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

Percentage Accurate: 99.4% → 99.4%

Time: 11.7s

Alternatives: 12

Speedup: N/A×

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

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((3.0d0 * y) + ((-3.0d0) + (0.3333333333333333d0 / x)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. associate--l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. *-commutative N/A

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. *-commutative N/A

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

   20. associate-/r* N/A

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

   21. associate-*r/ N/A

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

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

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

3. Simplified 99.5%

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

Alternative 2: 61.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-15}:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+265}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.12e-15)
   (/ (pow x -0.5) 3.0)
   (if (<= x 8e+149)
     (* (sqrt x) (* 3.0 y))
     (if (<= x 6.2e+205)
       (* (sqrt x) -3.0)
       (if (<= x 2.3e+265)
         (* 3.0 (* (sqrt x) y))
         (/ (sqrt x) -0.3333333333333333))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.12e-15) {
		tmp = pow(x, -0.5) / 3.0;
	} else if (x <= 8e+149) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (x <= 6.2e+205) {
		tmp = sqrt(x) * -3.0;
	} else if (x <= 2.3e+265) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) / -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.12d-15) then
        tmp = (x ** (-0.5d0)) / 3.0d0
    else if (x <= 8d+149) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (x <= 6.2d+205) then
        tmp = sqrt(x) * (-3.0d0)
    else if (x <= 2.3d+265) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) / (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.12e-15) {
		tmp = Math.pow(x, -0.5) / 3.0;
	} else if (x <= 8e+149) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (x <= 6.2e+205) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (x <= 2.3e+265) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) / -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.12e-15:
		tmp = math.pow(x, -0.5) / 3.0
	elif x <= 8e+149:
		tmp = math.sqrt(x) * (3.0 * y)
	elif x <= 6.2e+205:
		tmp = math.sqrt(x) * -3.0
	elif x <= 2.3e+265:
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) / -0.3333333333333333
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.12e-15)
		tmp = Float64((x ^ -0.5) / 3.0);
	elseif (x <= 8e+149)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (x <= 6.2e+205)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (x <= 2.3e+265)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) / -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.12e-15)
		tmp = (x ^ -0.5) / 3.0;
	elseif (x <= 8e+149)
		tmp = sqrt(x) * (3.0 * y);
	elseif (x <= 6.2e+205)
		tmp = sqrt(x) * -3.0;
	elseif (x <= 2.3e+265)
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) / -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.12e-15], N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[x, 8e+149], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e+205], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[x, 2.3e+265], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] / -0.3333333333333333), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.1200000000000001e-15

   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 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 81.1%

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

   7. Simplified 81.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. metadata-eval N/A

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

       2. associate-/r* N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

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

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

       6. pow1/2 N/A

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

       7. pow-flip N/A

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

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

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

       9. metadata-eval 81.3%

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

   11. Applied egg-rr 81.3%

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

   if 1.1200000000000001e-15 < x < 8.00000000000000039e149

   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 99.5%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 57.9%

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

   7. Simplified 57.9%

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

   if 8.00000000000000039e149 < x < 6.20000000000000035e205

   1. Initial program 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.9%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   9. 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 72.5%

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

   10. Simplified 72.5%

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

   if 6.20000000000000035e205 < x < 2.3e265

   1. Initial program 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

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

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

      2. sqrt-lowering-sqrt.f64 67.6%

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

   7. Simplified 67.6%

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

   if 2.3e265 < 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. 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 99.7%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 99.6%

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

   7. Taylor expanded in y around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 65.6%

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

   9. Simplified 65.6%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{1}{\frac{0.3333333333333333}{x} + -3}}} \]

   10. Taylor expanded in x around inf

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

   12. Simplified 65.6%

       \[\leadsto \frac{\sqrt{x}}{\color{blue}{-0.3333333333333333}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.12 \cdot 10^{-15}:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{+149}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+205}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{+265}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{-0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;x \leq 5.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.65 \cdot 10^{+196}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+266}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{-0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (* 3.0 y))))
   (if (<= x 5.7e-11)
     (/ (pow x -0.5) 3.0)
     (if (<= x 1.85e+149)
       t_0
       (if (<= x 1.65e+196)
         (* (sqrt x) -3.0)
         (if (<= x 3.2e+266) t_0 (/ (sqrt x) -0.3333333333333333)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (3.0 * y);
	double tmp;
	if (x <= 5.7e-11) {
		tmp = pow(x, -0.5) / 3.0;
	} else if (x <= 1.85e+149) {
		tmp = t_0;
	} else if (x <= 1.65e+196) {
		tmp = sqrt(x) * -3.0;
	} else if (x <= 3.2e+266) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) / -0.3333333333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (3.0d0 * y)
    if (x <= 5.7d-11) then
        tmp = (x ** (-0.5d0)) / 3.0d0
    else if (x <= 1.85d+149) then
        tmp = t_0
    else if (x <= 1.65d+196) then
        tmp = sqrt(x) * (-3.0d0)
    else if (x <= 3.2d+266) then
        tmp = t_0
    else
        tmp = sqrt(x) / (-0.3333333333333333d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (3.0 * y);
	double tmp;
	if (x <= 5.7e-11) {
		tmp = Math.pow(x, -0.5) / 3.0;
	} else if (x <= 1.85e+149) {
		tmp = t_0;
	} else if (x <= 1.65e+196) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (x <= 3.2e+266) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) / -0.3333333333333333;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (3.0 * y)
	tmp = 0
	if x <= 5.7e-11:
		tmp = math.pow(x, -0.5) / 3.0
	elif x <= 1.85e+149:
		tmp = t_0
	elif x <= 1.65e+196:
		tmp = math.sqrt(x) * -3.0
	elif x <= 3.2e+266:
		tmp = t_0
	else:
		tmp = math.sqrt(x) / -0.3333333333333333
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(3.0 * y))
	tmp = 0.0
	if (x <= 5.7e-11)
		tmp = Float64((x ^ -0.5) / 3.0);
	elseif (x <= 1.85e+149)
		tmp = t_0;
	elseif (x <= 1.65e+196)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (x <= 3.2e+266)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) / -0.3333333333333333);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (3.0 * y);
	tmp = 0.0;
	if (x <= 5.7e-11)
		tmp = (x ^ -0.5) / 3.0;
	elseif (x <= 1.85e+149)
		tmp = t_0;
	elseif (x <= 1.65e+196)
		tmp = sqrt(x) * -3.0;
	elseif (x <= 3.2e+266)
		tmp = t_0;
	else
		tmp = sqrt(x) / -0.3333333333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5.7e-11], N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision], If[LessEqual[x, 1.85e+149], t$95$0, If[LessEqual[x, 1.65e+196], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[x, 3.2e+266], t$95$0, N[(N[Sqrt[x], $MachinePrecision] / -0.3333333333333333), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 5.6999999999999997e-11

   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 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 81.1%

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

   7. Simplified 81.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. metadata-eval N/A

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

       2. associate-/r* N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

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

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

       6. pow1/2 N/A

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

       7. pow-flip N/A

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

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

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

       9. metadata-eval 81.3%

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

   11. Applied egg-rr 81.3%

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

   if 5.6999999999999997e-11 < x < 1.84999999999999989e149 or 1.6500000000000001e196 < x < 3.20000000000000021e266

   1. Initial program 99.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. associate-*r/ N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. *-lowering-*.f64 60.8%

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

   7. Simplified 60.8%

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

   if 1.84999999999999989e149 < x < 1.6500000000000001e196

   1. Initial program 99.7%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   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 99.9%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 72.5%

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

   7. Simplified 72.5%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   9. 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 72.5%

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

   10. Simplified 72.5%

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

   if 3.20000000000000021e266 < 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. 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 99.7%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]
   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 99.6%

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

   7. Taylor expanded in y around 0

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

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

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

      2. sub-neg N/A

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

      3. metadata-eval N/A

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

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

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. /-lowering-/.f64 65.6%

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

   9. Simplified 65.6%

      \[\leadsto \frac{\sqrt{x}}{\color{blue}{\frac{1}{\frac{0.3333333333333333}{x} + -3}}} \]

   10. Taylor expanded in x around inf

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

   12. Simplified 65.6%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\frac{y + \frac{0.1111111111111111}{x}}{\frac{0.3333333333333333}{\sqrt{x}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (/ (+ y (/ 0.1111111111111111 x)) (/ 0.3333333333333333 (sqrt x)))
   (* (sqrt x) (+ (* 3.0 y) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (y + (0.1111111111111111 / x)) / (0.3333333333333333 / sqrt(x));
	} else {
		tmp = sqrt(x) * ((3.0 * y) + -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 <= 0.11d0) then
        tmp = (y + (0.1111111111111111d0 / x)) / (0.3333333333333333d0 / sqrt(x))
    else
        tmp = sqrt(x) * ((3.0d0 * y) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = (y + (0.1111111111111111 / x)) / (0.3333333333333333 / Math.sqrt(x));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = (y + (0.1111111111111111 / x)) / (0.3333333333333333 / math.sqrt(x))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(Float64(y + Float64(0.1111111111111111 / x)) / Float64(0.3333333333333333 / sqrt(x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = (y + (0.1111111111111111 / x)) / (0.3333333333333333 / sqrt(x));
	else
		tmp = sqrt(x) * ((3.0 * y) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(N[(y + N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision] / N[(0.3333333333333333 / N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   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 99.3%

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

   4. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 3} \]
   5. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. remove-double-div N/A

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

      4. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. sqrt-lowering-sqrt.f64 99.1%

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

   6. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{1}{y + \left(\frac{0.1111111111111111}{x} + -1\right)}}{\sqrt{x} \cdot 3}}} \]
   7. Step-by-step derivation

      1. div-inv N/A

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

      2. associate-/r* N/A

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

      3. remove-double-div N/A

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

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

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

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

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

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

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

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

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

      8. *-commutative N/A

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

      9. associate-/r* N/A

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

      10. metadata-eval N/A

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

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

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

      12. sqrt-lowering-sqrt.f64 99.4%

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

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{y + \left(\frac{0.1111111111111111}{x} + -1\right)}{\frac{0.3333333333333333}{\sqrt{x}}}} \]

   9. Taylor expanded in x around 0

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

       1. /-lowering-/.f64 97.2%

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

   11. Simplified 97.2%

       \[\leadsto \frac{y + \color{blue}{\frac{0.1111111111111111}{x}}}{\frac{0.3333333333333333}{\sqrt{x}}} \]

   if 0.110000000000000001 < 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. 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 inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.038:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + 3 \cdot \frac{0.1111111111111111}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.038)
   (* (sqrt x) (+ -3.0 (* 3.0 (/ 0.1111111111111111 x))))
   (* (sqrt x) (+ (* 3.0 y) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.038) {
		tmp = sqrt(x) * (-3.0 + (3.0 * (0.1111111111111111 / x)));
	} else {
		tmp = sqrt(x) * ((3.0 * y) + -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 <= 0.038d0) then
        tmp = sqrt(x) * ((-3.0d0) + (3.0d0 * (0.1111111111111111d0 / x)))
    else
        tmp = sqrt(x) * ((3.0d0 * y) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.038) {
		tmp = Math.sqrt(x) * (-3.0 + (3.0 * (0.1111111111111111 / x)));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.038:
		tmp = math.sqrt(x) * (-3.0 + (3.0 * (0.1111111111111111 / x)))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.038)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(3.0 * Float64(0.1111111111111111 / x))));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.038)
		tmp = sqrt(x) * (-3.0 + (3.0 * (0.1111111111111111 / x)));
	else
		tmp = sqrt(x) * ((3.0 * y) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.038], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(3.0 * N[(0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0379999999999999991

   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-lft-in N/A

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

      10. metadata-eval N/A

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

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

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

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

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. /-lowering-/.f64 79.7%

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

   5. Simplified 79.7%

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

   if 0.0379999999999999991 < 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. 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 inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;\frac{-3 + \frac{0.3333333333333333}{x}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.42)
   (/ (+ -3.0 (/ 0.3333333333333333 x)) (pow x -0.5))
   (* (sqrt x) (+ (* 3.0 y) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.42) {
		tmp = (-3.0 + (0.3333333333333333 / x)) / pow(x, -0.5);
	} else {
		tmp = sqrt(x) * ((3.0 * y) + -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 <= 0.42d0) then
        tmp = ((-3.0d0) + (0.3333333333333333d0 / x)) / (x ** (-0.5d0))
    else
        tmp = sqrt(x) * ((3.0d0 * y) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.42) {
		tmp = (-3.0 + (0.3333333333333333 / x)) / Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.42:
		tmp = (-3.0 + (0.3333333333333333 / x)) / math.pow(x, -0.5)
	else:
		tmp = math.sqrt(x) * ((3.0 * y) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.42)
		tmp = Float64(Float64(-3.0 + Float64(0.3333333333333333 / x)) / (x ^ -0.5));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.42)
		tmp = (-3.0 + (0.3333333333333333 / x)) / (x ^ -0.5);
	else
		tmp = sqrt(x) * ((3.0 * y) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.42], N[(N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision] / N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.419999999999999984

   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 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 79.6%

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

   7. Simplified 79.6%

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

      1. *-commutative N/A

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

      2. remove-double-div N/A

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

      3. un-div-inv N/A

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

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

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

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

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

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

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

      7. pow1/2 N/A

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

      8. pow-flip N/A

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

      9. metadata-eval N/A

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

      10. pow-lowering-pow.f64 79.7%

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

   9. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{\frac{\frac{0.3333333333333333}{x} + -3}{{x}^{-0.5}}} \]

   if 0.419999999999999984 < 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. 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 inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.42:\\ \;\;\;\;\frac{-3 + \frac{0.3333333333333333}{x}}{{x}^{-0.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.16:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.16)
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
   (* (sqrt x) (+ (* 3.0 y) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.16) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * ((3.0 * y) + -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 <= 0.16d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * ((3.0d0 * y) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.16) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.16:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.16)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.16)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * ((3.0 * y) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.16], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.160000000000000003

   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 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 79.6%

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

   7. Simplified 79.6%

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

   if 0.160000000000000003 < 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. 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 inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 98.7%

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

   7. Simplified 98.7%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.16:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e-11) (/ (pow x -0.5) 3.0) (* (sqrt x) (+ (* 3.0 y) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-11) {
		tmp = pow(x, -0.5) / 3.0;
	} else {
		tmp = sqrt(x) * ((3.0 * y) + -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 <= 3.8d-11) then
        tmp = (x ** (-0.5d0)) / 3.0d0
    else
        tmp = sqrt(x) * ((3.0d0 * y) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-11) {
		tmp = Math.pow(x, -0.5) / 3.0;
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.8e-11:
		tmp = math.pow(x, -0.5) / 3.0
	else:
		tmp = math.sqrt(x) * ((3.0 * y) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.8e-11)
		tmp = Float64((x ^ -0.5) / 3.0);
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.8e-11)
		tmp = (x ^ -0.5) / 3.0;
	else
		tmp = sqrt(x) * ((3.0 * y) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.8e-11], N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999998e-11

   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 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 81.1%

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

   7. Simplified 81.1%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. un-div-inv N/A

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

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

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

      5. sqrt-lowering-sqrt.f64 81.1%

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

   9. Applied egg-rr 81.1%

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

       1. metadata-eval N/A

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

       2. associate-/r* N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

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

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

       6. pow1/2 N/A

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

       7. pow-flip N/A

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

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

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

       9. metadata-eval 81.3%

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

   11. Applied egg-rr 81.3%

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

   if 3.7999999999999998e-11 < 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. 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 inf

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

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

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

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

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

      3. sub-neg N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-lowering-*.f64 96.0%

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

   7. Simplified 96.0%

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

Alternative 9: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;\frac{{x}^{-0.5}}{3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2600.0) (/ (pow x -0.5) 3.0) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2600.0) {
		tmp = pow(x, -0.5) / 3.0;
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2600.0d0) then
        tmp = (x ** (-0.5d0)) / 3.0d0
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2600.0) {
		tmp = Math.pow(x, -0.5) / 3.0;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2600.0:
		tmp = math.pow(x, -0.5) / 3.0
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2600.0)
		tmp = Float64((x ^ -0.5) / 3.0);
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2600.0)
		tmp = (x ^ -0.5) / 3.0;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2600.0], N[(N[Power[x, -0.5], $MachinePrecision] / 3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2600

   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 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 76.2%

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

   7. Simplified 76.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 76.3%

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

   9. Applied egg-rr 76.3%

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

       1. metadata-eval N/A

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

       2. associate-/r* N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

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

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

       6. pow1/2 N/A

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

       7. pow-flip N/A

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

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

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

       9. metadata-eval 76.4%

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

   11. Applied egg-rr 76.4%

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

   if 2600 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. associate-*r/ N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 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 47.9%

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

   7. Simplified 47.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   9. 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 46.9%

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

   10. Simplified 46.9%

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

Alternative 10: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2600.0) (* 0.3333333333333333 (pow x -0.5)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2600.0) {
		tmp = 0.3333333333333333 * pow(x, -0.5);
	} 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 <= 2600.0d0) then
        tmp = 0.3333333333333333d0 * (x ** (-0.5d0))
    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 <= 2600.0) {
		tmp = 0.3333333333333333 * Math.pow(x, -0.5);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2600.0:
		tmp = 0.3333333333333333 * math.pow(x, -0.5)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2600.0)
		tmp = Float64(0.3333333333333333 * (x ^ -0.5));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2600.0)
		tmp = 0.3333333333333333 * (x ^ -0.5);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2600.0], N[(0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2600

   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 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 76.2%

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

   7. Simplified 76.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

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

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

      3. pow1/2 N/A

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

      4. inv-pow N/A

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

      5. pow-pow N/A

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

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

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

      7. metadata-eval 76.3%

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

   9. Applied egg-rr 76.3%

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

   if 2600 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. associate-*r/ N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 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 47.9%

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

   7. Simplified 47.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   9. 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 46.9%

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

   10. Simplified 46.9%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2600:\\ \;\;\;\;\frac{0.3333333333333333}{\sqrt{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2600.0) (/ 0.3333333333333333 (sqrt x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2600.0) {
		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 <= 2600.0d0) 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 <= 2600.0) {
		tmp = 0.3333333333333333 / Math.sqrt(x);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2600.0:
		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 <= 2600.0)
		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 <= 2600.0)
		tmp = 0.3333333333333333 / sqrt(x);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2600.0], 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 < 2600

   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 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 76.2%

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

   7. Simplified 76.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \sqrt{\frac{1}{x}}} \]
   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 76.3%

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

   9. Applied egg-rr 76.3%

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

   if 2600 < 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. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

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

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

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

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

      5. associate--l+ N/A

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

      6. distribute-lft-in N/A

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

      7. *-commutative N/A

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

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

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

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

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

      10. *-commutative N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. distribute-lft-in N/A

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

      14. *-commutative N/A

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

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

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. *-commutative N/A

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

      19. *-commutative N/A

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

      20. associate-/r* N/A

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

      21. associate-*r/ N/A

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

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

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 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 47.9%

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

   7. Simplified 47.9%

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

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   9. 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 46.9%

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

   10. Simplified 46.9%

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

Alternative 12: 25.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot -3 \end{array} \]

FPCore

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

C

double code(double x, double y) {
	return sqrt(x) * -3.0;
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

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

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

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

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

   5. associate--l+ N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

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

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

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

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

   10. *-commutative N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. distribute-lft-in N/A

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

   14. *-commutative N/A

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

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

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

   19. *-commutative N/A

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

   20. associate-/r* N/A

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

   21. associate-*r/ N/A

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

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

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 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 61.9%

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

7. Simplified 61.9%

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

8. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

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

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

   3. sqrt-lowering-sqrt.f64 26.1%

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

10. Simplified 26.1%

    \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
11. 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 2024160 
(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)))