sqrt E (should all be same)

Percentage Accurate: 53.6% → 100.0%

Time: 2.8s

Alternatives: 5

Speedup: 9.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 2.1× 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%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 99.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{x}{\sqrt{2}} \cdot \frac{x \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (/ x (sqrt 2.0)) (/ (* x -2.0) x)) (* x (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (x / sqrt(2.0)) * ((x * -2.0) / x);
	} else {
		tmp = x * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = (x / sqrt(2.0d0)) * ((x * (-2.0d0)) / x)
    else
        tmp = x * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = (x / Math.sqrt(2.0)) * ((x * -2.0) / x);
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = (x / math.sqrt(2.0)) * ((x * -2.0) / x)
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(Float64(x / sqrt(2.0)) * Float64(Float64(x * -2.0) / x));
	else
		tmp = Float64(x * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = (x / sqrt(2.0)) * ((x * -2.0) / x);
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[(x / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(x * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. neg-sub0 N/A

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

      4. flip-- N/A

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

      5. lower-/.f64 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 58.4

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

   7. Applied rewrites 58.4%

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

   8. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot \left({x}^{2} \cdot {\left(\sqrt{2}\right)}^{2}\right)}}{0 + x \cdot \sqrt{2}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. unpow2 N/A

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

      3. rem-square-sqrt N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval 58.4

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

   10. Applied rewrites 58.4%

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

       1. associate-*l* N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. +-lft-identity N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. times-frac N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 N/A

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

       10. lower-/.f64 N/A

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

       11. lower-*.f64 99.3

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

   12. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2e-310) (* (sqrt 2.0) (- x)) (* x (sqrt 2.0))))

C

double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = sqrt(2.0) * -x;
	} else {
		tmp = x * sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-2d-310)) then
        tmp = sqrt(2.0d0) * -x
    else
        tmp = x * sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -2e-310) {
		tmp = Math.sqrt(2.0) * -x;
	} else {
		tmp = x * Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2e-310:
		tmp = math.sqrt(2.0) * -x
	else:
		tmp = x * math.sqrt(2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2e-310)
		tmp = Float64(sqrt(2.0) * Float64(-x));
	else
		tmp = Float64(x * sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2e-310)
		tmp = sqrt(2.0) * -x;
	else
		tmp = x * sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2e-310], N[(N[Sqrt[2.0], $MachinePrecision] * (-x)), $MachinePrecision], N[(x * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.999999999999994e-310

   1. Initial program 58.7%

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

   3. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \sqrt{2}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

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

   if -1.999999999999994e-310 < x

   1. Initial program 53.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sqrt.f64 99.3

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

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{x \cdot \sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{2} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.3% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sqrt.f64 50.5

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

5. Applied rewrites 50.5%

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

Alternative 5: 3.9% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 56.1%

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. lower-hypot.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{hypot}\left(x, x\right)} \]
5. Step-by-step derivation

   1. pow1/2 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. flip-+ N/A

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

   5. +-rgt-identity N/A

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

   6. lift-fma.f64 N/A

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

   7. +-rgt-identity N/A

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

   8. lift-fma.f64 N/A

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

   9. +-inverses N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. +-inverses N/A

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

   14. metadata-eval N/A

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

   15. flip-- N/A

       \[\leadsto {\color{blue}{\left(0 - 0\right)}}^{\frac{1}{2}} \]

   16. metadata-eval N/A

       \[\leadsto {\color{blue}{0}}^{\frac{1}{2}} \]

   17. metadata-eval 3.9

       \[\leadsto \color{blue}{0} \]

6. Applied rewrites 3.9%

   \[\leadsto \color{blue}{0} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "sqrt E (should all be same)"
  :precision binary64
  (sqrt (+ (pow x 2.0) (pow x 2.0))))