sqrt D (should all be same)

Percentage Accurate: 54.9% → 99.4%

Time: 13.7s

Alternatives: 5

Speedup: 7.3×

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: 54.9% 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, 3.8× 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 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 51.9

      \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]

5. Simplified 51.9%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

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

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

   6. rem-square-sqrt N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. +-rgt-identity N/A

       \[\leadsto \sqrt{\sqrt{2} \cdot \color{blue}{\left(x \cdot \sqrt{2} + 0\right)}} \cdot {x}^{\frac{1}{2}} \]

   11. distribute-lft-in N/A

       \[\leadsto \sqrt{\color{blue}{\sqrt{2} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot 0}} \cdot {x}^{\frac{1}{2}} \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\sqrt{2} \cdot \color{blue}{\left(\sqrt{2} \cdot x\right)} + \sqrt{2} \cdot 0} \cdot {x}^{\frac{1}{2}} \]

   13. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot x} + \sqrt{2} \cdot 0} \cdot {x}^{\frac{1}{2}} \]

   14. rem-square-sqrt N/A

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

   15. mul0-rgt N/A

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

   16. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, x, 0\right)}} \cdot {x}^{\frac{1}{2}} \]

   17. pow1/2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \color{blue}{\sqrt{x}} \]

   18. sqrt-lowering-sqrt.f64 46.3

       \[\leadsto \sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \color{blue}{\sqrt{x}} \]

7. Applied egg-rr 46.3%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \sqrt{x}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 46.3

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

9. Applied egg-rr 46.3%

   \[\leadsto \sqrt{\color{blue}{x \cdot 2}} \cdot \sqrt{x} \]
10. Add Preprocessing

Alternative 2: 99.3% accurate, 5.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{x\_m}{\sqrt{0.5}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ x_m (sqrt 0.5)))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m / sqrt(0.5);
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m / sqrt(0.5d0)
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m / Math.sqrt(0.5);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m / math.sqrt(0.5)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m / sqrt(0.5))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m / sqrt(0.5);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 51.9

      \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]

5. Simplified 51.9%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. associate-*r* N/A

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

   3. sqrt-prod N/A

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

   4. pow1/2 N/A

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

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

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

   6. rem-square-sqrt N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. +-rgt-identity N/A

       \[\leadsto \sqrt{\sqrt{2} \cdot \color{blue}{\left(x \cdot \sqrt{2} + 0\right)}} \cdot {x}^{\frac{1}{2}} \]

   11. distribute-lft-in N/A

       \[\leadsto \sqrt{\color{blue}{\sqrt{2} \cdot \left(x \cdot \sqrt{2}\right) + \sqrt{2} \cdot 0}} \cdot {x}^{\frac{1}{2}} \]

   12. *-commutative N/A

       \[\leadsto \sqrt{\sqrt{2} \cdot \color{blue}{\left(\sqrt{2} \cdot x\right)} + \sqrt{2} \cdot 0} \cdot {x}^{\frac{1}{2}} \]

   13. associate-*r* N/A

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot x} + \sqrt{2} \cdot 0} \cdot {x}^{\frac{1}{2}} \]

   14. rem-square-sqrt N/A

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

   15. mul0-rgt N/A

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

   16. accelerator-lowering-fma.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(2, x, 0\right)}} \cdot {x}^{\frac{1}{2}} \]

   17. pow1/2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \color{blue}{\sqrt{x}} \]

   18. sqrt-lowering-sqrt.f64 46.3

       \[\leadsto \sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \color{blue}{\sqrt{x}} \]

7. Applied egg-rr 46.3%

   \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(2, x, 0\right)} \cdot \sqrt{x}} \]
8. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 46.3

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

9. Applied egg-rr 46.3%

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

    1. metadata-eval N/A

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

    2. div-inv N/A

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

    3. sqrt-div N/A

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

    4. pow1/2 N/A

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

    5. associate-*l/ N/A

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

    6. rem-square-sqrt N/A

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

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

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

    8. pow1/2 N/A

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

    9. sqrt-lowering-sqrt.f64 47.4

       \[\leadsto \frac{x}{\color{blue}{\sqrt{0.5}}} \]

11. Applied egg-rr 47.4%

    \[\leadsto \color{blue}{\frac{x}{\sqrt{0.5}}} \]
12. Add Preprocessing

Alternative 3: 99.3% accurate, 7.3× 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 51.9%

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

3. Taylor expanded in x around 0

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

   1. +-rgt-identity N/A

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

   2. unpow2 N/A

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

   3. accelerator-lowering-fma.f64 51.9

      \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]

5. Simplified 51.9%

   \[\leadsto \sqrt{2 \cdot \color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \]
6. Step-by-step derivation

   1. sqrt-prod N/A

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

   2. +-rgt-identity N/A

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt N/A

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

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

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

   6. sqrt-lowering-sqrt.f64 47.3

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

7. Applied egg-rr 47.3%

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

8. Final simplification 47.3%

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

Alternative 4: 7.0% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

   1. pow-to-exp N/A

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

   2. *-commutative N/A

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

   3. count-2 N/A

      \[\leadsto \sqrt{2 \cdot e^{\color{blue}{\log x + \log x}}} \]

   4. flip-+ N/A

      \[\leadsto \sqrt{2 \cdot e^{\color{blue}{\frac{\log x \cdot \log x - \log x \cdot \log x}{\log x - \log x}}}} \]

   5. +-inverses N/A

      \[\leadsto \sqrt{2 \cdot e^{\frac{\color{blue}{0}}{\log x - \log x}}} \]

   6. +-inverses N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. flip-+ N/A

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

   10. log-prod N/A

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

   11. sqr-pow N/A

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

   12. metadata-eval N/A

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

   13. unpow1 N/A

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

   14. rem-exp-log 3.3

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

4. Applied egg-rr 3.3%

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

5. Final simplification 3.3%

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

Alternative 5: 5.4% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.9%

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

4. Applied egg-rr 5.4%

   \[\leadsto \color{blue}{\sqrt{2}} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(FPCore (x)
  :name "sqrt D (should all be same)"
  :precision binary64
  (sqrt (* 2.0 (pow x 2.0))))