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

Percentage Accurate: 99.4% → 99.4%

Time: 10.5s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt (* x 9.0)) (+ (/ 0.1111111111111111 x) (+ y -1.0))))

C

double code(double x, double y) {
	return sqrt((x * 9.0)) * ((0.1111111111111111 / x) + (y + -1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0)) * ((0.1111111111111111d0 / x) + (y + (-1.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0)) * ((0.1111111111111111 / x) + (y + -1.0));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0)) * ((0.1111111111111111 / x) + (y + -1.0))

Julia

function code(x, y)
	return Float64(sqrt(Float64(x * 9.0)) * Float64(Float64(0.1111111111111111 / x) + Float64(y + -1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0)) * ((0.1111111111111111 / x) + (y + -1.0));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(N[(0.1111111111111111 / x), $MachinePrecision] + N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.5%

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

   11. distribute-neg-frac 99.5%

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

   12. *-commutative 99.5%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
6. Step-by-step derivation

   1. unpow2 99.1%

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

   2. add-sqr-sqrt 99.5%

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

   3. expm1-log1p-u 96.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   4. expm1-udef 55.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. add-sqr-sqrt 55.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{3 \cdot \sqrt{x}} \cdot \sqrt{3 \cdot \sqrt{x}}}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. sqrt-unprod 55.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(3 \cdot \sqrt{x}\right) \cdot \left(3 \cdot \sqrt{x}\right)}}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   7. *-commutative 55.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(3 \cdot \sqrt{x}\right)}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   8. *-commutative 55.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\left(\sqrt{x} \cdot 3\right) \cdot \color{blue}{\left(\sqrt{x} \cdot 3\right)}}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   9. swap-sqr 55.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(3 \cdot 3\right)}}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   10. add-sqr-sqrt 55.3%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{x} \cdot \left(3 \cdot 3\right)}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   11. metadata-eval 55.3%

       \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x \cdot \color{blue}{9}}\right)} - 1\right) \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

7. Applied egg-rr 55.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]
8. Step-by-step derivation

   1. expm1-def 96.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   2. expm1-log1p 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

9. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

10. Final simplification 99.6%

    \[\leadsto \sqrt{x \cdot 9} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

Alternative 2: 61.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ t_1 := \sqrt{x} \cdot -3\\ t_2 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-272}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-194}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* y (sqrt x))))
        (t_1 (* (sqrt x) -3.0))
        (t_2 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -0.0005)
     t_0
     (if (<= y -1.6e-172)
       t_1
       (if (<= y -8.2e-187)
         (sqrt (+ (/ 0.1111111111111111 x) -2.0))
         (if (<= y -1.45e-218)
           t_1
           (if (<= y -1.02e-272)
             t_2
             (if (<= y 1.6e-280)
               t_1
               (if (<= y 1.9e-194) t_2 (if (<= y 2.1e-6) t_1 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (y * sqrt(x));
	double t_1 = sqrt(x) * -3.0;
	double t_2 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -0.0005) {
		tmp = t_0;
	} else if (y <= -1.6e-172) {
		tmp = t_1;
	} else if (y <= -8.2e-187) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (y <= -1.45e-218) {
		tmp = t_1;
	} else if (y <= -1.02e-272) {
		tmp = t_2;
	} else if (y <= 1.6e-280) {
		tmp = t_1;
	} else if (y <= 1.9e-194) {
		tmp = t_2;
	} else if (y <= 2.1e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 3.0d0 * (y * sqrt(x))
    t_1 = sqrt(x) * (-3.0d0)
    t_2 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-0.0005d0)) then
        tmp = t_0
    else if (y <= (-1.6d-172)) then
        tmp = t_1
    else if (y <= (-8.2d-187)) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else if (y <= (-1.45d-218)) then
        tmp = t_1
    else if (y <= (-1.02d-272)) then
        tmp = t_2
    else if (y <= 1.6d-280) then
        tmp = t_1
    else if (y <= 1.9d-194) then
        tmp = t_2
    else if (y <= 2.1d-6) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (y * Math.sqrt(x));
	double t_1 = Math.sqrt(x) * -3.0;
	double t_2 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -0.0005) {
		tmp = t_0;
	} else if (y <= -1.6e-172) {
		tmp = t_1;
	} else if (y <= -8.2e-187) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (y <= -1.45e-218) {
		tmp = t_1;
	} else if (y <= -1.02e-272) {
		tmp = t_2;
	} else if (y <= 1.6e-280) {
		tmp = t_1;
	} else if (y <= 1.9e-194) {
		tmp = t_2;
	} else if (y <= 2.1e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (y * math.sqrt(x))
	t_1 = math.sqrt(x) * -3.0
	t_2 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -0.0005:
		tmp = t_0
	elif y <= -1.6e-172:
		tmp = t_1
	elif y <= -8.2e-187:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	elif y <= -1.45e-218:
		tmp = t_1
	elif y <= -1.02e-272:
		tmp = t_2
	elif y <= 1.6e-280:
		tmp = t_1
	elif y <= 1.9e-194:
		tmp = t_2
	elif y <= 2.1e-6:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(y * sqrt(x)))
	t_1 = Float64(sqrt(x) * -3.0)
	t_2 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -0.0005)
		tmp = t_0;
	elseif (y <= -1.6e-172)
		tmp = t_1;
	elseif (y <= -8.2e-187)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	elseif (y <= -1.45e-218)
		tmp = t_1;
	elseif (y <= -1.02e-272)
		tmp = t_2;
	elseif (y <= 1.6e-280)
		tmp = t_1;
	elseif (y <= 1.9e-194)
		tmp = t_2;
	elseif (y <= 2.1e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (y * sqrt(x));
	t_1 = sqrt(x) * -3.0;
	t_2 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -0.0005)
		tmp = t_0;
	elseif (y <= -1.6e-172)
		tmp = t_1;
	elseif (y <= -8.2e-187)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	elseif (y <= -1.45e-218)
		tmp = t_1;
	elseif (y <= -1.02e-272)
		tmp = t_2;
	elseif (y <= 1.6e-280)
		tmp = t_1;
	elseif (y <= 1.9e-194)
		tmp = t_2;
	elseif (y <= 2.1e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -0.0005], t$95$0, If[LessEqual[y, -1.6e-172], t$95$1, If[LessEqual[y, -8.2e-187], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, -1.45e-218], t$95$1, If[LessEqual[y, -1.02e-272], t$95$2, If[LessEqual[y, 1.6e-280], t$95$1, If[LessEqual[y, 1.9e-194], t$95$2, If[LessEqual[y, 2.1e-6], t$95$1, t$95$0]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.0000000000000001e-4 or 2.0999999999999998e-6 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 75.9%

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

      1. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   if -5.0000000000000001e-4 < y < -1.6000000000000001e-172 or -8.2000000000000004e-187 < y < -1.4500000000000001e-218 or -1.01999999999999993e-272 < y < 1.6e-280 or 1.9000000000000001e-194 < y < 2.0999999999999998e-6

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 97.1%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      2. *-commutative 97.1%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right)\right)} \]

      3. unpow2 97.1%

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

      4. rem-square-sqrt 97.9%

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

      5. sub-neg 97.9%

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

      6. associate-*r/ 97.9%

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

      7. metadata-eval 97.9%

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

      8. metadata-eval 97.9%

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

      9. +-commutative 97.9%

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

   8. Simplified 97.9%

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

   9. Taylor expanded in x around inf 67.8%

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

   if -1.6000000000000001e-172 < y < -8.2000000000000004e-187

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate--l+ 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. remove-double-neg 99.1%

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

      5. distribute-lft-neg-in 99.1%

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

      6. distribute-rgt-neg-in 99.1%

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

      7. mul-1-neg 99.1%

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

      8. metadata-eval 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/r* 98.7%

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

      11. distribute-neg-frac 98.7%

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

      12. *-commutative 98.7%

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

      13. associate-/r/ 99.0%

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

      14. associate-/l/ 99.0%

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

      15. associate-/r/ 98.7%

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

   3. Simplified 98.7%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 98.0%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 98.3%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 98.3%

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

      4. rem-square-sqrt 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*l* 99.5%

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

      9. sub-neg 99.5%

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

      10. associate-*r/ 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.5%

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

      3. swap-sqr 99.3%

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

      4. metadata-eval 99.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 99.3%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 99.3%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 52.1%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 51.8%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 51.8%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 51.8%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 51.8%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 52.1%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 52.1%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 99.3%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 99.3%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 99.3%

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

       6. swap-sqr 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

       9. associate-*r* 99.2%

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

       10. metadata-eval 99.2%

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

       11. associate-*r/ 99.5%

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

       12. metadata-eval 99.5%

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

       13. sub-neg 99.5%

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

       14. *-commutative 99.5%

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

       15. *-commutative 99.5%

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

       16. associate-*r* 99.5%

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

   12. Simplified 51.8%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 100.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]
   14. Step-by-step derivation

       1. sub-neg 100.0%

          \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} + \left(-2\right)}} \]

       2. associate-*r/ 100.0%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} + \left(-2\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111}}{x} + \left(-2\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \sqrt{\frac{0.1111111111111111}{x} + \color{blue}{-2}} \]

   15. Simplified 100.0%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} + -2}} \]

   if -1.4500000000000001e-218 < y < -1.01999999999999993e-272 or 1.6e-280 < y < 1.9000000000000001e-194

   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 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.2%

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

      14. associate-/l/ 99.2%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.1%

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

      2. pow2 99.1%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 98.4%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 98.4%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 98.4%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 98.4%

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

      4. rem-square-sqrt 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.2%

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

      7. *-commutative 99.2%

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

      8. associate-*l* 99.3%

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

      9. sub-neg 99.3%

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

      10. associate-*r/ 99.4%

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

      11. metadata-eval 99.4%

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

      12. metadata-eval 99.4%

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

      13. +-commutative 99.4%

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

   8. Simplified 99.4%

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

      1. add-sqr-sqrt 74.2%

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

      2. sqrt-unprod 74.8%

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

      3. swap-sqr 74.7%

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

      4. metadata-eval 74.7%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 74.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 74.7%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 30.1%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 30.2%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 30.2%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 30.2%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 30.2%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 30.1%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 30.1%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 74.7%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 74.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 74.7%

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

       6. swap-sqr 74.8%

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

       7. *-commutative 74.8%

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

       8. *-commutative 74.8%

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

       9. associate-*r* 74.7%

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

       10. metadata-eval 74.7%

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

       11. associate-*r/ 74.6%

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

       12. metadata-eval 74.6%

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

       13. sub-neg 74.6%

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

       14. *-commutative 74.6%

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

       15. *-commutative 74.6%

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

       16. associate-*r* 74.7%

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

   12. Simplified 30.2%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 75.0%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{-172}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-187}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-280}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-194}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 3: 61.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-173}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-218}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* y (sqrt x)))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -0.0005)
     t_0
     (if (<= y -4.6e-173)
       t_1
       (if (<= y -1.2e-188)
         (sqrt (+ (/ 0.1111111111111111 x) -2.0))
         (if (<= y -3.5e-218)
           t_1
           (if (<= y -2.6e-261)
             (* (sqrt x) (/ 0.3333333333333333 x))
             (if (<= y 6.2e-281)
               t_1
               (if (<= y 5.5e-193)
                 (sqrt (/ 0.1111111111111111 x))
                 (if (<= y 2.1e-6) t_1 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (y * sqrt(x));
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0005) {
		tmp = t_0;
	} else if (y <= -4.6e-173) {
		tmp = t_1;
	} else if (y <= -1.2e-188) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (y <= -3.5e-218) {
		tmp = t_1;
	} else if (y <= -2.6e-261) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (y <= 6.2e-281) {
		tmp = t_1;
	} else if (y <= 5.5e-193) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (y <= 2.1e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 3.0d0 * (y * sqrt(x))
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-0.0005d0)) then
        tmp = t_0
    else if (y <= (-4.6d-173)) then
        tmp = t_1
    else if (y <= (-1.2d-188)) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else if (y <= (-3.5d-218)) then
        tmp = t_1
    else if (y <= (-2.6d-261)) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (y <= 6.2d-281) then
        tmp = t_1
    else if (y <= 5.5d-193) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (y <= 2.1d-6) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (y * Math.sqrt(x));
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -0.0005) {
		tmp = t_0;
	} else if (y <= -4.6e-173) {
		tmp = t_1;
	} else if (y <= -1.2e-188) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (y <= -3.5e-218) {
		tmp = t_1;
	} else if (y <= -2.6e-261) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (y <= 6.2e-281) {
		tmp = t_1;
	} else if (y <= 5.5e-193) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (y <= 2.1e-6) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (y * math.sqrt(x))
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -0.0005:
		tmp = t_0
	elif y <= -4.6e-173:
		tmp = t_1
	elif y <= -1.2e-188:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	elif y <= -3.5e-218:
		tmp = t_1
	elif y <= -2.6e-261:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif y <= 6.2e-281:
		tmp = t_1
	elif y <= 5.5e-193:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif y <= 2.1e-6:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(y * sqrt(x)))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -0.0005)
		tmp = t_0;
	elseif (y <= -4.6e-173)
		tmp = t_1;
	elseif (y <= -1.2e-188)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	elseif (y <= -3.5e-218)
		tmp = t_1;
	elseif (y <= -2.6e-261)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (y <= 6.2e-281)
		tmp = t_1;
	elseif (y <= 5.5e-193)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (y <= 2.1e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (y * sqrt(x));
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -0.0005)
		tmp = t_0;
	elseif (y <= -4.6e-173)
		tmp = t_1;
	elseif (y <= -1.2e-188)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	elseif (y <= -3.5e-218)
		tmp = t_1;
	elseif (y <= -2.6e-261)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (y <= 6.2e-281)
		tmp = t_1;
	elseif (y <= 5.5e-193)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (y <= 2.1e-6)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -0.0005], t$95$0, If[LessEqual[y, -4.6e-173], t$95$1, If[LessEqual[y, -1.2e-188], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, -3.5e-218], t$95$1, If[LessEqual[y, -2.6e-261], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.2e-281], t$95$1, If[LessEqual[y, 5.5e-193], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 2.1e-6], t$95$1, t$95$0]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.0000000000000001e-4 or 2.0999999999999998e-6 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 75.9%

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

      1. *-commutative 75.9%

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

   6. Simplified 75.9%

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

   if -5.0000000000000001e-4 < y < -4.59999999999999976e-173 or -1.2e-188 < y < -3.5e-218 or -2.6000000000000001e-261 < y < 6.2000000000000004e-281 or 5.50000000000000014e-193 < y < 2.0999999999999998e-6

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 97.2%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. *-commutative 97.2%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      2. *-commutative 97.2%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right)\right)} \]

      3. unpow2 97.2%

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

      4. rem-square-sqrt 98.0%

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

      5. sub-neg 98.0%

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

      6. associate-*r/ 98.0%

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

      7. metadata-eval 98.0%

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

      8. metadata-eval 98.0%

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

      9. +-commutative 98.0%

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

   8. Simplified 98.0%

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

   9. Taylor expanded in x around inf 67.0%

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

   if -4.59999999999999976e-173 < y < -1.2e-188

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate--l+ 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. remove-double-neg 99.1%

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

      5. distribute-lft-neg-in 99.1%

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

      6. distribute-rgt-neg-in 99.1%

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

      7. mul-1-neg 99.1%

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

      8. metadata-eval 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-/r* 98.7%

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

      11. distribute-neg-frac 98.7%

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

      12. *-commutative 98.7%

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

      13. associate-/r/ 99.0%

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

      14. associate-/l/ 99.0%

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

      15. associate-/r/ 98.7%

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

   3. Simplified 98.7%

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

      1. add-sqr-sqrt 99.0%

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

      2. pow2 99.0%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 98.0%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 98.3%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 98.3%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 98.3%

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

      4. rem-square-sqrt 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-*r* 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*l* 99.5%

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

      9. sub-neg 99.5%

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

      10. associate-*r/ 99.5%

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

      11. metadata-eval 99.5%

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

      12. metadata-eval 99.5%

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

      13. +-commutative 99.5%

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

   8. Simplified 99.5%

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

      1. add-sqr-sqrt 99.2%

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

      2. sqrt-unprod 99.5%

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

      3. swap-sqr 99.3%

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

      4. metadata-eval 99.3%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 99.3%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 99.3%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 52.1%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 51.8%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 51.8%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 51.8%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 51.8%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 52.1%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 52.1%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 99.3%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 99.3%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 99.3%

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

       6. swap-sqr 99.5%

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

       7. *-commutative 99.5%

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

       8. *-commutative 99.5%

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

       9. associate-*r* 99.2%

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

       10. metadata-eval 99.2%

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

       11. associate-*r/ 99.5%

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

       12. metadata-eval 99.5%

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

       13. sub-neg 99.5%

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

       14. *-commutative 99.5%

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

       15. *-commutative 99.5%

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

       16. associate-*r* 99.5%

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

   12. Simplified 51.8%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 100.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]
   14. Step-by-step derivation

       1. sub-neg 100.0%

          \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} + \left(-2\right)}} \]

       2. associate-*r/ 100.0%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} + \left(-2\right)} \]

       3. metadata-eval 100.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111}}{x} + \left(-2\right)} \]

       4. metadata-eval 100.0%

          \[\leadsto \sqrt{\frac{0.1111111111111111}{x} + \color{blue}{-2}} \]

   15. Simplified 100.0%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} + -2}} \]

   if -3.5e-218 < y < -2.6000000000000001e-261

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. associate--l+ 99.1%

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

      3. distribute-rgt-in 99.1%

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

      4. +-commutative 99.1%

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

      5. fma-def 99.1%

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

      6. remove-double-neg 99.1%

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

      7. fma-neg 99.1%

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

      8. associate-*r* 99.1%

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

      9. *-commutative 99.1%

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

      10. associate-*r* 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r* 99.1%

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

      14. associate-/r/ 99.6%

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

      15. mul-1-neg 99.6%

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

      16. metadata-eval 99.6%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 99.8%

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

   if 6.2000000000000004e-281 < y < 5.50000000000000014e-193

   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 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.1%

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

      14. associate-/l/ 99.1%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 98.5%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 98.4%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 98.4%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 98.4%

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

      4. rem-square-sqrt 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.1%

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

      7. *-commutative 99.1%

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

      8. associate-*l* 99.1%

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

      9. sub-neg 99.1%

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

      10. associate-*r/ 99.2%

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

      11. metadata-eval 99.2%

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

      12. metadata-eval 99.2%

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

      13. +-commutative 99.2%

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

   8. Simplified 99.2%

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

      1. add-sqr-sqrt 70.6%

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

      2. sqrt-unprod 71.1%

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

      3. swap-sqr 71.1%

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

      4. metadata-eval 71.1%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 71.1%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 71.1%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 30.5%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 30.5%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 30.5%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 30.5%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 30.5%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 30.5%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 30.5%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 71.1%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 71.1%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 71.1%

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

       6. swap-sqr 71.1%

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

       7. *-commutative 71.1%

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

       8. *-commutative 71.1%

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

       9. associate-*r* 71.2%

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

       10. metadata-eval 71.2%

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

       11. associate-*r/ 71.1%

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

       12. metadata-eval 71.1%

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

       13. sub-neg 71.1%

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

       14. *-commutative 71.1%

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

       15. *-commutative 71.1%

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

       16. associate-*r* 71.2%

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

   12. Simplified 30.4%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 71.5%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.0005:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -4.6 \cdot 10^{-173}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-188}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{-261}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-281}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-193}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]

Alternative 4: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot 3\\ \mathbf{if}\;y \leq -220000000000 \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;\left(y + -1\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) 3.0)))
   (if (or (<= y -220000000000.0) (not (<= y 17000000000.0)))
     (* (+ y -1.0) t_0)
     (* t_0 (+ (/ 0.1111111111111111 x) -1.0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * 3.0;
	double tmp;
	if ((y <= -220000000000.0) || !(y <= 17000000000.0)) {
		tmp = (y + -1.0) * t_0;
	} else {
		tmp = t_0 * ((0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * 3.0d0
    if ((y <= (-220000000000.0d0)) .or. (.not. (y <= 17000000000.0d0))) then
        tmp = (y + (-1.0d0)) * t_0
    else
        tmp = t_0 * ((0.1111111111111111d0 / x) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * 3.0;
	double tmp;
	if ((y <= -220000000000.0) || !(y <= 17000000000.0)) {
		tmp = (y + -1.0) * t_0;
	} else {
		tmp = t_0 * ((0.1111111111111111 / x) + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * 3.0
	tmp = 0
	if (y <= -220000000000.0) or not (y <= 17000000000.0):
		tmp = (y + -1.0) * t_0
	else:
		tmp = t_0 * ((0.1111111111111111 / x) + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * 3.0)
	tmp = 0.0
	if ((y <= -220000000000.0) || !(y <= 17000000000.0))
		tmp = Float64(Float64(y + -1.0) * t_0);
	else
		tmp = Float64(t_0 * Float64(Float64(0.1111111111111111 / x) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * 3.0;
	tmp = 0.0;
	if ((y <= -220000000000.0) || ~((y <= 17000000000.0)))
		tmp = (y + -1.0) * t_0;
	else
		tmp = t_0 * ((0.1111111111111111 / x) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]}, If[Or[LessEqual[y, -220000000000.0], N[Not[LessEqual[y, 17000000000.0]], $MachinePrecision]], N[(N[(y + -1.0), $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.2e11 or 1.7e10 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 79.4%

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

   if -2.2e11 < y < 1.7e10

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. *-commutative 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

      6. associate-*l* 98.5%

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

   6. Simplified 98.5%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -220000000000 \lor \neg \left(y \leq 17000000000\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\\ \end{array} \]

Alternative 5: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 160000000000\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -33000000000.0) (not (<= y 160000000000.0)))
   (* 3.0 (* y (sqrt x)))
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 160000000000.0)) {
		tmp = 3.0 * (y * sqrt(x));
	} else {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-33000000000.0d0)) .or. (.not. (y <= 160000000000.0d0))) then
        tmp = 3.0d0 * (y * sqrt(x))
    else
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -33000000000.0) || !(y <= 160000000000.0)) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -33000000000.0) or not (y <= 160000000000.0):
		tmp = 3.0 * (y * math.sqrt(x))
	else:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -33000000000.0) || !(y <= 160000000000.0))
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -33000000000.0) || ~((y <= 160000000000.0)))
		tmp = 3.0 * (y * sqrt(x));
	else
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -33000000000.0], N[Not[LessEqual[y, 160000000000.0]], $MachinePrecision]], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e10 or 1.6e11 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 78.8%

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

      1. *-commutative 78.8%

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

   6. Simplified 78.8%

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

   if -3.3e10 < y < 1.6e11

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. associate-*l/ 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. +-commutative 98.5%

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

      11. *-commutative 98.5%

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

      12. +-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -33000000000 \lor \neg \left(y \leq 160000000000\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 6: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -950000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -950000000.0) (not (<= y 200000000.0)))
   (* (+ y -1.0) (* (sqrt x) 3.0))
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -950000000.0) || !(y <= 200000000.0)) {
		tmp = (y + -1.0) * (sqrt(x) * 3.0);
	} else {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-950000000.0d0)) .or. (.not. (y <= 200000000.0d0))) then
        tmp = (y + (-1.0d0)) * (sqrt(x) * 3.0d0)
    else
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -950000000.0) || !(y <= 200000000.0)) {
		tmp = (y + -1.0) * (Math.sqrt(x) * 3.0);
	} else {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -950000000.0) or not (y <= 200000000.0):
		tmp = (y + -1.0) * (math.sqrt(x) * 3.0)
	else:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -950000000.0) || !(y <= 200000000.0))
		tmp = Float64(Float64(y + -1.0) * Float64(sqrt(x) * 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -950000000.0) || ~((y <= 200000000.0)))
		tmp = (y + -1.0) * (sqrt(x) * 3.0);
	else
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -950000000.0], N[Not[LessEqual[y, 200000000.0]], $MachinePrecision]], N[(N[(y + -1.0), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.5e8 or 2e8 < y

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 79.4%

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

   if -9.5e8 < y < 2e8

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. associate-*l/ 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. +-commutative 98.5%

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

      11. *-commutative 98.5%

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

      12. +-commutative 98.5%

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

   6. Simplified 98.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -950000000 \lor \neg \left(y \leq 200000000\right):\\ \;\;\;\;\left(y + -1\right) \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 7: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -12600000000:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 72000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -12600000000.0)
   (* 3.0 (* y (sqrt x)))
   (if (<= y 72000000000.0)
     (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
     (* (sqrt x) (- (* y 3.0) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -12600000000.0) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (y <= 72000000000.0) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * ((y * 3.0) - 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 (y <= (-12600000000.0d0)) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (y <= 72000000000.0d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * ((y * 3.0d0) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -12600000000.0) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (y <= 72000000000.0) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * ((y * 3.0) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -12600000000.0:
		tmp = 3.0 * (y * math.sqrt(x))
	elif y <= 72000000000.0:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * ((y * 3.0) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -12600000000.0)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (y <= 72000000000.0)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(y * 3.0) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -12600000000.0)
		tmp = 3.0 * (y * sqrt(x));
	elseif (y <= 72000000000.0)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * ((y * 3.0) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -12600000000.0], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 72000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.26e10

   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 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.4%

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

      14. associate-/l/ 99.4%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 74.2%

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

      1. *-commutative 74.2%

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

   6. Simplified 74.2%

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

   if -1.26e10 < y < 7.2e10

   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 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-rgt-in 99.4%

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

      4. remove-double-neg 99.4%

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

      5. distribute-lft-neg-in 99.4%

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

      6. distribute-rgt-neg-in 99.4%

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

      7. mul-1-neg 99.4%

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

      8. metadata-eval 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.3%

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

      14. associate-/l/ 99.3%

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

      15. associate-/r/ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 98.4%

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

      1. associate-*r* 98.4%

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

      2. sub-neg 98.4%

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

      3. associate-*r/ 98.4%

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

      4. metadata-eval 98.4%

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

      5. metadata-eval 98.4%

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

      6. distribute-rgt-in 98.4%

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

      7. associate-*l/ 98.5%

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

      8. metadata-eval 98.5%

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

      9. metadata-eval 98.5%

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

      10. +-commutative 98.5%

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

      11. *-commutative 98.5%

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

      12. +-commutative 98.5%

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

   6. Simplified 98.5%

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

   if 7.2e10 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. +-commutative 99.5%

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

      5. fma-def 99.5%

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

      6. remove-double-neg 99.5%

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

      7. fma-neg 99.5%

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

      8. associate-*r* 99.4%

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

      9. *-commutative 99.4%

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

      10. associate-*r* 99.3%

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

      11. distribute-rgt-neg-in 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r* 99.4%

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

      14. associate-/r/ 99.4%

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

      15. mul-1-neg 99.4%

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

      16. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 85.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -12600000000:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq 72000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3 - 3\right)\\ \end{array} \]

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.0%

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

   10. *-commutative 99.0%

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

   11. distribute-rgt-out-- 99.0%

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

   12. distribute-lft-neg-in 99.0%

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

   13. cancel-sign-sub 99.0%

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

   14. +-commutative 99.0%

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

   15. *-commutative 99.0%

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

   16. distribute-rgt-in 99.0%

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

3. Simplified 99.0%

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

   1. fma-udef 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 9: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e-11)
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-11) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} 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 <= 3.8d-11) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    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 <= 3.8e-11) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.8e-11:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.8e-11)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.8e-11)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.8e-11], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999998e-11

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r/ 99.2%

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

      14. associate-/l/ 99.2%

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

      15. associate-/r/ 99.3%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 72.9%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 73.0%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 73.0%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 73.0%

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

      4. rem-square-sqrt 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

      7. *-commutative 73.7%

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

      8. associate-*l* 73.7%

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

      9. sub-neg 73.7%

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

      10. associate-*r/ 73.8%

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

      11. metadata-eval 73.8%

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

      12. metadata-eval 73.8%

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

      13. +-commutative 73.8%

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

   8. Simplified 73.8%

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

      1. add-sqr-sqrt 73.5%

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

      2. sqrt-unprod 73.8%

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

      3. swap-sqr 73.7%

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

      4. metadata-eval 73.7%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 73.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 73.7%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 33.7%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 33.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 33.7%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 33.7%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 33.7%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 33.7%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 33.7%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 73.7%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 73.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 73.7%

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

       6. swap-sqr 73.8%

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

       7. *-commutative 73.8%

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

       8. *-commutative 73.8%

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

       9. associate-*r* 73.7%

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

       10. metadata-eval 73.7%

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

       11. associate-*r/ 73.7%

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

       12. metadata-eval 73.7%

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

       13. sub-neg 73.7%

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

       14. *-commutative 73.7%

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

       15. *-commutative 73.7%

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

       16. associate-*r* 73.7%

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

   12. Simplified 33.7%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 74.0%

       \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} - 2}} \]
   14. Step-by-step derivation

       1. sub-neg 74.0%

          \[\leadsto \sqrt{\color{blue}{0.1111111111111111 \cdot \frac{1}{x} + \left(-2\right)}} \]

       2. associate-*r/ 74.0%

          \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111 \cdot 1}{x}} + \left(-2\right)} \]

       3. metadata-eval 74.0%

          \[\leadsto \sqrt{\frac{\color{blue}{0.1111111111111111}}{x} + \left(-2\right)} \]

       4. metadata-eval 74.0%

          \[\leadsto \sqrt{\frac{0.1111111111111111}{x} + \color{blue}{-2}} \]

   15. Simplified 74.0%

       \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x} + -2}} \]

   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 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 48.9%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. *-commutative 48.9%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      2. *-commutative 48.9%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right)\right)} \]

      3. unpow2 48.9%

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

      4. rem-square-sqrt 49.3%

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

      5. sub-neg 49.3%

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

      6. associate-*r/ 49.3%

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

      7. metadata-eval 49.3%

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

      8. metadata-eval 49.3%

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

      9. +-commutative 49.3%

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

   8. Simplified 49.3%

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

   9. Taylor expanded in x around inf 47.6%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e-11) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-11) {
		tmp = sqrt((0.1111111111111111 / 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 <= 3.8d-11) then
        tmp = sqrt((0.1111111111111111d0 / 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 <= 3.8e-11) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.8e-11:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.8e-11)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.8e-11)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.8e-11], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.7999999999999998e-11

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-rgt-in 99.3%

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

      4. remove-double-neg 99.3%

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

      5. distribute-lft-neg-in 99.3%

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

      6. distribute-rgt-neg-in 99.3%

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

      7. mul-1-neg 99.3%

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

      8. metadata-eval 99.3%

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

      9. *-commutative 99.3%

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

      10. associate-/r* 99.3%

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

      11. distribute-neg-frac 99.3%

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

      12. *-commutative 99.3%

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

      13. associate-/r/ 99.2%

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

      14. associate-/l/ 99.2%

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

      15. associate-/r/ 99.3%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 98.9%

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

      2. pow2 98.9%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 72.9%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. associate-*l* 73.0%

         \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

      2. *-commutative 73.0%

         \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      3. unpow2 73.0%

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

      4. rem-square-sqrt 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*r* 73.7%

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

      7. *-commutative 73.7%

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

      8. associate-*l* 73.7%

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

      9. sub-neg 73.7%

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

      10. associate-*r/ 73.8%

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

      11. metadata-eval 73.8%

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

      12. metadata-eval 73.8%

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

      13. +-commutative 73.8%

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

   8. Simplified 73.8%

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

      1. add-sqr-sqrt 73.5%

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

      2. sqrt-unprod 73.8%

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

      3. swap-sqr 73.7%

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

      4. metadata-eval 73.7%

         \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      5. *-commutative 73.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

      6. *-commutative 73.7%

         \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

      7. swap-sqr 33.7%

         \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

      8. add-sqr-sqrt 33.7%

         \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

      9. pow2 33.7%

         \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

      10. +-commutative 33.7%

          \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

   10. Applied egg-rr 33.7%

       \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 33.7%

          \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

       2. unpow2 33.7%

          \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

       3. swap-sqr 73.7%

          \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

       4. *-commutative 73.7%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

       5. metadata-eval 73.7%

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

       6. swap-sqr 73.8%

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

       7. *-commutative 73.8%

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

       8. *-commutative 73.8%

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

       9. associate-*r* 73.7%

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

       10. metadata-eval 73.7%

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

       11. associate-*r/ 73.7%

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

       12. metadata-eval 73.7%

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

       13. sub-neg 73.7%

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

       14. *-commutative 73.7%

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

       15. *-commutative 73.7%

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

       16. associate-*r* 73.7%

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

   12. Simplified 33.7%

       \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

   13. Taylor expanded in x around 0 73.7%

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

   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 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 99.2%

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

      2. pow2 99.2%

         \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   5. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

   6. Taylor expanded in y around 0 48.9%

      \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
   7. Step-by-step derivation

      1. *-commutative 48.9%

         \[\leadsto \color{blue}{\sqrt{x} \cdot \left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

      2. *-commutative 48.9%

         \[\leadsto \sqrt{x} \cdot \color{blue}{\left({\left(\sqrt{3}\right)}^{2} \cdot \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right)\right)} \]

      3. unpow2 48.9%

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

      4. rem-square-sqrt 49.3%

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

      5. sub-neg 49.3%

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

      6. associate-*r/ 49.3%

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

      7. metadata-eval 49.3%

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

      8. metadata-eval 49.3%

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

      9. +-commutative 49.3%

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

   8. Simplified 49.3%

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

   9. Taylor expanded in x around inf 47.6%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 11: 3.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* x 9.0)))

C

double code(double x, double y) {
	return sqrt((x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0))

Julia

function code(x, y)
	return sqrt(Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.5%

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

   11. distribute-neg-frac 99.5%

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

   12. *-commutative 99.5%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

6. Taylor expanded in y around 0 59.6%

   \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
7. Step-by-step derivation

   1. associate-*l* 59.6%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

   2. *-commutative 59.6%

      \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

   3. unpow2 59.6%

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

   4. rem-square-sqrt 60.1%

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

   5. *-commutative 60.1%

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

   6. associate-*r* 60.1%

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

   7. *-commutative 60.1%

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

   8. associate-*l* 60.1%

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

   9. sub-neg 60.1%

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

   10. associate-*r/ 60.2%

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

   11. metadata-eval 60.2%

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

   12. metadata-eval 60.2%

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

   13. +-commutative 60.2%

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

8. Simplified 60.2%

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

   1. add-sqr-sqrt 32.7%

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

   2. sqrt-unprod 34.0%

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

   3. swap-sqr 33.9%

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

   4. metadata-eval 33.9%

      \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

   5. *-commutative 33.9%

      \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

   6. *-commutative 33.9%

      \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

   7. swap-sqr 16.1%

      \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

   8. add-sqr-sqrt 16.1%

      \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

   9. pow2 16.1%

      \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

   10. +-commutative 16.1%

       \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

10. Applied egg-rr 16.1%

    \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
11. Step-by-step derivation

    1. rem-square-sqrt 16.1%

       \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

    2. unpow2 16.1%

       \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

    3. swap-sqr 33.9%

       \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

    4. *-commutative 33.9%

       \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

    5. metadata-eval 33.9%

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

    6. swap-sqr 34.0%

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

    7. *-commutative 34.0%

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

    8. *-commutative 34.0%

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

    9. associate-*r* 33.9%

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

    10. metadata-eval 33.9%

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

    11. associate-*r/ 33.9%

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

    12. metadata-eval 33.9%

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

    13. sub-neg 33.9%

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

    14. *-commutative 33.9%

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

    15. *-commutative 33.9%

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

    16. associate-*r* 33.9%

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

12. Simplified 16.1%

    \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

13. Taylor expanded in x around inf 3.1%

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

    1. *-commutative 3.1%

       \[\leadsto \sqrt{\color{blue}{x \cdot 9}} \]

15. Simplified 3.1%

    \[\leadsto \sqrt{\color{blue}{x \cdot 9}} \]

16. Final simplification 3.1%

    \[\leadsto \sqrt{x \cdot 9} \]

Alternative 12: 37.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((0.1111111111111111d0 / x))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((0.1111111111111111 / x));
}

Python

def code(x, y):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x, y)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. distribute-rgt-in 99.4%

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

   4. remove-double-neg 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. mul-1-neg 99.4%

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

   8. metadata-eval 99.4%

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

   9. *-commutative 99.4%

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

   10. associate-/r* 99.5%

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

   11. distribute-neg-frac 99.5%

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

   12. *-commutative 99.5%

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

   13. associate-/r/ 99.4%

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

   14. associate-/l/ 99.4%

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

   15. associate-/r/ 99.5%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.1%

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

   2. pow2 99.1%

      \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{{\left(\sqrt{3 \cdot \sqrt{x}}\right)}^{2}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y - 1\right)\right) \]

6. Taylor expanded in y around 0 59.6%

   \[\leadsto \color{blue}{\left(\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot {\left(\sqrt{3}\right)}^{2}\right) \cdot \sqrt{x}} \]
7. Step-by-step derivation

   1. associate-*l* 59.6%

      \[\leadsto \color{blue}{\left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \left({\left(\sqrt{3}\right)}^{2} \cdot \sqrt{x}\right)} \]

   2. *-commutative 59.6%

      \[\leadsto \left(0.1111111111111111 \cdot \frac{1}{x} - 1\right) \cdot \color{blue}{\left(\sqrt{x} \cdot {\left(\sqrt{3}\right)}^{2}\right)} \]

   3. unpow2 59.6%

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

   4. rem-square-sqrt 60.1%

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

   5. *-commutative 60.1%

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

   6. associate-*r* 60.1%

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

   7. *-commutative 60.1%

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

   8. associate-*l* 60.1%

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

   9. sub-neg 60.1%

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

   10. associate-*r/ 60.2%

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

   11. metadata-eval 60.2%

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

   12. metadata-eval 60.2%

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

   13. +-commutative 60.2%

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

8. Simplified 60.2%

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

   1. add-sqr-sqrt 32.7%

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

   2. sqrt-unprod 34.0%

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

   3. swap-sqr 33.9%

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

   4. metadata-eval 33.9%

      \[\leadsto \sqrt{\color{blue}{9} \cdot \left(\left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

   5. *-commutative 33.9%

      \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \sqrt{x}\right)\right)} \]

   6. *-commutative 33.9%

      \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right) \cdot \color{blue}{\left(\sqrt{x} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)}\right)} \]

   7. swap-sqr 16.1%

      \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)}} \]

   8. add-sqr-sqrt 16.1%

      \[\leadsto \sqrt{9 \cdot \left(\color{blue}{x} \cdot \left(\left(-1 + \frac{0.1111111111111111}{x}\right) \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)\right)\right)} \]

   9. pow2 16.1%

      \[\leadsto \sqrt{9 \cdot \left(x \cdot \color{blue}{{\left(-1 + \frac{0.1111111111111111}{x}\right)}^{2}}\right)} \]

   10. +-commutative 16.1%

       \[\leadsto \sqrt{9 \cdot \left(x \cdot {\color{blue}{\left(\frac{0.1111111111111111}{x} + -1\right)}}^{2}\right)} \]

10. Applied egg-rr 16.1%

    \[\leadsto \color{blue}{\sqrt{9 \cdot \left(x \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)}} \]
11. Step-by-step derivation

    1. rem-square-sqrt 16.1%

       \[\leadsto \sqrt{9 \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot {\left(\frac{0.1111111111111111}{x} + -1\right)}^{2}\right)} \]

    2. unpow2 16.1%

       \[\leadsto \sqrt{9 \cdot \left(\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \color{blue}{\left(\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)}\right)} \]

    3. swap-sqr 33.9%

       \[\leadsto \sqrt{9 \cdot \color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)}} \]

    4. *-commutative 33.9%

       \[\leadsto \sqrt{\color{blue}{\left(\left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \left(\sqrt{x} \cdot \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right) \cdot 9}} \]

    5. metadata-eval 33.9%

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

    6. swap-sqr 34.0%

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

    7. *-commutative 34.0%

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

    8. *-commutative 34.0%

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

    9. associate-*r* 33.9%

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

    10. metadata-eval 33.9%

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

    11. associate-*r/ 33.9%

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

    12. metadata-eval 33.9%

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

    13. sub-neg 33.9%

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

    14. *-commutative 33.9%

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

    15. *-commutative 33.9%

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

    16. associate-*r* 33.9%

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

12. Simplified 16.1%

    \[\leadsto \color{blue}{\sqrt{x \cdot {\left(\frac{0.3333333333333333}{x} + -3\right)}^{2}}} \]

13. Taylor expanded in x around 0 33.9%

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

14. Final simplification 33.9%

    \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]

Developer target: 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 2023176 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))