Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B

Percentage Accurate: 99.4% → 99.4%

Time: 11.9s

Alternatives: 16

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) - 1.0d0)
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) - 1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) - 1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* 3.0 (sqrt x)) (+ (+ y (/ 1.0 (* x 9.0))) -1.0)))

C

double code(double x, double y) {
	return (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (3.0d0 * sqrt(x)) * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
end function

Java

public static double code(double x, double y) {
	return (3.0 * Math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
}

Python

def code(x, y):
	return (3.0 * math.sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0)

Julia

function code(x, y)
	return Float64(Float64(3.0 * sqrt(x)) * Float64(Float64(y + Float64(1.0 / Float64(x * 9.0))) + -1.0))
end

MATLAB

function tmp = code(x, y)
	tmp = (3.0 * sqrt(x)) * ((y + (1.0 / (x * 9.0))) + -1.0);
end

Wolfram

code[x_, y_] := N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(N[(y + N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing

3. Final simplification 99.5%

   \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \]
4. Add Preprocessing

Alternative 2: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.25e+43)
   (* 3.0 (/ y (pow x -0.5)))
   (if (<= y -4e-305)
     (/ (sqrt x) (* 3.0 x))
     (if (<= y 3.7e-65)
       (* (sqrt x) -3.0)
       (if (<= y 1200000.0)
         (/ (/ 0.3333333333333333 x) (pow x -0.5))
         (* y (/ 3.0 (pow x -0.5))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+43) {
		tmp = 3.0 * (y / pow(x, -0.5));
	} else if (y <= -4e-305) {
		tmp = sqrt(x) / (3.0 * x);
	} else if (y <= 3.7e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1200000.0) {
		tmp = (0.3333333333333333 / x) / pow(x, -0.5);
	} else {
		tmp = y * (3.0 / pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.25d+43)) then
        tmp = 3.0d0 * (y / (x ** (-0.5d0)))
    else if (y <= (-4d-305)) then
        tmp = sqrt(x) / (3.0d0 * x)
    else if (y <= 3.7d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1200000.0d0) then
        tmp = (0.3333333333333333d0 / x) / (x ** (-0.5d0))
    else
        tmp = y * (3.0d0 / (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.25e+43) {
		tmp = 3.0 * (y / Math.pow(x, -0.5));
	} else if (y <= -4e-305) {
		tmp = Math.sqrt(x) / (3.0 * x);
	} else if (y <= 3.7e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1200000.0) {
		tmp = (0.3333333333333333 / x) / Math.pow(x, -0.5);
	} else {
		tmp = y * (3.0 / Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.25e+43:
		tmp = 3.0 * (y / math.pow(x, -0.5))
	elif y <= -4e-305:
		tmp = math.sqrt(x) / (3.0 * x)
	elif y <= 3.7e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1200000.0:
		tmp = (0.3333333333333333 / x) / math.pow(x, -0.5)
	else:
		tmp = y * (3.0 / math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.25e+43)
		tmp = Float64(3.0 * Float64(y / (x ^ -0.5)));
	elseif (y <= -4e-305)
		tmp = Float64(sqrt(x) / Float64(3.0 * x));
	elseif (y <= 3.7e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1200000.0)
		tmp = Float64(Float64(0.3333333333333333 / x) / (x ^ -0.5));
	else
		tmp = Float64(y * Float64(3.0 / (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.25e+43)
		tmp = 3.0 * (y / (x ^ -0.5));
	elseif (y <= -4e-305)
		tmp = sqrt(x) / (3.0 * x);
	elseif (y <= 3.7e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1200000.0)
		tmp = (0.3333333333333333 / x) / (x ^ -0.5);
	else
		tmp = y * (3.0 / (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.25e+43], N[(3.0 * N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -4e-305], N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1200000.0], N[(N[(0.3333333333333333 / x), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(y * N[(3.0 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -3.2499999999999999e43

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \sqrt{x}\right), 3\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{\sqrt{x}}\right)\right), 3\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), 3\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 3\right) \]

      8. metadata-eval 86.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), 3\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \cdot 3 \]

   if -3.2499999999999999e43 < y < -3.99999999999999999e-305

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \sqrt{x} \cdot \frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \sqrt{x} \cdot \frac{1}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{\color{blue}{\frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}}\right)\right) \]

      7. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3 \cdot y + \color{blue}{\left(-3 + \frac{\frac{1}{3}}{x}\right)}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(3 \cdot y + \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}\right)\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1}{3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(3 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(x \cdot \color{blue}{3}\right)\right) \]

      2. *-lowering-*.f64 55.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   9. Simplified 55.8%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{x \cdot 3}} \]

   if -3.99999999999999999e-305 < y < 3.7e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   if 3.7e-65 < y < 1.2e6

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{\frac{1}{x}}{9}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x}\right), 9\right)\right), 1\right)\right) \]

      3. /-lowering-/.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), 9\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{\frac{1}{x}}{9}}\right) - 1\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot \color{blue}{\left(3 \cdot \sqrt{x}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \color{blue}{\sqrt{x}} \]

      3. remove-double-div N/A

         \[\leadsto \left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \frac{1}{\color{blue}{\frac{1}{\sqrt{x}}}} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3}{\color{blue}{\frac{1}{\sqrt{x}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right), \color{blue}{\left(\frac{1}{\sqrt{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right), 3\right), \left(\frac{\color{blue}{1}}{\sqrt{x}}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{x}}{9} + y\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{x}}{9} + y\right) + -1\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{x}}{9} + \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{x}}{9}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      17. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left(\frac{1}{{x}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

      18. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left({x}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      20. metadata-eval 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot 3}{{x}^{-0.5}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{\frac{1}{3}}{x}\right)}, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 64.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), \mathsf{pow.f64}\left(\color{blue}{x}, \frac{-1}{2}\right)\right) \]

   9. Simplified 64.1%

      \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{x}}}{{x}^{-0.5}} \]

   if 1.2e6 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot y \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right) \cdot y \]

      5. div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\frac{1}{3}} \cdot y \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{x}}{\frac{1}{3}}\right), \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot 3\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \sqrt{x}\right), y\right) \]

      10. remove-double-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{3}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{\sqrt{x}}\right)\right), y\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), y\right) \]

      14. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), y\right) \]

      16. metadata-eval 86.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), y\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{3}{{x}^{-0.5}} \cdot y} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1200000:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{x}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-302}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 4200000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt x) (* 3.0 x))))
   (if (<= y -5.5e+43)
     (* 3.0 (/ y (pow x -0.5)))
     (if (<= y -1.05e-302)
       t_0
       (if (<= y 3.8e-65)
         (* (sqrt x) -3.0)
         (if (<= y 4200000000.0) t_0 (* y (/ 3.0 (pow x -0.5)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -5.5e+43) {
		tmp = 3.0 * (y / pow(x, -0.5));
	} else if (y <= -1.05e-302) {
		tmp = t_0;
	} else if (y <= 3.8e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 4200000000.0) {
		tmp = t_0;
	} else {
		tmp = y * (3.0 / pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) / (3.0d0 * x)
    if (y <= (-5.5d+43)) then
        tmp = 3.0d0 * (y / (x ** (-0.5d0)))
    else if (y <= (-1.05d-302)) then
        tmp = t_0
    else if (y <= 3.8d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 4200000000.0d0) then
        tmp = t_0
    else
        tmp = y * (3.0d0 / (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -5.5e+43) {
		tmp = 3.0 * (y / Math.pow(x, -0.5));
	} else if (y <= -1.05e-302) {
		tmp = t_0;
	} else if (y <= 3.8e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 4200000000.0) {
		tmp = t_0;
	} else {
		tmp = y * (3.0 / Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) / (3.0 * x)
	tmp = 0
	if y <= -5.5e+43:
		tmp = 3.0 * (y / math.pow(x, -0.5))
	elif y <= -1.05e-302:
		tmp = t_0
	elif y <= 3.8e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 4200000000.0:
		tmp = t_0
	else:
		tmp = y * (3.0 / math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) / Float64(3.0 * x))
	tmp = 0.0
	if (y <= -5.5e+43)
		tmp = Float64(3.0 * Float64(y / (x ^ -0.5)));
	elseif (y <= -1.05e-302)
		tmp = t_0;
	elseif (y <= 3.8e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 4200000000.0)
		tmp = t_0;
	else
		tmp = Float64(y * Float64(3.0 / (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) / (3.0 * x);
	tmp = 0.0;
	if (y <= -5.5e+43)
		tmp = 3.0 * (y / (x ^ -0.5));
	elseif (y <= -1.05e-302)
		tmp = t_0;
	elseif (y <= 3.8e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 4200000000.0)
		tmp = t_0;
	else
		tmp = y * (3.0 / (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+43], N[(3.0 * N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.05e-302], t$95$0, If[LessEqual[y, 3.8e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 4200000000.0], t$95$0, N[(y * N[(3.0 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.49999999999999989e43

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \sqrt{x}\right), 3\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{\sqrt{x}}\right)\right), 3\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), 3\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 3\right) \]

      8. metadata-eval 86.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), 3\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \cdot 3 \]

   if -5.49999999999999989e43 < y < -1.05000000000000006e-302 or 3.8000000000000002e-65 < y < 4.2e9

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \sqrt{x} \cdot \frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \sqrt{x} \cdot \frac{1}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{\color{blue}{\frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}}\right)\right) \]

      7. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3 \cdot y + \color{blue}{\left(-3 + \frac{\frac{1}{3}}{x}\right)}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(3 \cdot y + \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1}{3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(3 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(x \cdot \color{blue}{3}\right)\right) \]

      2. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   9. Simplified 56.8%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{x \cdot 3}} \]

   if -1.05000000000000006e-302 < y < 3.8000000000000002e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   if 4.2e9 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot y \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right) \cdot y \]

      5. div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\frac{1}{3}} \cdot y \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{x}}{\frac{1}{3}}\right), \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot 3\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \sqrt{x}\right), y\right) \]

      10. remove-double-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{3}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{\sqrt{x}}\right)\right), y\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), y\right) \]

      14. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), y\right) \]

      16. metadata-eval 86.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), y\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{3}{{x}^{-0.5}} \cdot y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{-302}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 4200000000:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{3}{{x}^{-0.5}}\\ t_1 := \frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-304}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 840000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (/ 3.0 (pow x -0.5)))) (t_1 (/ (sqrt x) (* 3.0 x))))
   (if (<= y -3.25e+43)
     t_0
     (if (<= y -3.5e-304)
       t_1
       (if (<= y 3.8e-65)
         (* (sqrt x) -3.0)
         (if (<= y 840000000000.0) t_1 t_0))))))

C

double code(double x, double y) {
	double t_0 = y * (3.0 / pow(x, -0.5));
	double t_1 = sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -3.25e+43) {
		tmp = t_0;
	} else if (y <= -3.5e-304) {
		tmp = t_1;
	} else if (y <= 3.8e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 840000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (3.0d0 / (x ** (-0.5d0)))
    t_1 = sqrt(x) / (3.0d0 * x)
    if (y <= (-3.25d+43)) then
        tmp = t_0
    else if (y <= (-3.5d-304)) then
        tmp = t_1
    else if (y <= 3.8d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 840000000000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (3.0 / Math.pow(x, -0.5));
	double t_1 = Math.sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -3.25e+43) {
		tmp = t_0;
	} else if (y <= -3.5e-304) {
		tmp = t_1;
	} else if (y <= 3.8e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 840000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (3.0 / math.pow(x, -0.5))
	t_1 = math.sqrt(x) / (3.0 * x)
	tmp = 0
	if y <= -3.25e+43:
		tmp = t_0
	elif y <= -3.5e-304:
		tmp = t_1
	elif y <= 3.8e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 840000000000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(3.0 / (x ^ -0.5)))
	t_1 = Float64(sqrt(x) / Float64(3.0 * x))
	tmp = 0.0
	if (y <= -3.25e+43)
		tmp = t_0;
	elseif (y <= -3.5e-304)
		tmp = t_1;
	elseif (y <= 3.8e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 840000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (3.0 / (x ^ -0.5));
	t_1 = sqrt(x) / (3.0 * x);
	tmp = 0.0;
	if (y <= -3.25e+43)
		tmp = t_0;
	elseif (y <= -3.5e-304)
		tmp = t_1;
	elseif (y <= 3.8e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 840000000000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(3.0 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.25e+43], t$95$0, If[LessEqual[y, -3.5e-304], t$95$1, If[LessEqual[y, 3.8e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 840000000000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2499999999999999e43 or 8.4e11 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot y \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right) \cdot y \]

      5. div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\frac{1}{3}} \cdot y \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{x}}{\frac{1}{3}}\right), \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot 3\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \sqrt{x}\right), y\right) \]

      10. remove-double-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{3}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{\sqrt{x}}\right)\right), y\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), y\right) \]

      14. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), y\right) \]

      16. metadata-eval 86.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), y\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{3}{{x}^{-0.5}} \cdot y} \]

   if -3.2499999999999999e43 < y < -3.5e-304 or 3.8000000000000002e-65 < y < 8.4e11

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \sqrt{x} \cdot \frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \sqrt{x} \cdot \frac{1}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{\color{blue}{\frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}}\right)\right) \]

      7. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3 \cdot y + \color{blue}{\left(-3 + \frac{\frac{1}{3}}{x}\right)}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(3 \cdot y + \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1}{3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(3 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(x \cdot \color{blue}{3}\right)\right) \]

      2. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   9. Simplified 56.8%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{x \cdot 3}} \]

   if -3.5e-304 < y < 3.8000000000000002e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 840000000000:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-305}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1200000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{y}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sqrt x) (* 3.0 x))))
   (if (<= y -3.25e+43)
     (* (sqrt x) (* 3.0 y))
     (if (<= y -1.6e-305)
       t_0
       (if (<= y 3.65e-65)
         (* (sqrt x) -3.0)
         (if (<= y 1200000000000.0)
           t_0
           (* (sqrt x) (/ y 0.3333333333333333))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -3.25e+43) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= -1.6e-305) {
		tmp = t_0;
	} else if (y <= 3.65e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1200000000000.0) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * (y / 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) / (3.0d0 * x)
    if (y <= (-3.25d+43)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= (-1.6d-305)) then
        tmp = t_0
    else if (y <= 3.65d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1200000000000.0d0) then
        tmp = t_0
    else
        tmp = sqrt(x) * (y / 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) / (3.0 * x);
	double tmp;
	if (y <= -3.25e+43) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= -1.6e-305) {
		tmp = t_0;
	} else if (y <= 3.65e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1200000000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (y / 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) / (3.0 * x)
	tmp = 0
	if y <= -3.25e+43:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= -1.6e-305:
		tmp = t_0
	elif y <= 3.65e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1200000000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (y / 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) / Float64(3.0 * x))
	tmp = 0.0
	if (y <= -3.25e+43)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= -1.6e-305)
		tmp = t_0;
	elseif (y <= 3.65e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1200000000000.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(y / 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) / (3.0 * x);
	tmp = 0.0;
	if (y <= -3.25e+43)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= -1.6e-305)
		tmp = t_0;
	elseif (y <= 3.65e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1200000000000.0)
		tmp = t_0;
	else
		tmp = sqrt(x) * (y / 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] / N[(3.0 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.25e+43], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.6e-305], t$95$0, If[LessEqual[y, 3.65e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1200000000000.0], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(y / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.2499999999999999e43

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot y\right)\right) \]

      6. *-lowering-*.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(3, \color{blue}{y}\right)\right) \]

   7. Simplified 86.0%

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

   if -3.2499999999999999e43 < y < -1.60000000000000004e-305 or 3.6499999999999999e-65 < y < 1.2e12

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto \sqrt{x} \cdot \frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}} \]

      2. clear-num N/A

         \[\leadsto \sqrt{x} \cdot \frac{1}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{\color{blue}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}\right)\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{\color{blue}{\frac{{\left(3 \cdot y\right)}^{3} + {\left(-3 + \frac{\frac{1}{3}}{x}\right)}^{3}}{\left(3 \cdot y\right) \cdot \left(3 \cdot y\right) + \left(\left(-3 + \frac{\frac{1}{3}}{x}\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right) - \left(3 \cdot y\right) \cdot \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}}}\right)\right) \]

      7. flip3-+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3 \cdot y + \color{blue}{\left(-3 + \frac{\frac{1}{3}}{x}\right)}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(1, \color{blue}{\left(3 \cdot y + \left(-3 + \frac{\frac{1}{3}}{x}\right)\right)}\right)\right) \]

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{\frac{\sqrt{x}}{\frac{1}{3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)}}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(3 \cdot x\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(x \cdot \color{blue}{3}\right)\right) \]

      2. *-lowering-*.f64 56.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(x, \color{blue}{3}\right)\right) \]

   9. Simplified 56.8%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{x \cdot 3}} \]

   if -1.60000000000000004e-305 < y < 3.6499999999999999e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   if 1.2e12 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\sqrt{x}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot 3\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{3}}\right), \left(\sqrt{x}\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{3}}\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{3}\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{3}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

   9. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{y}{0.3333333333333333} \cdot \sqrt{x}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+43}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1200000000000:\\ \;\;\;\;\frac{\sqrt{x}}{3 \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{y}{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1920000000000:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{y}{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4.5e+38)
   (* (sqrt x) (* 3.0 y))
   (if (<= y -1.1e-302)
     (/ 0.3333333333333333 (sqrt x))
     (if (<= y 3.65e-65)
       (* (sqrt x) -3.0)
       (if (<= y 1920000000000.0)
         (* 0.3333333333333333 (pow x -0.5))
         (* (sqrt x) (/ y 0.3333333333333333)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+38) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= -1.1e-302) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else if (y <= 3.65e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1920000000000.0) {
		tmp = 0.3333333333333333 * pow(x, -0.5);
	} else {
		tmp = sqrt(x) * (y / 0.3333333333333333);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-4.5d+38)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= (-1.1d-302)) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else if (y <= 3.65d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1920000000000.0d0) then
        tmp = 0.3333333333333333d0 * (x ** (-0.5d0))
    else
        tmp = sqrt(x) * (y / 0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -4.5e+38) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= -1.1e-302) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else if (y <= 3.65e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1920000000000.0) {
		tmp = 0.3333333333333333 * Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(x) * (y / 0.3333333333333333);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4.5e+38:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= -1.1e-302:
		tmp = 0.3333333333333333 / math.sqrt(x)
	elif y <= 3.65e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1920000000000.0:
		tmp = 0.3333333333333333 * math.pow(x, -0.5)
	else:
		tmp = math.sqrt(x) * (y / 0.3333333333333333)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4.5e+38)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= -1.1e-302)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	elseif (y <= 3.65e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1920000000000.0)
		tmp = Float64(0.3333333333333333 * (x ^ -0.5));
	else
		tmp = Float64(sqrt(x) * Float64(y / 0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4.5e+38)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= -1.1e-302)
		tmp = 0.3333333333333333 / sqrt(x);
	elseif (y <= 3.65e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1920000000000.0)
		tmp = 0.3333333333333333 * (x ^ -0.5);
	else
		tmp = sqrt(x) * (y / 0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4.5e+38], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.1e-302], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.65e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1920000000000.0], N[(0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y / 0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.4999999999999998e38

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot y\right)\right) \]

      6. *-lowering-*.f64 84.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(3, \color{blue}{y}\right)\right) \]

   7. Simplified 84.7%

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

   if -4.4999999999999998e38 < y < -1.10000000000000004e-302

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt{\color{blue}{x}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt{x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 55.9%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(x\right)\right) \]

   9. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if -1.10000000000000004e-302 < y < 3.6499999999999999e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   if 3.6499999999999999e-65 < y < 1.92e12

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\frac{1}{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{-1}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \frac{1}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(-1 \cdot \frac{1}{2}\right)\right), \frac{1}{3}\right) \]

      7. metadata-eval 63.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{3}\right) \]

   9. Applied egg-rr 63.6%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.3333333333333333} \]

   if 1.92e12 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\sqrt{x}\right)}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot 3\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{3}}\right), \left(\sqrt{x}\right)\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{3}}\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{3}\right), \left(\sqrt{\color{blue}{x}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 86.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \frac{1}{3}\right), \mathsf{sqrt.f64}\left(x\right)\right) \]

   9. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{y}{0.3333333333333333} \cdot \sqrt{x}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{+38}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-302}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.65 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1920000000000:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \frac{y}{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1700000:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (* 3.0 y))))
   (if (<= y -3.1e+35)
     t_0
     (if (<= y -5.5e-304)
       (/ 0.3333333333333333 (sqrt x))
       (if (<= y 4.8e-65)
         (* (sqrt x) -3.0)
         (if (<= y 1700000.0) (* 0.3333333333333333 (pow x -0.5)) t_0))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (3.0 * y);
	double tmp;
	if (y <= -3.1e+35) {
		tmp = t_0;
	} else if (y <= -5.5e-304) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else if (y <= 4.8e-65) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 1700000.0) {
		tmp = 0.3333333333333333 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (3.0d0 * y)
    if (y <= (-3.1d+35)) then
        tmp = t_0
    else if (y <= (-5.5d-304)) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else if (y <= 4.8d-65) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 1700000.0d0) then
        tmp = 0.3333333333333333d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (3.0 * y);
	double tmp;
	if (y <= -3.1e+35) {
		tmp = t_0;
	} else if (y <= -5.5e-304) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else if (y <= 4.8e-65) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 1700000.0) {
		tmp = 0.3333333333333333 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (3.0 * y)
	tmp = 0
	if y <= -3.1e+35:
		tmp = t_0
	elif y <= -5.5e-304:
		tmp = 0.3333333333333333 / math.sqrt(x)
	elif y <= 4.8e-65:
		tmp = math.sqrt(x) * -3.0
	elif y <= 1700000.0:
		tmp = 0.3333333333333333 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(3.0 * y))
	tmp = 0.0
	if (y <= -3.1e+35)
		tmp = t_0;
	elseif (y <= -5.5e-304)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	elseif (y <= 4.8e-65)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 1700000.0)
		tmp = Float64(0.3333333333333333 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (3.0 * y);
	tmp = 0.0;
	if (y <= -3.1e+35)
		tmp = t_0;
	elseif (y <= -5.5e-304)
		tmp = 0.3333333333333333 / sqrt(x);
	elseif (y <= 4.8e-65)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 1700000.0)
		tmp = 0.3333333333333333 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.1e+35], t$95$0, If[LessEqual[y, -5.5e-304], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-65], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 1700000.0], N[(0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.09999999999999987e35 or 1.7e6 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot y\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot y\right)\right) \]

      6. *-lowering-*.f64 85.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{*.f64}\left(3, \color{blue}{y}\right)\right) \]

   7. Simplified 85.4%

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

   if -3.09999999999999987e35 < y < -5.50000000000000035e-304

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 55.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt{\color{blue}{x}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt{x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 55.9%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(x\right)\right) \]

   9. Applied egg-rr 55.9%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if -5.50000000000000035e-304 < y < 4.8000000000000003e-65

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.4%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 99.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 61.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 61.5%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   if 4.8000000000000003e-65 < y < 1.7e6

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 63.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\frac{1}{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{-1}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \frac{1}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(-1 \cdot \frac{1}{2}\right)\right), \frac{1}{3}\right) \]

      7. metadata-eval 63.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{3}\right) \]

   9. Applied egg-rr 63.6%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.3333333333333333} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1700000:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;3 \cdot \frac{0.1111111111111111 - x}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7e+43)
   (* 3.0 (/ y (pow x -0.5)))
   (if (<= y 3.25e-9)
     (* 3.0 (/ (- 0.1111111111111111 x) (sqrt x)))
     (* (* 3.0 (sqrt x)) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7e+43) {
		tmp = 3.0 * (y / pow(x, -0.5));
	} else if (y <= 3.25e-9) {
		tmp = 3.0 * ((0.1111111111111111 - x) / sqrt(x));
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7d+43)) then
        tmp = 3.0d0 * (y / (x ** (-0.5d0)))
    else if (y <= 3.25d-9) then
        tmp = 3.0d0 * ((0.1111111111111111d0 - x) / sqrt(x))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7e+43) {
		tmp = 3.0 * (y / Math.pow(x, -0.5));
	} else if (y <= 3.25e-9) {
		tmp = 3.0 * ((0.1111111111111111 - x) / Math.sqrt(x));
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7e+43:
		tmp = 3.0 * (y / math.pow(x, -0.5))
	elif y <= 3.25e-9:
		tmp = 3.0 * ((0.1111111111111111 - x) / math.sqrt(x))
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7e+43)
		tmp = Float64(3.0 * Float64(y / (x ^ -0.5)));
	elseif (y <= 3.25e-9)
		tmp = Float64(3.0 * Float64(Float64(0.1111111111111111 - x) / sqrt(x)));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7e+43)
		tmp = 3.0 * (y / (x ^ -0.5));
	elseif (y <= 3.25e-9)
		tmp = 3.0 * ((0.1111111111111111 - x) / sqrt(x));
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7e+43], N[(3.0 * N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.25e-9], N[(3.0 * N[(N[(0.1111111111111111 - x), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000002e43

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \sqrt{x}\right), 3\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{\sqrt{x}}\right)\right), 3\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), 3\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 3\right) \]

      8. metadata-eval 86.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), 3\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \cdot 3 \]

   if -7.0000000000000002e43 < y < 3.2500000000000002e-9

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 99.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 99.5%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 96.4%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]
   11. Step-by-step derivation

       1. associate-*l* N/A

          \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\left(\left(\frac{1}{9} - x\right) \cdot 3\right)} \]

       2. inv-pow N/A

          \[\leadsto \sqrt{{x}^{-1}} \cdot \left(\left(\color{blue}{\frac{1}{9}} - x\right) \cdot 3\right) \]

       3. sqrt-pow1 N/A

          \[\leadsto {x}^{\left(\frac{-1}{2}\right)} \cdot \left(\color{blue}{\left(\frac{1}{9} - x\right)} \cdot 3\right) \]

       4. metadata-eval N/A

          \[\leadsto {x}^{\frac{-1}{2}} \cdot \left(\left(\frac{1}{9} - \color{blue}{x}\right) \cdot 3\right) \]

       5. associate-*r* N/A

          \[\leadsto \left({x}^{\frac{-1}{2}} \cdot \left(\frac{1}{9} - x\right)\right) \cdot \color{blue}{3} \]

       6. metadata-eval N/A

          \[\leadsto \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \left(\frac{1}{9} - x\right)\right) \cdot 3 \]

       7. pow-flip N/A

          \[\leadsto \left(\frac{1}{{x}^{\frac{1}{2}}} \cdot \left(\frac{1}{9} - x\right)\right) \cdot 3 \]

       8. pow1/2 N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \left(\frac{1}{9} - x\right)\right) \cdot 3 \]

       9. flip3-- N/A

          \[\leadsto \left(\frac{1}{\sqrt{x}} \cdot \frac{{\frac{1}{9}}^{3} - {x}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(x \cdot x + \frac{1}{9} \cdot x\right)}\right) \cdot 3 \]

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

           \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\sqrt{x}} \cdot \frac{{\frac{1}{9}}^{3} - {x}^{3}}{\frac{1}{9} \cdot \frac{1}{9} + \left(x \cdot x + \frac{1}{9} \cdot x\right)}\right), \color{blue}{3}\right) \]

   12. Applied egg-rr 96.5%

       \[\leadsto \color{blue}{\frac{0.1111111111111111 - x}{\sqrt{x}} \cdot 3} \]

   if 3.2500000000000002e-9 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\color{blue}{y}, 1\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.5%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+43}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;3 \cdot \frac{0.1111111111111111 - x}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+44}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e+44)
   (* 3.0 (/ y (pow x -0.5)))
   (if (<= y 3.25e-9)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (* 3.0 (sqrt x)) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+44) {
		tmp = 3.0 * (y / pow(x, -0.5));
	} else if (y <= 3.25e-9) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.9d+44)) then
        tmp = 3.0d0 * (y / (x ** (-0.5d0)))
    else if (y <= 3.25d-9) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.9e+44) {
		tmp = 3.0 * (y / Math.pow(x, -0.5));
	} else if (y <= 3.25e-9) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e+44:
		tmp = 3.0 * (y / math.pow(x, -0.5))
	elif y <= 3.25e-9:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e+44)
		tmp = Float64(3.0 * Float64(y / (x ^ -0.5)));
	elseif (y <= 3.25e-9)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.9e+44)
		tmp = 3.0 * (y / (x ^ -0.5));
	elseif (y <= 3.25e-9)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e+44], N[(3.0 * N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.25e-9], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9000000000000002e44

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \sqrt{x}\right), 3\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{\sqrt{x}}\right)\right), 3\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), 3\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 3\right) \]

      8. metadata-eval 86.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), 3\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \cdot 3 \]

   if -2.9000000000000002e44 < y < 3.2500000000000002e-9

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{1}{3} \cdot \frac{1}{x} - 3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{x} - 3\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{\frac{1}{3} \cdot \frac{1}{x}} - 3\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3} \cdot \frac{1}{x} + -3\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{1}{x}\right), \color{blue}{-3}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3} \cdot 1}{x}\right), -3\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), -3\right)\right) \]

      8. /-lowering-/.f64 96.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), -3\right)\right) \]

   7. Simplified 96.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]

   if 3.2500000000000002e-9 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\color{blue}{y}, 1\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 86.5%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+44}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+44}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 7500000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2e+44)
   (* 3.0 (/ y (pow x -0.5)))
   (if (<= y 7500000000000.0)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* y (/ 3.0 (pow x -0.5))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2e+44) {
		tmp = 3.0 * (y / pow(x, -0.5));
	} else if (y <= 7500000000000.0) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (3.0 / pow(x, -0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2d+44)) then
        tmp = 3.0d0 * (y / (x ** (-0.5d0)))
    else if (y <= 7500000000000.0d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y * (3.0d0 / (x ** (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2e+44) {
		tmp = 3.0 * (y / Math.pow(x, -0.5));
	} else if (y <= 7500000000000.0) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y * (3.0 / Math.pow(x, -0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2e+44:
		tmp = 3.0 * (y / math.pow(x, -0.5))
	elif y <= 7500000000000.0:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y * (3.0 / math.pow(x, -0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2e+44)
		tmp = Float64(3.0 * Float64(y / (x ^ -0.5)));
	elseif (y <= 7500000000000.0)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y * Float64(3.0 / (x ^ -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2e+44)
		tmp = 3.0 * (y / (x ^ -0.5));
	elseif (y <= 7500000000000.0)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y * (3.0 / (x ^ -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2e+44], N[(3.0 * N[(y / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7500000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(3.0 / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000002e44

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \sqrt{x}\right), 3\right) \]

      2. remove-double-div N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y}{\frac{1}{\sqrt{x}}}\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{\sqrt{x}}\right)\right), 3\right) \]

      5. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), 3\right) \]

      6. pow-flip N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), 3\right) \]

      8. metadata-eval 86.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), 3\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{y}{{x}^{-0.5}}} \cdot 3 \]

   if -2.0000000000000002e44 < y < 7.5e12

   1. Initial program 99.4%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{1}{3} \cdot \frac{1}{x} - 3\right)} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(\frac{1}{3} \cdot \frac{1}{x} - 3\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{\frac{1}{3} \cdot \frac{1}{x}} - 3\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3} \cdot \frac{1}{x} + \color{blue}{\left(\mathsf{neg}\left(3\right)\right)}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\frac{1}{3} \cdot \frac{1}{x} + -3\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{1}{3} \cdot \frac{1}{x}\right), \color{blue}{-3}\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3} \cdot 1}{x}\right), -3\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(\frac{\frac{1}{3}}{x}\right), -3\right)\right) \]

      8. /-lowering-/.f64 95.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, x\right), -3\right)\right) \]

   7. Simplified 95.5%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]

   if 7.5e12 < y

   1. Initial program 99.5%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{x} \cdot y\right)}, 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), y\right), 3\right) \]

      2. sqrt-lowering-sqrt.f64 85.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), y\right), 3\right) \]

   7. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot y \]

      4. metadata-eval N/A

         \[\leadsto \left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right) \cdot y \]

      5. div-inv N/A

         \[\leadsto \frac{\sqrt{x}}{\frac{1}{3}} \cdot y \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{x}}{\frac{1}{3}}\right), \color{blue}{y}\right) \]

      7. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \frac{1}{\frac{1}{3}}\right), y\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot 3\right), y\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \sqrt{x}\right), y\right) \]

      10. remove-double-div N/A

          \[\leadsto \mathsf{*.f64}\left(\left(3 \cdot \frac{1}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{3}{\frac{1}{\sqrt{x}}}\right), y\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{\sqrt{x}}\right)\right), y\right) \]

      13. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left(\frac{1}{{x}^{\frac{1}{2}}}\right)\right), y\right) \]

      14. pow-flip N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \left({x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right), y\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right), y\right) \]

      16. metadata-eval 86.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(3, \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right), y\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{3}{{x}^{-0.5}} \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+44}:\\ \;\;\;\;3 \cdot \frac{y}{{x}^{-0.5}}\\ \mathbf{elif}\;y \leq 7500000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{3}{{x}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8000000:\\ \;\;\;\;\frac{3 \cdot \left(y + \frac{0.1111111111111111}{x}\right)}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 8000000.0)
   (/ (* 3.0 (+ y (/ 0.1111111111111111 x))) (pow x -0.5))
   (* (* 3.0 (sqrt x)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8000000.0) {
		tmp = (3.0 * (y + (0.1111111111111111 / x))) / pow(x, -0.5);
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 8000000.0d0) then
        tmp = (3.0d0 * (y + (0.1111111111111111d0 / x))) / (x ** (-0.5d0))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8000000.0) {
		tmp = (3.0 * (y + (0.1111111111111111 / x))) / Math.pow(x, -0.5);
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8000000.0:
		tmp = (3.0 * (y + (0.1111111111111111 / x))) / math.pow(x, -0.5)
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8000000.0)
		tmp = Float64(Float64(3.0 * Float64(y + Float64(0.1111111111111111 / x))) / (x ^ -0.5));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8000000.0)
		tmp = (3.0 * (y + (0.1111111111111111 / x))) / (x ^ -0.5);
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8000000.0], N[(N[(3.0 * N[(y + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8e6

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \left(\frac{\frac{1}{x}}{9}\right)\right), 1\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\left(\frac{1}{x}\right), 9\right)\right), 1\right)\right) \]

      3. /-lowering-/.f64 99.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\mathsf{+.f64}\left(y, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, x\right), 9\right)\right), 1\right)\right) \]

   4. Applied egg-rr 99.3%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \color{blue}{\frac{\frac{1}{x}}{9}}\right) - 1\right) \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot \color{blue}{\left(3 \cdot \sqrt{x}\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \color{blue}{\sqrt{x}} \]

      3. remove-double-div N/A

         \[\leadsto \left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right) \cdot \frac{1}{\color{blue}{\frac{1}{\sqrt{x}}}} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3}{\color{blue}{\frac{1}{\sqrt{x}}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right) \cdot 3\right), \color{blue}{\left(\frac{1}{\sqrt{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) - 1\right), 3\right), \left(\frac{\color{blue}{1}}{\sqrt{x}}\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(y + \frac{\frac{1}{x}}{9}\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{x}}{9} + y\right) + \left(\mathsf{neg}\left(1\right)\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{\frac{1}{x}}{9} + y\right) + -1\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      10. associate-+l+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\frac{1}{x}}{9} + \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{x}}{9}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      12. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      13. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \left(y + -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left(\frac{1}{\sqrt{x}}\right)\right) \]

      17. pow1/2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left(\frac{1}{{x}^{\color{blue}{\frac{1}{2}}}}\right)\right) \]

      18. pow-flip N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \left({x}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \mathsf{pow.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \]

      20. metadata-eval 99.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \mathsf{+.f64}\left(y, -1\right)\right), 3\right), \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \cdot 3}{{x}^{-0.5}}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), \color{blue}{y}\right), 3\right), \mathsf{pow.f64}\left(x, \frac{-1}{2}\right)\right) \]
   8. Step-by-step derivation

   9. Simplified 98.0%

      \[\leadsto \frac{\left(\frac{0.1111111111111111}{x} + \color{blue}{y}\right) \cdot 3}{{x}^{-0.5}} \]

   if 8e6 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(3, \mathsf{sqrt.f64}\left(x\right)\right), \mathsf{\_.f64}\left(\color{blue}{y}, 1\right)\right) \]
   4. Step-by-step derivation

   5. Simplified 99.5%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8000000:\\ \;\;\;\;\frac{3 \cdot \left(y + \frac{0.1111111111111111}{x}\right)}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (* (sqrt x) (+ y (+ (/ 0.1111111111111111 x) -1.0)))))

C

double code(double x, double y) {
	return 3.0 * (sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * (sqrt(x) * (y + ((0.1111111111111111d0 / x) + (-1.0d0))))
end function

Java

public static double code(double x, double y) {
	return 3.0 * (Math.sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0)));
}

Python

def code(x, y):
	return 3.0 * (math.sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0)))

Julia

function code(x, y)
	return Float64(3.0 * Float64(sqrt(x) * Float64(y + Float64(Float64(0.1111111111111111 / x) + -1.0))))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * (sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0)));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

   16. metadata-eval 99.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

5. Final simplification 99.5%

   \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \]
6. Add Preprocessing

Alternative 13: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (+ (* 3.0 y) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	return sqrt(x) * ((3.0 * y) + (-3.0 + (0.3333333333333333 / x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((3.0d0 * y) + ((-3.0d0) + (0.3333333333333333d0 / x)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * ((3.0 * y) + (-3.0 + (0.3333333333333333 / x)));
}

Python

def code(x, y):
	return math.sqrt(x) * ((3.0 * y) + (-3.0 + (0.3333333333333333 / x)))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(3.0 * y) + Float64(-3.0 + Float64(0.3333333333333333 / x))))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * ((3.0 * y) + (-3.0 + (0.3333333333333333 / x)));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

   2. associate-*l* N/A

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

   5. associate--l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

   6. distribute-lft-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

   11. sub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

   13. distribute-lft-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

   15. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

   18. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

   20. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

   21. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

   22. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 14: 61.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00185:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00185) (/ (pow x -0.5) 3.0) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00185) {
		tmp = pow(x, -0.5) / 3.0;
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00185d0) then
        tmp = (x ** (-0.5d0)) / 3.0d0
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00185) {
		tmp = Math.pow(x, -0.5) / 3.0;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00185:
		tmp = math.pow(x, -0.5) / 3.0
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00185)
		tmp = Float64((x ^ -0.5) / 3.0);
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00185)
		tmp = (x ^ -0.5) / 3.0;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00185], N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0018500000000000001

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{x}} \cdot \color{blue}{\frac{1}{3}} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \color{blue}{\frac{1}{3}}\right) \]

      3. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{x}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      4. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({x}^{-1}\right)}^{\frac{1}{2}}\right), \frac{1}{3}\right) \]

      5. pow-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({x}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \frac{1}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \left(-1 \cdot \frac{1}{2}\right)\right), \frac{1}{3}\right) \]

      7. metadata-eval 69.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), \frac{1}{3}\right) \]

   9. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{{x}^{-0.5} \cdot 0.3333333333333333} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto {x}^{\frac{-1}{2}} \cdot \frac{1}{\color{blue}{3}} \]

       2. div-inv N/A

          \[\leadsto \frac{{x}^{\frac{-1}{2}}}{\color{blue}{3}} \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left({x}^{\frac{-1}{2}}\right), \color{blue}{3}\right) \]

       4. pow-lowering-pow.f64 69.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(x, \frac{-1}{2}\right), 3\right) \]

   11. Applied egg-rr 69.1%

       \[\leadsto \color{blue}{\frac{{x}^{-0.5}}{3}} \]

   if 0.0018500000000000001 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 92.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 92.7%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 49.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 49.2%

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

Alternative 15: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00185:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00185) (/ 0.3333333333333333 (sqrt x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00185) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00185d0) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00185) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00185:
		tmp = 0.3333333333333333 / math.sqrt(x)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00185)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00185)
		tmp = 0.3333333333333333 / sqrt(x);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00185], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0018500000000000001

   1. Initial program 99.3%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} - 1\right) \]

      2. associate-*l* N/A

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{\left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\color{blue}{3} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)\right) \]

      5. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      6. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \color{blue}{3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(3 \cdot y + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \color{blue}{3}\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\left(3 \cdot y\right), \color{blue}{\left(\left(\frac{1}{x \cdot 9} - 1\right) \cdot 3\right)}\right)\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(\color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)} \cdot 3\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} - 1\right)}\right)\right)\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\frac{1}{x \cdot 9} + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\left(\mathsf{neg}\left(1\right)\right) + \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      13. distribute-lft-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \color{blue}{3 \cdot \frac{1}{x \cdot 9}}\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \left(3 \cdot \left(\mathsf{neg}\left(1\right)\right) + \frac{1}{x \cdot 9} \cdot \color{blue}{3}\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot \left(\mathsf{neg}\left(1\right)\right)\right), \color{blue}{\left(\frac{1}{x \cdot 9} \cdot 3\right)}\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(\left(3 \cdot -1\right), \left(\frac{1}{\color{blue}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      17. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\color{blue}{\frac{1}{x \cdot 9}} \cdot 3\right)\right)\right)\right) \]

      18. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \color{blue}{\frac{1}{x \cdot 9}}\right)\right)\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{1}{9 \cdot \color{blue}{x}}\right)\right)\right)\right) \]

      20. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(3 \cdot \frac{\frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      21. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \left(\frac{3 \cdot \frac{1}{9}}{\color{blue}{x}}\right)\right)\right)\right) \]

      22. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(3, y\right), \mathsf{+.f64}\left(-3, \mathsf{/.f64}\left(\left(3 \cdot \frac{1}{9}\right), \color{blue}{x}\right)\right)\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{3} \cdot \sqrt{\frac{1}{x}}} \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{\frac{1}{x}}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right)\right) \]

      3. /-lowering-/.f64 69.1%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right)\right) \]

   7. Simplified 69.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

         \[\leadsto \frac{1}{3} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{x}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{1}{3} \cdot \frac{1}{\sqrt{\color{blue}{x}}} \]

      3. un-div-inv N/A

         \[\leadsto \frac{\frac{1}{3}}{\color{blue}{\sqrt{x}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \color{blue}{\left(\sqrt{x}\right)}\right) \]

      5. sqrt-lowering-sqrt.f64 69.1%

         \[\leadsto \mathsf{/.f64}\left(\frac{1}{3}, \mathsf{sqrt.f64}\left(x\right)\right) \]

   9. Applied egg-rr 69.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if 0.0018500000000000001 < x

   1. Initial program 99.6%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

      6. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

      12. associate-/r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

      16. metadata-eval 99.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

   4. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
   6. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

      6. +-lowering-+.f64 92.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

   7. Simplified 92.7%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

      2. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

      10. remove-double-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

      13. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

      14. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

      15. associate-*l/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

      16. associate-/l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

      17. *-inverses N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

      18. *-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

      19. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

      21. distribute-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   10. Simplified 49.2%

       \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

   11. Taylor expanded in x around inf

       \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

       3. sqrt-lowering-sqrt.f64 49.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

   13. Simplified 49.2%

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

Alternative 16: 25.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sqrt x) -3.0))

C

double code(double x, double y) {
	return sqrt(x) * -3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (-3.0d0)
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * -3.0;
}

Python

def code(x, y):
	return math.sqrt(x) * -3.0

Julia

function code(x, y)
	return Float64(sqrt(x) * -3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * -3.0;
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto 3 \cdot \color{blue}{\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right) \cdot \color{blue}{3} \]

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

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), \color{blue}{3}\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{x}\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right), 3\right) \]

   6. associate--l+ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} - 1\right)\right)\right), 3\right) \]

   8. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 3\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \left(\frac{1}{x \cdot 9} + -1\right)\right)\right), 3\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{x \cdot 9}\right), -1\right)\right)\right), 3\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{1}{9 \cdot x}\right), -1\right)\right)\right), 3\right) \]

   12. associate-/r* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{\frac{1}{9}}{x}\right), -1\right)\right)\right), 3\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\left(\frac{{9}^{-1}}{x}\right), -1\right)\right)\right), 3\right) \]

   15. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\left({9}^{-1}\right), x\right), -1\right)\right)\right), 3\right) \]

   16. metadata-eval 99.5%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{+.f64}\left(y, \mathsf{+.f64}\left(\mathsf{/.f64}\left(\frac{1}{9}, x\right), -1\right)\right)\right), 3\right) \]

4. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \color{blue}{\left(\frac{\frac{1}{9} + x \cdot \left(y - 1\right)}{x}\right)}\right), 3\right) \]
6. Step-by-step derivation

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\left(\frac{1}{9} + x \cdot \left(y - 1\right)\right), x\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \left(x \cdot \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y - 1\right)\right)\right), x\right)\right), 3\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), x\right)\right), 3\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \left(y + -1\right)\right)\right), x\right)\right), 3\right) \]

   6. +-lowering-+.f64 95.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\frac{1}{9}, \mathsf{*.f64}\left(x, \mathsf{+.f64}\left(y, -1\right)\right)\right), x\right)\right), 3\right) \]

7. Simplified 95.8%

   \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111 + x \cdot \left(y + -1\right)}{x}}\right) \cdot 3 \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)}, 3\right) \]
9. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right), 3\right) \]

   2. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right), 3\right) \]

   3. associate-/l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right), 3\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\sqrt{\frac{1}{x}} \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right), 3\right) \]

   5. associate-*r/ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\left(\sqrt{\frac{1}{x}} \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right), 3\right) \]

   6. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}} \cdot \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{1}{x}}\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{1}{x}\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\frac{\frac{1}{9} + -1 \cdot x}{y} \cdot y\right)\right), 3\right) \]

   10. remove-double-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right)\right)\right) \cdot y\right)\right), 3\right) \]

   11. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\left(\mathsf{neg}\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right)\right) \cdot y\right)\right), 3\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right)\right), 3\right) \]

   13. neg-sub0 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \frac{\frac{1}{9} + -1 \cdot x}{y}\right) \cdot y\right)\right), 3\right) \]

   14. associate-*r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)}{y} \cdot y\right)\right), 3\right) \]

   15. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \frac{\left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot y}{y}\right)\right), 3\right) \]

   16. associate-/l* N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot \frac{y}{y}\right)\right), 3\right) \]

   17. *-inverses N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(-1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right) \cdot 1\right)\right), 3\right) \]

   18. *-rgt-identity N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - -1 \cdot \left(\frac{1}{9} + -1 \cdot x\right)\right)\right), 3\right) \]

   19. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(\frac{1}{9} + -1 \cdot x\right)\right)\right)\right)\right), 3\right) \]

   20. +-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\mathsf{neg}\left(\left(-1 \cdot x + \frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

   21. distribute-neg-in N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(1, x\right)\right), \left(0 - \left(\left(\mathsf{neg}\left(-1 \cdot x\right)\right) + \left(\mathsf{neg}\left(\frac{1}{9}\right)\right)\right)\right)\right), 3\right) \]

10. Simplified 59.3%

    \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{x}} \cdot \left(0.1111111111111111 - x\right)\right)} \cdot 3 \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \sqrt{x} \cdot \color{blue}{-3} \]

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

       \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{x}\right), \color{blue}{-3}\right) \]

    3. sqrt-lowering-sqrt.f64 26.6%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(x\right), -3\right) \]

13. Simplified 26.6%

    \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
14. Add Preprocessing

Developer Target 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(y \cdot \sqrt{x} + \left(\frac{1}{x \cdot 9} - 1\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x)))))

C

double code(double x, double y) {
	return 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 3.0d0 * ((y * sqrt(x)) + (((1.0d0 / (x * 9.0d0)) - 1.0d0) * sqrt(x)))
end function

Java

public static double code(double x, double y) {
	return 3.0 * ((y * Math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * Math.sqrt(x)));
}

Python

def code(x, y):
	return 3.0 * ((y * math.sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * math.sqrt(x)))

Julia

function code(x, y)
	return Float64(3.0 * Float64(Float64(y * sqrt(x)) + Float64(Float64(Float64(1.0 / Float64(x * 9.0)) - 1.0) * sqrt(x))))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * ((y * sqrt(x)) + (((1.0 / (x * 9.0)) - 1.0) * sqrt(x)));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.0 / N[(x * 9.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024152 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x)))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))