)}}\right)}^{3} \]

   7. associate-*l* 51.8%

      \[\leadsto {\left(e^{\color{blue}{\left(\log \left(x \cdot \left(2 \cdot x\right)\right) \cdot 0.5\right) \cdot 0.3333333333333333}}\right)}^{3} \]

   8. *-commutative 51.8%

      \[\leadsto {\left(e^{\color{blue}{\left(0.5 \cdot \log \left(x \cdot \left(2 \cdot x\right)\right)\right)} \cdot 0.3333333333333333}\right)}^{3} \]

   9. exp-prod 52.1%

      \[\leadsto {\color{blue}{\left({\left(e^{0.5 \cdot \log \left(x \cdot \left(2 \cdot x\right)\right)}\right)}^{0.3333333333333333}\right)}}^{3} \]

   10. unpow1/3 52.2%

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

   11. rem-cube-cbrt 52.3%

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

   12. *-commutative 52.3%

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

   13. exp-to-pow 56.1%

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

   14. unpow1/2 56.1%

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

   15. *-commutative 56.1%

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

   16. associate-*l* 56.1%

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

   17. count-2 56.1%

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

   18. hypot-define 100.0%

       \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]

6. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
7. Add Preprocessing

Alternative 2: 11.4% accurate, 35.0× speedup

Math

\[\begin{array}{l} \\ x + x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ x x))

C

double code(double x) {
	return x + x;
}

Fortran

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

Java

public static double code(double x) {
	return x + x;
}

Python

def code(x):
	return x + x

Julia

function code(x)
	return Float64(x + x)
end

MATLAB

function tmp = code(x)
	tmp = x + x;
end

Wolfram

code[x_] := N[(x + x), $MachinePrecision]

Derivation

1. Initial program 56.1%

   \[\sqrt{\left(2 \cdot x\right) \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 51.1%

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

   1. *-commutative 51.1%

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

   2. add-sqr-sqrt 49.9%

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

   3. associate-*r* 49.8%

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

   4. sqrt-prod 50.1%

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

   5. count-2 50.1%

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

   6. add-log-exp 3.2%

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

   7. *-commutative 3.2%

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

   8. exp-prod 3.2%

      \[\leadsto \log \color{blue}{\left({\left(e^{\sqrt{x}}\right)}^{\left(\sqrt{x + x}\right)}\right)} \]

   9. flip-+ 1.2%

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

   10. difference-of-squares 1.2%

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

   11. associate-*r/ 1.2%

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

   12. +-inverses 1.2%

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

   13. +-inverses 1.2%

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

   14. flip-+ 3.5%

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

   15. sqrt-unprod 2.7%

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

   16. add-sqr-sqrt 2.7%

       \[\leadsto \log \left({\left(e^{\sqrt{x}}\right)}^{\color{blue}{\left(x + x\right)}}\right) \]

   17. flip-+ 1.2%

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

   18. div-inv 1.2%

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

   19. +-inverses 1.2%

       \[\leadsto \log \left({\left(e^{\sqrt{x}}\right)}^{\left(\color{blue}{0} \cdot \frac{1}{x - x}\right)}\right) \]

   20. +-inverses 1.2%

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

5. Applied egg-rr 11.3%

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

Reproduce

herbie shell --seed 2024097 
(FPCore (x)
  :name "sqrt B (should all be same)"
  :precision binary64
  (sqrt (* (* 2.0 x) x)))