sqrt D (should all be same)

Percentage Accurate: 53.5% → 99.4%

Time: 4.0s

Alternatives: 4

Speedup: 2.0×

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (sqrt (* 2.0 (pow x 2.0))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Sqrt[N[(2.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× 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 57.4%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 57.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 57.4%

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

   1. pow1/2 N/A

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

   2. rem-square-sqrt N/A

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

   3. pow1/2 N/A

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

   4. pow1/2 N/A

      \[\leadsto {\left(2 \cdot \left(x \cdot \left({x}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}\right)\right)\right)}^{\frac{1}{2}} \]

   5. associate-*r* N/A

      \[\leadsto {\left(2 \cdot \left(\left(x \cdot {x}^{\frac{1}{2}}\right) \cdot {x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}} \]

   6. *-commutative N/A

      \[\leadsto {\left(2 \cdot \left({x}^{\frac{1}{2}} \cdot \left(x \cdot {x}^{\frac{1}{2}}\right)\right)\right)}^{\frac{1}{2}} \]

   7. associate-*r* N/A

      \[\leadsto {\left(\left(2 \cdot {x}^{\frac{1}{2}}\right) \cdot \left(x \cdot {x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}} \]

   8. unpow-prod-down N/A

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

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(x \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}\right) \]

   10. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left({x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(\sqrt{x}\right)}^{\frac{1}{2}}\right), \left({\left(x \cdot {\color{blue}{x}}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   12. sqrt-pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\frac{1}{2}}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   14. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   15. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(2 \cdot 2\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   16. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   17. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot \left(2 \cdot 2\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \left(2 \cdot 2\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\left(\color{blue}{x} \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, 4\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\left(x \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, 4\right), \frac{1}{4}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

6. Applied egg-rr 50.2%

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

Alternative 2: 99.5% accurate, 1.0× 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 57.4%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 57.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 57.4%

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

   1. pow1/2 N/A

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

   2. associate-*r* N/A

      \[\leadsto {\left(\left(2 \cdot x\right) \cdot x\right)}^{\frac{1}{2}} \]

   3. unpow-prod-down N/A

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

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot x\right)}^{\frac{1}{2}}\right), \color{blue}{\left({x}^{\frac{1}{2}}\right)}\right) \]

   5. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2 \cdot x}\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot x\right)\right), \left({\color{blue}{x}}^{\frac{1}{2}}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left({x}^{\frac{1}{2}}\right)\right) \]

   8. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \left(\sqrt{x}\right)\right) \]

   9. sqrt-lowering-sqrt.f64 50.1%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, x\right)\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

6. Applied egg-rr 50.1%

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

7. Final simplification 50.1%

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

Alternative 3: 99.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{{0.25}^{-0.25}}{\frac{1}{x\_m}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (pow 0.25 -0.25) (/ 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(0.25, -0.25) / (1.0 / x_m);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (0.25d0 ** (-0.25d0)) / (1.0d0 / x_m)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(0.25, -0.25) / (1.0 / x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(0.25, -0.25) / (1.0 / x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64((0.25 ^ -0.25) / Float64(1.0 / x_m))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (0.25 ^ -0.25) / (1.0 / x_m);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[0.25, -0.25], $MachinePrecision] / N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.4%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 57.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 57.4%

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

   1. pow1/2 N/A

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

   2. rem-square-sqrt N/A

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

   3. pow1/2 N/A

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

   4. pow1/2 N/A

      \[\leadsto {\left(2 \cdot \left(x \cdot \left({x}^{\frac{1}{2}} \cdot {x}^{\frac{1}{2}}\right)\right)\right)}^{\frac{1}{2}} \]

   5. associate-*r* N/A

      \[\leadsto {\left(2 \cdot \left(\left(x \cdot {x}^{\frac{1}{2}}\right) \cdot {x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}} \]

   6. *-commutative N/A

      \[\leadsto {\left(2 \cdot \left({x}^{\frac{1}{2}} \cdot \left(x \cdot {x}^{\frac{1}{2}}\right)\right)\right)}^{\frac{1}{2}} \]

   7. associate-*r* N/A

      \[\leadsto {\left(\left(2 \cdot {x}^{\frac{1}{2}}\right) \cdot \left(x \cdot {x}^{\frac{1}{2}}\right)\right)}^{\frac{1}{2}} \]

   8. unpow-prod-down N/A

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

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(x \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}\right) \]

   10. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left({x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   11. pow1/2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {\left(\sqrt{x}\right)}^{\frac{1}{2}}\right), \left({\left(x \cdot {\color{blue}{x}}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   12. sqrt-pow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}} \cdot {x}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\frac{1}{2}}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   14. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot \left({2}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {2}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   15. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(2 \cdot 2\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

   16. pow-prod-down N/A

       \[\leadsto \mathsf{*.f64}\left(\left({\left(x \cdot \left(2 \cdot 2\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   17. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x \cdot \left(2 \cdot 2\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\color{blue}{\left(x \cdot {x}^{\frac{1}{2}}\right)}}^{\frac{1}{2}}\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, \left(2 \cdot 2\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\left(\color{blue}{x} \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, 4\right), \left(\frac{\frac{1}{2}}{2}\right)\right), \left({\left(x \cdot {x}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x, 4\right), \frac{1}{4}\right), \left({\left(x \cdot \color{blue}{{x}^{\frac{1}{2}}}\right)}^{\frac{1}{2}}\right)\right) \]

6. Applied egg-rr 50.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{4}^{\frac{1}{4}} \cdot x} \]
8. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \frac{{4}^{\frac{1}{4}} \cdot x}{\color{blue}{1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{x \cdot {4}^{\frac{1}{4}}}{1} \]

   3. associate-*l/ N/A

      \[\leadsto \frac{x}{1} \cdot \color{blue}{{4}^{\frac{1}{4}}} \]

   4. associate-/r/ N/A

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{{4}^{\frac{1}{4}}}}} \]

   5. exp-to-pow N/A

      \[\leadsto \frac{x}{\frac{1}{e^{\log 4 \cdot \frac{1}{4}}}} \]

   6. exp-neg N/A

      \[\leadsto \frac{x}{e^{\mathsf{neg}\left(\log 4 \cdot \frac{1}{4}\right)}} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(e^{\mathsf{neg}\left(\log 4 \cdot \frac{1}{4}\right)}\right)}\right) \]

   8. distribute-lft-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left(e^{\left(\mathsf{neg}\left(\log 4\right)\right) \cdot \frac{1}{4}}\right)\right) \]

   9. exp-prod N/A

      \[\leadsto \mathsf{/.f64}\left(x, \left({\left(e^{\mathsf{neg}\left(\log 4\right)}\right)}^{\color{blue}{\frac{1}{4}}}\right)\right) \]

   10. exp-neg N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left({\left(\frac{1}{e^{\log 4}}\right)}^{\frac{1}{4}}\right)\right) \]

   11. rem-exp-log N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left({\left(\frac{1}{4}\right)}^{\frac{1}{4}}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(x, \left({\frac{1}{4}}^{\frac{1}{4}}\right)\right) \]

   13. pow-lowering-pow.f64 51.2%

       \[\leadsto \mathsf{/.f64}\left(x, \mathsf{pow.f64}\left(\frac{1}{4}, \color{blue}{\frac{1}{4}}\right)\right) \]

9. Simplified 51.2%

   \[\leadsto \color{blue}{\frac{x}{{0.25}^{0.25}}} \]
10. Step-by-step derivation

    1. clear-num N/A

       \[\leadsto \frac{1}{\color{blue}{\frac{{\frac{1}{4}}^{\frac{1}{4}}}{x}}} \]

    2. div-inv N/A

       \[\leadsto \frac{1}{{\frac{1}{4}}^{\frac{1}{4}} \cdot \color{blue}{\frac{1}{x}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{1}{{\frac{1}{4}}^{\frac{1}{4}}}}{\color{blue}{\frac{1}{x}}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{{\frac{1}{4}}^{\frac{1}{4}}}\right), \color{blue}{\left(\frac{1}{x}\right)}\right) \]

    5. pow-flip N/A

       \[\leadsto \mathsf{/.f64}\left(\left({\frac{1}{4}}^{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right)}\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{4}, \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right), \left(\frac{\color{blue}{1}}{x}\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{4}, \frac{-1}{4}\right), \left(\frac{1}{x}\right)\right) \]

    8. /-lowering-/.f64 51.3%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\frac{1}{4}, \frac{-1}{4}\right), \mathsf{/.f64}\left(1, \color{blue}{x}\right)\right) \]

11. Applied egg-rr 51.3%

    \[\leadsto \color{blue}{\frac{{0.25}^{-0.25}}{\frac{1}{x}}} \]
12. Add Preprocessing

Alternative 4: 99.3% accurate, 2.0× 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 57.4%

   \[\sqrt{2 \cdot {x}^{2}} \]
2. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot {x}^{2}\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left({x}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(x \cdot x\right)\right)\right) \]

   4. *-lowering-*.f64 57.4%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

3. Simplified 57.4%

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

   1. sqrt-prod N/A

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

   2. pow1/2 N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

      \[\leadsto {2}^{\frac{1}{2}} \cdot x \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({2}^{\frac{1}{2}}\right), \color{blue}{x}\right) \]

   6. pow1/2 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{2}\right), x\right) \]

   7. sqrt-lowering-sqrt.f64 51.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(2\right), x\right) \]

6. Applied egg-rr 51.2%

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

7. Final simplification 51.2%

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

Reproduce

herbie shell --seed 2024150 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))