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

Percentage Accurate: 99.4% → 99.4%

Time: 11.9s

Alternatives: 13

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

\[\begin{array}{l} \\ \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3 \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (* (sqrt x) (+ y (+ (/ 0.1111111111111111 x) -1.0))) 3.0))

C

double code(double x, double y) {
	return (sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0))) * 3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (sqrt(x) * (y + ((0.1111111111111111d0 / x) + (-1.0d0)))) * 3.0d0
end function

Java

public static double code(double x, double y) {
	return (Math.sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0))) * 3.0;
}

Python

def code(x, y):
	return (math.sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0))) * 3.0

Julia

function code(x, y)
	return Float64(Float64(sqrt(x) * Float64(y + Float64(Float64(0.1111111111111111 / x) + -1.0))) * 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = (sqrt(x) * (y + ((0.1111111111111111 / x) + -1.0))) * 3.0;
end

Wolfram

code[x_, y_] := N[(N[(N[Sqrt[x], $MachinePrecision] * N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 3.0), $MachinePrecision]

Derivation

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 99.4%

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

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. Add Preprocessing

Alternative 2: 60.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-165}:\\ \;\;\;\;\left(x \cdot -3\right) \cdot {x}^{-0.5}\\ \mathbf{elif}\;y \leq -9.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))))
   (if (<= y -1000.0)
     t_0
     (if (<= y -9.5e-165)
       (* (* x -3.0) (pow x -0.5))
       (if (<= y -9.6e-226)
         (/ 0.3333333333333333 (sqrt x))
         (if (<= y 3.9e-140)
           (* (sqrt x) -3.0)
           (if (<= y 6.5e+37) (/ (sqrt x) (* x 3.0)) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double tmp;
	if (y <= -1000.0) {
		tmp = t_0;
	} else if (y <= -9.5e-165) {
		tmp = (x * -3.0) * pow(x, -0.5);
	} else if (y <= -9.6e-226) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else if (y <= 3.9e-140) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 6.5e+37) {
		tmp = sqrt(x) / (x * 3.0);
	} 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 = 3.0d0 * (sqrt(x) * y)
    if (y <= (-1000.0d0)) then
        tmp = t_0
    else if (y <= (-9.5d-165)) then
        tmp = (x * (-3.0d0)) * (x ** (-0.5d0))
    else if (y <= (-9.6d-226)) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else if (y <= 3.9d-140) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 6.5d+37) then
        tmp = sqrt(x) / (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double tmp;
	if (y <= -1000.0) {
		tmp = t_0;
	} else if (y <= -9.5e-165) {
		tmp = (x * -3.0) * Math.pow(x, -0.5);
	} else if (y <= -9.6e-226) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else if (y <= 3.9e-140) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 6.5e+37) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	tmp = 0
	if y <= -1000.0:
		tmp = t_0
	elif y <= -9.5e-165:
		tmp = (x * -3.0) * math.pow(x, -0.5)
	elif y <= -9.6e-226:
		tmp = 0.3333333333333333 / math.sqrt(x)
	elif y <= 3.9e-140:
		tmp = math.sqrt(x) * -3.0
	elif y <= 6.5e+37:
		tmp = math.sqrt(x) / (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -1000.0)
		tmp = t_0;
	elseif (y <= -9.5e-165)
		tmp = Float64(Float64(x * -3.0) * (x ^ -0.5));
	elseif (y <= -9.6e-226)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	elseif (y <= 3.9e-140)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 6.5e+37)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -1000.0)
		tmp = t_0;
	elseif (y <= -9.5e-165)
		tmp = (x * -3.0) * (x ^ -0.5);
	elseif (y <= -9.6e-226)
		tmp = 0.3333333333333333 / sqrt(x);
	elseif (y <= 3.9e-140)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 6.5e+37)
		tmp = sqrt(x) / (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1000.0], t$95$0, If[LessEqual[y, -9.5e-165], N[(N[(x * -3.0), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -9.6e-226], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e-140], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 6.5e+37], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1e3 or 6.4999999999999998e37 < y

   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 99.3%

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

   5. Taylor expanded in y around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 77.1%

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

   7. Simplified 77.1%

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

   if -1e3 < y < -9.49999999999999973e-165

   1. Initial program 99.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 95.6%

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

   5. Simplified 95.6%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 95.6%

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

   8. Simplified 95.6%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{0.3333333333333333 + x \cdot -3}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{3} + x \cdot -3}{x} \cdot \color{blue}{\sqrt{x}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow1/2 N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot \left({x}^{-1} \cdot {x}^{\color{blue}{\frac{1}{2}}}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\frac{-1}{2}} \]

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

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

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

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

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

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

      11. pow-lowering-pow.f64 95.8%

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

   10. Applied egg-rr 95.8%

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

   11. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. *-lowering-*.f64 56.7%

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

   13. Simplified 56.7%

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

   if -9.49999999999999973e-165 < y < -9.5999999999999998e-226

   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 99.6%

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

   5. Taylor expanded in 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 90.8%

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

   7. Simplified 90.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 90.8%

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

   9. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if -9.5999999999999998e-226 < y < 3.90000000000000019e-140

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   7. 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 62.8%

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

   8. Simplified 62.8%

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

   if 3.90000000000000019e-140 < y < 6.4999999999999998e37

   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 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 62.8%

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

   9. Simplified 62.8%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{x \cdot 3}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 60.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -1000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -8.4 \cdot 10^{-164}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-226}:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -1000.0)
     t_0
     (if (<= y -8.4e-164)
       t_1
       (if (<= y -2.8e-226)
         (/ 0.3333333333333333 (sqrt x))
         (if (<= y 5.1e-140)
           t_1
           (if (<= y 1.3e+37) (/ (sqrt x) (* x 3.0)) t_0)))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -1000.0) {
		tmp = t_0;
	} else if (y <= -8.4e-164) {
		tmp = t_1;
	} else if (y <= -2.8e-226) {
		tmp = 0.3333333333333333 / sqrt(x);
	} else if (y <= 5.1e-140) {
		tmp = t_1;
	} else if (y <= 1.3e+37) {
		tmp = sqrt(x) / (x * 3.0);
	} 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 = 3.0d0 * (sqrt(x) * y)
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-1000.0d0)) then
        tmp = t_0
    else if (y <= (-8.4d-164)) then
        tmp = t_1
    else if (y <= (-2.8d-226)) then
        tmp = 0.3333333333333333d0 / sqrt(x)
    else if (y <= 5.1d-140) then
        tmp = t_1
    else if (y <= 1.3d+37) then
        tmp = sqrt(x) / (x * 3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -1000.0) {
		tmp = t_0;
	} else if (y <= -8.4e-164) {
		tmp = t_1;
	} else if (y <= -2.8e-226) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else if (y <= 5.1e-140) {
		tmp = t_1;
	} else if (y <= 1.3e+37) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -1000.0:
		tmp = t_0
	elif y <= -8.4e-164:
		tmp = t_1
	elif y <= -2.8e-226:
		tmp = 0.3333333333333333 / math.sqrt(x)
	elif y <= 5.1e-140:
		tmp = t_1
	elif y <= 1.3e+37:
		tmp = math.sqrt(x) / (x * 3.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -1000.0)
		tmp = t_0;
	elseif (y <= -8.4e-164)
		tmp = t_1;
	elseif (y <= -2.8e-226)
		tmp = Float64(0.3333333333333333 / sqrt(x));
	elseif (y <= 5.1e-140)
		tmp = t_1;
	elseif (y <= 1.3e+37)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -1000.0)
		tmp = t_0;
	elseif (y <= -8.4e-164)
		tmp = t_1;
	elseif (y <= -2.8e-226)
		tmp = 0.3333333333333333 / sqrt(x);
	elseif (y <= 5.1e-140)
		tmp = t_1;
	elseif (y <= 1.3e+37)
		tmp = sqrt(x) / (x * 3.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -1000.0], t$95$0, If[LessEqual[y, -8.4e-164], t$95$1, If[LessEqual[y, -2.8e-226], N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.1e-140], t$95$1, If[LessEqual[y, 1.3e+37], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1e3 or 1.3e37 < y

   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 99.3%

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

   5. Taylor expanded in y around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 77.1%

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

   7. Simplified 77.1%

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

   if -1e3 < y < -8.3999999999999996e-164 or -2.80000000000000008e-226 < y < 5.1000000000000004e-140

   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. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   7. 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 60.1%

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

   8. Simplified 60.1%

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

   if -8.3999999999999996e-164 < y < -2.80000000000000008e-226

   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 99.6%

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

   5. Taylor expanded in 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 90.8%

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

   7. Simplified 90.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 90.8%

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

   9. Applied egg-rr 90.8%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if 5.1000000000000004e-140 < y < 1.3e37

   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 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 62.8%

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

   9. Simplified 62.8%

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

Alternative 4: 60.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333}{\sqrt{x}}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -1000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.32 \cdot 10^{-166}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-225}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 0.3333333333333333 (sqrt x)))
        (t_1 (* 3.0 (* (sqrt x) y)))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -1000.0)
     t_1
     (if (<= y -2.32e-166)
       t_2
       (if (<= y -5.2e-225)
         t_0
         (if (<= y 2.1e-140) t_2 (if (<= y 2.6e+37) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 / sqrt(x);
	double t_1 = 3.0 * (sqrt(x) * y);
	double t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -1000.0) {
		tmp = t_1;
	} else if (y <= -2.32e-166) {
		tmp = t_2;
	} else if (y <= -5.2e-225) {
		tmp = t_0;
	} else if (y <= 2.1e-140) {
		tmp = t_2;
	} else if (y <= 2.6e+37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 / sqrt(x)
    t_1 = 3.0d0 * (sqrt(x) * y)
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-1000.0d0)) then
        tmp = t_1
    else if (y <= (-2.32d-166)) then
        tmp = t_2
    else if (y <= (-5.2d-225)) then
        tmp = t_0
    else if (y <= 2.1d-140) then
        tmp = t_2
    else if (y <= 2.6d+37) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.3333333333333333 / Math.sqrt(x);
	double t_1 = 3.0 * (Math.sqrt(x) * y);
	double t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -1000.0) {
		tmp = t_1;
	} else if (y <= -2.32e-166) {
		tmp = t_2;
	} else if (y <= -5.2e-225) {
		tmp = t_0;
	} else if (y <= 2.1e-140) {
		tmp = t_2;
	} else if (y <= 2.6e+37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.3333333333333333 / math.sqrt(x)
	t_1 = 3.0 * (math.sqrt(x) * y)
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -1000.0:
		tmp = t_1
	elif y <= -2.32e-166:
		tmp = t_2
	elif y <= -5.2e-225:
		tmp = t_0
	elif y <= 2.1e-140:
		tmp = t_2
	elif y <= 2.6e+37:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 / sqrt(x))
	t_1 = Float64(3.0 * Float64(sqrt(x) * y))
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -1000.0)
		tmp = t_1;
	elseif (y <= -2.32e-166)
		tmp = t_2;
	elseif (y <= -5.2e-225)
		tmp = t_0;
	elseif (y <= 2.1e-140)
		tmp = t_2;
	elseif (y <= 2.6e+37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.3333333333333333 / sqrt(x);
	t_1 = 3.0 * (sqrt(x) * y);
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -1000.0)
		tmp = t_1;
	elseif (y <= -2.32e-166)
		tmp = t_2;
	elseif (y <= -5.2e-225)
		tmp = t_0;
	elseif (y <= 2.1e-140)
		tmp = t_2;
	elseif (y <= 2.6e+37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -1000.0], t$95$1, If[LessEqual[y, -2.32e-166], t$95$2, If[LessEqual[y, -5.2e-225], t$95$0, If[LessEqual[y, 2.1e-140], t$95$2, If[LessEqual[y, 2.6e+37], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1e3 or 2.5999999999999999e37 < y

   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 99.3%

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

   5. Taylor expanded in y around inf

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

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

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

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

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

      3. sqrt-lowering-sqrt.f64 77.1%

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

   7. Simplified 77.1%

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

   if -1e3 < y < -2.32000000000000002e-166 or -5.20000000000000027e-225 < y < 2.10000000000000017e-140

   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. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 97.9%

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

   5. Simplified 97.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   7. 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 60.1%

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

   8. Simplified 60.1%

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

   if -2.32000000000000002e-166 < y < -5.20000000000000027e-225 or 2.10000000000000017e-140 < y < 2.5999999999999999e37

   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 69.0%

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

   7. Simplified 69.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 69.2%

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

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{if}\;y \leq -700000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\left(0.3333333333333333 + x \cdot -3\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ y -1.0))))
   (if (<= y -700000.0)
     t_0
     (if (<= y 1.75e+28)
       (* (+ 0.3333333333333333 (* x -3.0)) (pow x -0.5))
       t_0))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * (y + -1.0);
	double tmp;
	if (y <= -700000.0) {
		tmp = t_0;
	} else if (y <= 1.75e+28) {
		tmp = (0.3333333333333333 + (x * -3.0)) * 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 + (-1.0d0))
    if (y <= (-700000.0d0)) then
        tmp = t_0
    else if (y <= 1.75d+28) then
        tmp = (0.3333333333333333d0 + (x * (-3.0d0))) * (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 + -1.0);
	double tmp;
	if (y <= -700000.0) {
		tmp = t_0;
	} else if (y <= 1.75e+28) {
		tmp = (0.3333333333333333 + (x * -3.0)) * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(x) * 3.0) * (y + -1.0)
	tmp = 0
	if y <= -700000.0:
		tmp = t_0
	elif y <= 1.75e+28:
		tmp = (0.3333333333333333 + (x * -3.0)) * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0))
	tmp = 0.0
	if (y <= -700000.0)
		tmp = t_0;
	elseif (y <= 1.75e+28)
		tmp = Float64(Float64(0.3333333333333333 + Float64(x * -3.0)) * (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 + -1.0);
	tmp = 0.0;
	if (y <= -700000.0)
		tmp = t_0;
	elseif (y <= 1.75e+28)
		tmp = (0.3333333333333333 + (x * -3.0)) * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -700000.0], t$95$0, If[LessEqual[y, 1.75e+28], N[(N[(0.3333333333333333 + N[(x * -3.0), $MachinePrecision]), $MachinePrecision] * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -7e5 or 1.75e28 < y

   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. 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 77.3%

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

   if -7e5 < y < 1.75e28

   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. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.5%

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

   8. Simplified 96.5%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{0.3333333333333333 + x \cdot -3}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{3} + x \cdot -3}{x} \cdot \color{blue}{\sqrt{x}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow1/2 N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot \left({x}^{-1} \cdot {x}^{\color{blue}{\frac{1}{2}}}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\frac{-1}{2}} \]

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

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

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

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

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

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

      11. pow-lowering-pow.f64 96.7%

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

   10. Applied egg-rr 96.7%

       \[\leadsto \color{blue}{\left(0.3333333333333333 + x \cdot -3\right) \cdot {x}^{-0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -700000:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+28}:\\ \;\;\;\;\left(0.3333333333333333 + x \cdot -3\right) \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{if}\;y \leq -1200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{0.3333333333333333 + x \cdot -3}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ y -1.0))))
   (if (<= y -1200000.0)
     t_0
     (if (<= y 4e+26) (/ (+ 0.3333333333333333 (* x -3.0)) (sqrt x)) t_0))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * (y + -1.0);
	double tmp;
	if (y <= -1200000.0) {
		tmp = t_0;
	} else if (y <= 4e+26) {
		tmp = (0.3333333333333333 + (x * -3.0)) / sqrt(x);
	} 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 + (-1.0d0))
    if (y <= (-1200000.0d0)) then
        tmp = t_0
    else if (y <= 4d+26) then
        tmp = (0.3333333333333333d0 + (x * (-3.0d0))) / sqrt(x)
    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 + -1.0);
	double tmp;
	if (y <= -1200000.0) {
		tmp = t_0;
	} else if (y <= 4e+26) {
		tmp = (0.3333333333333333 + (x * -3.0)) / Math.sqrt(x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(x) * 3.0) * (y + -1.0)
	tmp = 0
	if y <= -1200000.0:
		tmp = t_0
	elif y <= 4e+26:
		tmp = (0.3333333333333333 + (x * -3.0)) / math.sqrt(x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0))
	tmp = 0.0
	if (y <= -1200000.0)
		tmp = t_0;
	elseif (y <= 4e+26)
		tmp = Float64(Float64(0.3333333333333333 + Float64(x * -3.0)) / sqrt(x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sqrt(x) * 3.0) * (y + -1.0);
	tmp = 0.0;
	if (y <= -1200000.0)
		tmp = t_0;
	elseif (y <= 4e+26)
		tmp = (0.3333333333333333 + (x * -3.0)) / sqrt(x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1200000.0], t$95$0, If[LessEqual[y, 4e+26], N[(N[(0.3333333333333333 + N[(x * -3.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.2e6 or 4.00000000000000019e26 < y

   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. 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 77.3%

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

   if -1.2e6 < y < 4.00000000000000019e26

   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. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 96.6%

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

   5. Simplified 96.6%

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

   6. Taylor expanded in x around 0

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

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

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

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

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

      3. *-commutative N/A

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

      4. *-lowering-*.f64 96.5%

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

   8. Simplified 96.5%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{0.3333333333333333 + x \cdot -3}{x}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{3} + x \cdot -3}{x} \cdot \color{blue}{\sqrt{x}} \]

      2. div-inv N/A

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

      3. associate-*l* N/A

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

      4. inv-pow N/A

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

      5. pow1/2 N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot \left({x}^{-1} \cdot {x}^{\color{blue}{\frac{1}{2}}}\right) \]

      6. pow-prod-up N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\color{blue}{\left(-1 + \frac{1}{2}\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\frac{-1}{2}} \]

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

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

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

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

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

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

      11. pow-lowering-pow.f64 96.7%

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

   10. Applied egg-rr 96.7%

       \[\leadsto \color{blue}{\left(0.3333333333333333 + x \cdot -3\right) \cdot {x}^{-0.5}} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot {x}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

       2. pow-flip N/A

          \[\leadsto \left(\frac{1}{3} + x \cdot -3\right) \cdot \frac{1}{\color{blue}{{x}^{\frac{1}{2}}}} \]

       3. pow1/2 N/A

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

       4. un-div-inv N/A

          \[\leadsto \frac{\frac{1}{3} + x \cdot -3}{\color{blue}{\sqrt{x}}} \]

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

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

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

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

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

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

       8. sqrt-lowering-sqrt.f64 96.6%

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

   12. Applied egg-rr 96.6%

       \[\leadsto \color{blue}{\frac{0.3333333333333333 + x \cdot -3}{\sqrt{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1200000:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;\frac{0.3333333333333333 + x \cdot -3}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{if}\;y \leq -5400:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* (sqrt x) 3.0) (+ y -1.0))))
   (if (<= y -5400.0)
     t_0
     (if (<= y 4.2e+31) (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x))) t_0))))

C

double code(double x, double y) {
	double t_0 = (sqrt(x) * 3.0) * (y + -1.0);
	double tmp;
	if (y <= -5400.0) {
		tmp = t_0;
	} else if (y <= 4.2e+31) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} 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 + (-1.0d0))
    if (y <= (-5400.0d0)) then
        tmp = t_0
    else if (y <= 4.2d+31) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    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 + -1.0);
	double tmp;
	if (y <= -5400.0) {
		tmp = t_0;
	} else if (y <= 4.2e+31) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (math.sqrt(x) * 3.0) * (y + -1.0)
	tmp = 0
	if y <= -5400.0:
		tmp = t_0
	elif y <= 4.2e+31:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0))
	tmp = 0.0
	if (y <= -5400.0)
		tmp = t_0;
	elseif (y <= 4.2e+31)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (sqrt(x) * 3.0) * (y + -1.0);
	tmp = 0.0;
	if (y <= -5400.0)
		tmp = t_0;
	elseif (y <= 4.2e+31)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5400.0], t$95$0, If[LessEqual[y, 4.2e+31], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5400 or 4.19999999999999958e31 < y

   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. 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 77.3%

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

   if -5400 < y < 4.19999999999999958e31

   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 96.6%

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

   7. Simplified 96.6%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5400:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+31}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -490000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{0.3333333333333333}{\sqrt{x}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -490000.0)
   (* (sqrt x) (* 3.0 (+ y -1.0)))
   (if (<= y 1.05e+27)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (/ y (/ 0.3333333333333333 (sqrt x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -490000.0) {
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	} else if (y <= 1.05e+27) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y / (0.3333333333333333 / sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-490000.0d0)) then
        tmp = sqrt(x) * (3.0d0 * (y + (-1.0d0)))
    else if (y <= 1.05d+27) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = y / (0.3333333333333333d0 / sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -490000.0) {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	} else if (y <= 1.05e+27) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = y / (0.3333333333333333 / Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -490000.0:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	elif y <= 1.05e+27:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = y / (0.3333333333333333 / math.sqrt(x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -490000.0)
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	elseif (y <= 1.05e+27)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(y / Float64(0.3333333333333333 / sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -490000.0)
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	elseif (y <= 1.05e+27)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = y / (0.3333333333333333 / sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -490000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+27], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.9e5

   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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 76.1%

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

   7. Simplified 76.1%

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

   if -4.9e5 < y < 1.04999999999999997e27

   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 96.6%

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

   7. Simplified 96.6%

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

   if 1.04999999999999997e27 < y

   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 99.3%

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

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

      1. /-lowering-/.f64 78.6%

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

   9. Simplified 78.6%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{0.3333333333333333}{y}}} \]
   10. Step-by-step derivation

       1. associate-/r/ N/A

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

       2. *-commutative N/A

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

       3. clear-num N/A

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

       4. un-div-inv N/A

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

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

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

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

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

       7. sqrt-lowering-sqrt.f64 78.7%

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

   11. Applied egg-rr 78.7%

       \[\leadsto \color{blue}{\frac{y}{\frac{0.3333333333333333}{\sqrt{x}}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -490000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{0.3333333333333333}{\sqrt{x}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (* (sqrt x) (+ (* y 3.0) (/ 0.3333333333333333 x)))
   (* (* (sqrt x) 3.0) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt(x) * ((y * 3.0) + (0.3333333333333333 / x));
	} else {
		tmp = (sqrt(x) * 3.0) * (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 <= 0.112d0) then
        tmp = sqrt(x) * ((y * 3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = (sqrt(x) * 3.0d0) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = Math.sqrt(x) * ((y * 3.0) + (0.3333333333333333 / x));
	} else {
		tmp = (Math.sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt(x) * ((y * 3.0) + (0.3333333333333333 / x))
	else:
		tmp = (math.sqrt(x) * 3.0) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = sqrt(x) * ((y * 3.0) + (0.3333333333333333 / x));
	else
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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

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

      1. /-lowering-/.f64 98.1%

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

   7. Simplified 98.1%

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

   if 0.112000000000000002 < x

   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 98.8%

      \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \left(\color{blue}{y} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{x}}{x \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.6e-19) (/ (sqrt x) (* x 3.0)) (* (sqrt x) (* 3.0 (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.6e-19) {
		tmp = sqrt(x) / (x * 3.0);
	} else {
		tmp = sqrt(x) * (3.0 * (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 <= 2.6d-19) then
        tmp = sqrt(x) / (x * 3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.6e-19) {
		tmp = Math.sqrt(x) / (x * 3.0);
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.6e-19:
		tmp = math.sqrt(x) / (x * 3.0)
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.6e-19)
		tmp = Float64(sqrt(x) / Float64(x * 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.6e-19)
		tmp = sqrt(x) / (x * 3.0);
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.6e-19], N[(N[Sqrt[x], $MachinePrecision] / N[(x * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000013e-19

   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. 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 72.7%

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

   9. Simplified 72.7%

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

   if 2.60000000000000013e-19 < x

   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 inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

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

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 96.6%

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

   7. Simplified 96.6%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(y \cdot 3 + \left(-3 + \frac{0.3333333333333333}{x}\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (+ (* y 3.0) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	return sqrt(x) * ((y * 3.0) + (-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) * ((y * 3.0d0) + ((-3.0d0) + (0.3333333333333333d0 / x)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * ((y * 3.0) + (-3.0 + (0.3333333333333333 / x)));
}

Python

def code(x, y):
	return math.sqrt(x) * ((y * 3.0) + (-3.0 + (0.3333333333333333 / x)))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(y * 3.0) + Float64(-3.0 + Float64(0.3333333333333333 / x))))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * ((y * 3.0) + (-3.0 + (0.3333333333333333 / x)));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] + N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 99.3%

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

5. Final simplification 99.3%

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

Alternative 12: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (/ 0.3333333333333333 (sqrt x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		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 <= 0.112d0) 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 <= 0.112) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		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 <= 0.112)
		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 <= 0.112)
		tmp = 0.3333333333333333 / sqrt(x);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], 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 < 0.112000000000000002

   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 70.4%

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

   7. Simplified 70.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 70.5%

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

   9. Applied egg-rr 70.5%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if 0.112000000000000002 < x

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

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

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

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

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

      6. sub-neg N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 52.5%

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

   5. Simplified 52.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   7. 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 52.0%

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

   8. Simplified 52.0%

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

Alternative 13: 25.5% 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 99.4%

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

3. Taylor expanded in y around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

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

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

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

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

   6. sub-neg N/A

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

   7. metadata-eval N/A

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

   8. +-commutative N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

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

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

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

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

   13. associate-*r/ N/A

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

   14. metadata-eval N/A

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

   15. /-lowering-/.f64 62.0%

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

5. Simplified 62.0%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
7. 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 27.1%

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

8. Simplified 27.1%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
9. 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 2024161 
(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)))