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

Percentage Accurate: 99.4% → 99.4%

Time: 16.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

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. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

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. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.5%

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

   4. pow1/2 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
7. Step-by-step derivation

   1. unpow1/2 99.5%

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

8. Simplified 99.5%

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

Alternative 2: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sqrt{x \cdot 9}\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 1.6 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+195}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+219}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+255} \lor \neg \left(x \leq 8.2 \cdot 10^{+267}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt (* x 9.0)))) (t_1 (* 3.0 (* y (sqrt x)))))
   (if (<= x 1.6e-17)
     (sqrt (/ 0.1111111111111111 x))
     (if (<= x 9e+90)
       t_1
       (if (<= x 3.9e+111)
         t_0
         (if (<= x 9.5e+139)
           t_1
           (if (<= x 7.6e+195)
             t_0
             (if (<= x 1.7e+213)
               t_1
               (if (<= x 3e+219)
                 t_0
                 (if (or (<= x 7e+255) (not (<= x 8.2e+267)))
                   t_1
                   (* (sqrt x) -3.0)))))))))))

C

double code(double x, double y) {
	double t_0 = -sqrt((x * 9.0));
	double t_1 = 3.0 * (y * sqrt(x));
	double tmp;
	if (x <= 1.6e-17) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 9e+90) {
		tmp = t_1;
	} else if (x <= 3.9e+111) {
		tmp = t_0;
	} else if (x <= 9.5e+139) {
		tmp = t_1;
	} else if (x <= 7.6e+195) {
		tmp = t_0;
	} else if (x <= 1.7e+213) {
		tmp = t_1;
	} else if (x <= 3e+219) {
		tmp = t_0;
	} else if ((x <= 7e+255) || !(x <= 8.2e+267)) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -sqrt((x * 9.0d0))
    t_1 = 3.0d0 * (y * sqrt(x))
    if (x <= 1.6d-17) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 9d+90) then
        tmp = t_1
    else if (x <= 3.9d+111) then
        tmp = t_0
    else if (x <= 9.5d+139) then
        tmp = t_1
    else if (x <= 7.6d+195) then
        tmp = t_0
    else if (x <= 1.7d+213) then
        tmp = t_1
    else if (x <= 3d+219) then
        tmp = t_0
    else if ((x <= 7d+255) .or. (.not. (x <= 8.2d+267))) then
        tmp = t_1
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -Math.sqrt((x * 9.0));
	double t_1 = 3.0 * (y * Math.sqrt(x));
	double tmp;
	if (x <= 1.6e-17) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 9e+90) {
		tmp = t_1;
	} else if (x <= 3.9e+111) {
		tmp = t_0;
	} else if (x <= 9.5e+139) {
		tmp = t_1;
	} else if (x <= 7.6e+195) {
		tmp = t_0;
	} else if (x <= 1.7e+213) {
		tmp = t_1;
	} else if (x <= 3e+219) {
		tmp = t_0;
	} else if ((x <= 7e+255) || !(x <= 8.2e+267)) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -math.sqrt((x * 9.0))
	t_1 = 3.0 * (y * math.sqrt(x))
	tmp = 0
	if x <= 1.6e-17:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 9e+90:
		tmp = t_1
	elif x <= 3.9e+111:
		tmp = t_0
	elif x <= 9.5e+139:
		tmp = t_1
	elif x <= 7.6e+195:
		tmp = t_0
	elif x <= 1.7e+213:
		tmp = t_1
	elif x <= 3e+219:
		tmp = t_0
	elif (x <= 7e+255) or not (x <= 8.2e+267):
		tmp = t_1
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-sqrt(Float64(x * 9.0)))
	t_1 = Float64(3.0 * Float64(y * sqrt(x)))
	tmp = 0.0
	if (x <= 1.6e-17)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 9e+90)
		tmp = t_1;
	elseif (x <= 3.9e+111)
		tmp = t_0;
	elseif (x <= 9.5e+139)
		tmp = t_1;
	elseif (x <= 7.6e+195)
		tmp = t_0;
	elseif (x <= 1.7e+213)
		tmp = t_1;
	elseif (x <= 3e+219)
		tmp = t_0;
	elseif ((x <= 7e+255) || !(x <= 8.2e+267))
		tmp = t_1;
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -sqrt((x * 9.0));
	t_1 = 3.0 * (y * sqrt(x));
	tmp = 0.0;
	if (x <= 1.6e-17)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 9e+90)
		tmp = t_1;
	elseif (x <= 3.9e+111)
		tmp = t_0;
	elseif (x <= 9.5e+139)
		tmp = t_1;
	elseif (x <= 7.6e+195)
		tmp = t_0;
	elseif (x <= 1.7e+213)
		tmp = t_1;
	elseif (x <= 3e+219)
		tmp = t_0;
	elseif ((x <= 7e+255) || ~((x <= 8.2e+267)))
		tmp = t_1;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.6e-17], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 9e+90], t$95$1, If[LessEqual[x, 3.9e+111], t$95$0, If[LessEqual[x, 9.5e+139], t$95$1, If[LessEqual[x, 7.6e+195], t$95$0, If[LessEqual[x, 1.7e+213], t$95$1, If[LessEqual[x, 3e+219], t$95$0, If[Or[LessEqual[x, 7e+255], N[Not[LessEqual[x, 8.2e+267]], $MachinePrecision]], t$95$1, N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.6000000000000001e-17

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. metadata-eval 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.8%

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

      4. pow1/2 78.8%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.8%

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

   9. Simplified 78.8%

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

   if 1.6000000000000001e-17 < x < 9e90 or 3.89999999999999979e111 < x < 9.5000000000000002e139 or 7.6e195 < x < 1.69999999999999996e213 or 2.9999999999999997e219 < x < 6.99999999999999971e255 or 8.19999999999999997e267 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 64.9%

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

   if 9e90 < x < 3.89999999999999979e111 or 9.5000000000000002e139 < x < 7.6e195 or 1.69999999999999996e213 < x < 2.9999999999999997e219

   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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 87.4%

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

   6. Taylor expanded in y around 0 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(0.3333333333333333 + -3 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 65.3%

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

      2. *-commutative 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in x around inf 65.3%

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

       1. *-commutative 65.3%

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

   11. Simplified 65.3%

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

       1. sqrt-div 65.2%

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

       2. metadata-eval 65.2%

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

       3. un-div-inv 65.2%

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

       4. *-commutative 65.2%

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

   13. Applied egg-rr 65.2%

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

       1. *-commutative 65.2%

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

   15. Simplified 65.2%

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

       1. *-un-lft-identity 65.2%

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

       2. *-commutative 65.2%

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

       3. clear-num 65.3%

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

       4. *-un-lft-identity 65.3%

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

       5. *-commutative 65.3%

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

       6. times-frac 65.3%

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

       7. metadata-eval 65.3%

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

       8. pow1/2 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot 1 \]

       9. pow1 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot 1 \]

       10. pow-div 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot \color{blue}{{x}^{\left(0.5 - 1\right)}}} \cdot 1 \]

       11. metadata-eval 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}} \cdot 1 \]

   17. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}} \cdot 1} \]
   18. Step-by-step derivation

       1. *-rgt-identity 65.4%

          \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}} \]

       2. associate-/r* 65.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333}}{{x}^{-0.5}}} \]

       3. metadata-eval 65.6%

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

   19. Simplified 65.6%

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

   21. Applied egg-rr 65.6%

       \[\leadsto \color{blue}{-1 \cdot \sqrt{x \cdot 9}} \]
   22. Step-by-step derivation

       1. neg-mul-1 65.6%

          \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   23. Simplified 65.6%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   if 6.99999999999999971e255 < x < 8.19999999999999997e267

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 100.0%

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

      3. associate--l+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. fma-define 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-/r* 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 67.6%

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

      1. *-commutative 67.6%

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

   8. Simplified 67.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{+90}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+111}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+139}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{+195}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{+213}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+219}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+255} \lor \neg \left(x \leq 8.2 \cdot 10^{+267}\right):\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := -t\_0\\ t_2 := t\_0 \cdot y\\ \mathbf{if}\;x \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+213}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+219}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+255}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))) (t_1 (- t_0)) (t_2 (* t_0 y)))
   (if (<= x 1.55e-17)
     (sqrt (/ 0.1111111111111111 x))
     (if (<= x 1.1e+93)
       t_2
       (if (<= x 2.5e+111)
         t_1
         (if (<= x 6.2e+140)
           t_2
           (if (<= x 1.6e+193)
             t_1
             (if (<= x 7.5e+213)
               t_2
               (if (<= x 1.6e+219)
                 t_1
                 (if (<= x 9.2e+255)
                   (* 3.0 (* y (sqrt x)))
                   (if (<= x 9.5e+267)
                     (* (sqrt x) -3.0)
                     (* (sqrt x) (* y 3.0)))))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double tmp;
	if (x <= 1.55e-17) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 1.1e+93) {
		tmp = t_2;
	} else if (x <= 2.5e+111) {
		tmp = t_1;
	} else if (x <= 6.2e+140) {
		tmp = t_2;
	} else if (x <= 1.6e+193) {
		tmp = t_1;
	} else if (x <= 7.5e+213) {
		tmp = t_2;
	} else if (x <= 1.6e+219) {
		tmp = t_1;
	} else if (x <= 9.2e+255) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (x <= 9.5e+267) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (y * 3.0);
	}
	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 = sqrt((x * 9.0d0))
    t_1 = -t_0
    t_2 = t_0 * y
    if (x <= 1.55d-17) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 1.1d+93) then
        tmp = t_2
    else if (x <= 2.5d+111) then
        tmp = t_1
    else if (x <= 6.2d+140) then
        tmp = t_2
    else if (x <= 1.6d+193) then
        tmp = t_1
    else if (x <= 7.5d+213) then
        tmp = t_2
    else if (x <= 1.6d+219) then
        tmp = t_1
    else if (x <= 9.2d+255) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (x <= 9.5d+267) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double tmp;
	if (x <= 1.55e-17) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 1.1e+93) {
		tmp = t_2;
	} else if (x <= 2.5e+111) {
		tmp = t_1;
	} else if (x <= 6.2e+140) {
		tmp = t_2;
	} else if (x <= 1.6e+193) {
		tmp = t_1;
	} else if (x <= 7.5e+213) {
		tmp = t_2;
	} else if (x <= 1.6e+219) {
		tmp = t_1;
	} else if (x <= 9.2e+255) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (x <= 9.5e+267) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	t_1 = -t_0
	t_2 = t_0 * y
	tmp = 0
	if x <= 1.55e-17:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 1.1e+93:
		tmp = t_2
	elif x <= 2.5e+111:
		tmp = t_1
	elif x <= 6.2e+140:
		tmp = t_2
	elif x <= 1.6e+193:
		tmp = t_1
	elif x <= 7.5e+213:
		tmp = t_2
	elif x <= 1.6e+219:
		tmp = t_1
	elif x <= 9.2e+255:
		tmp = 3.0 * (y * math.sqrt(x))
	elif x <= 9.5e+267:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	t_1 = Float64(-t_0)
	t_2 = Float64(t_0 * y)
	tmp = 0.0
	if (x <= 1.55e-17)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 1.1e+93)
		tmp = t_2;
	elseif (x <= 2.5e+111)
		tmp = t_1;
	elseif (x <= 6.2e+140)
		tmp = t_2;
	elseif (x <= 1.6e+193)
		tmp = t_1;
	elseif (x <= 7.5e+213)
		tmp = t_2;
	elseif (x <= 1.6e+219)
		tmp = t_1;
	elseif (x <= 9.2e+255)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (x <= 9.5e+267)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	t_1 = -t_0;
	t_2 = t_0 * y;
	tmp = 0.0;
	if (x <= 1.55e-17)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 1.1e+93)
		tmp = t_2;
	elseif (x <= 2.5e+111)
		tmp = t_1;
	elseif (x <= 6.2e+140)
		tmp = t_2;
	elseif (x <= 1.6e+193)
		tmp = t_1;
	elseif (x <= 7.5e+213)
		tmp = t_2;
	elseif (x <= 1.6e+219)
		tmp = t_1;
	elseif (x <= 9.2e+255)
		tmp = 3.0 * (y * sqrt(x));
	elseif (x <= 9.5e+267)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(t$95$0 * y), $MachinePrecision]}, If[LessEqual[x, 1.55e-17], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1e+93], t$95$2, If[LessEqual[x, 2.5e+111], t$95$1, If[LessEqual[x, 6.2e+140], t$95$2, If[LessEqual[x, 1.6e+193], t$95$1, If[LessEqual[x, 7.5e+213], t$95$2, If[LessEqual[x, 1.6e+219], t$95$1, If[LessEqual[x, 9.2e+255], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e+267], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if x < 1.5499999999999999e-17

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. metadata-eval 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.8%

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

      4. pow1/2 78.8%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.8%

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

   9. Simplified 78.8%

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

   if 1.5499999999999999e-17 < x < 1.10000000000000011e93 or 2.4999999999999998e111 < x < 6.2000000000000001e140 or 1.60000000000000007e193 < x < 7.5000000000000003e213

   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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 64.8%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{y} \]

   if 1.10000000000000011e93 < x < 2.4999999999999998e111 or 6.2000000000000001e140 < x < 1.60000000000000007e193 or 7.5000000000000003e213 < x < 1.60000000000000013e219

   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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 87.4%

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

   6. Taylor expanded in y around 0 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(0.3333333333333333 + -3 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 65.3%

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

      2. *-commutative 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in x around inf 65.3%

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

       1. *-commutative 65.3%

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

   11. Simplified 65.3%

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

       1. sqrt-div 65.2%

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

       2. metadata-eval 65.2%

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

       3. un-div-inv 65.2%

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

       4. *-commutative 65.2%

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

   13. Applied egg-rr 65.2%

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

       1. *-commutative 65.2%

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

   15. Simplified 65.2%

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

       1. *-un-lft-identity 65.2%

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

       2. *-commutative 65.2%

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

       3. clear-num 65.3%

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

       4. *-un-lft-identity 65.3%

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

       5. *-commutative 65.3%

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

       6. times-frac 65.3%

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

       7. metadata-eval 65.3%

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

       8. pow1/2 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot 1 \]

       9. pow1 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot 1 \]

       10. pow-div 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot \color{blue}{{x}^{\left(0.5 - 1\right)}}} \cdot 1 \]

       11. metadata-eval 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}} \cdot 1 \]

   17. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}} \cdot 1} \]
   18. Step-by-step derivation

       1. *-rgt-identity 65.4%

          \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}} \]

       2. associate-/r* 65.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333}}{{x}^{-0.5}}} \]

       3. metadata-eval 65.6%

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

   19. Simplified 65.6%

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

   21. Applied egg-rr 65.6%

       \[\leadsto \color{blue}{-1 \cdot \sqrt{x \cdot 9}} \]
   22. Step-by-step derivation

       1. neg-mul-1 65.6%

          \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   23. Simplified 65.6%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   if 1.60000000000000013e219 < x < 9.2000000000000001e255

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.7%

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

      5. fma-define 99.7%

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

      6. sub-neg 99.7%

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

      7. +-commutative 99.7%

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

      8. distribute-lft-in 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

      11. *-commutative 99.7%

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

      12. associate-/r* 99.7%

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

      13. associate-*r/ 99.7%

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

      14. metadata-eval 99.7%

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

      15. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 72.1%

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

   if 9.2000000000000001e255 < x < 9.50000000000000066e267

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 100.0%

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

      3. associate--l+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. fma-define 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-/r* 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 67.6%

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

      1. *-commutative 67.6%

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

   8. Simplified 67.6%

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

   if 9.50000000000000066e267 < x

   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%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-*l* 59.4%

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

      3. *-commutative 59.4%

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

   7. Simplified 59.4%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y\right)} \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+111}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+140}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+193}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{+213}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{+219}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{+255}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \left(y \cdot 3\right)\\ t_1 := -\sqrt{x \cdot 9}\\ t_2 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+91}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+193}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+212}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+219}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (* y 3.0)))
        (t_1 (- (sqrt (* x 9.0))))
        (t_2 (* 3.0 (* y (sqrt x)))))
   (if (<= x 1.55e-17)
     (sqrt (/ 0.1111111111111111 x))
     (if (<= x 5.3e+91)
       t_2
       (if (<= x 7.2e+110)
         t_1
         (if (<= x 2.85e+138)
           t_2
           (if (<= x 1.56e+193)
             t_1
             (if (<= x 1.9e+212)
               t_0
               (if (<= x 2.05e+219)
                 t_1
                 (if (<= x 8.2e+254)
                   t_2
                   (if (<= x 9.5e+267) (* (sqrt x) -3.0) t_0)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (y * 3.0);
	double t_1 = -sqrt((x * 9.0));
	double t_2 = 3.0 * (y * sqrt(x));
	double tmp;
	if (x <= 1.55e-17) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 5.3e+91) {
		tmp = t_2;
	} else if (x <= 7.2e+110) {
		tmp = t_1;
	} else if (x <= 2.85e+138) {
		tmp = t_2;
	} else if (x <= 1.56e+193) {
		tmp = t_1;
	} else if (x <= 1.9e+212) {
		tmp = t_0;
	} else if (x <= 2.05e+219) {
		tmp = t_1;
	} else if (x <= 8.2e+254) {
		tmp = t_2;
	} else if (x <= 9.5e+267) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(x) * (y * 3.0d0)
    t_1 = -sqrt((x * 9.0d0))
    t_2 = 3.0d0 * (y * sqrt(x))
    if (x <= 1.55d-17) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 5.3d+91) then
        tmp = t_2
    else if (x <= 7.2d+110) then
        tmp = t_1
    else if (x <= 2.85d+138) then
        tmp = t_2
    else if (x <= 1.56d+193) then
        tmp = t_1
    else if (x <= 1.9d+212) then
        tmp = t_0
    else if (x <= 2.05d+219) then
        tmp = t_1
    else if (x <= 8.2d+254) then
        tmp = t_2
    else if (x <= 9.5d+267) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (y * 3.0);
	double t_1 = -Math.sqrt((x * 9.0));
	double t_2 = 3.0 * (y * Math.sqrt(x));
	double tmp;
	if (x <= 1.55e-17) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 5.3e+91) {
		tmp = t_2;
	} else if (x <= 7.2e+110) {
		tmp = t_1;
	} else if (x <= 2.85e+138) {
		tmp = t_2;
	} else if (x <= 1.56e+193) {
		tmp = t_1;
	} else if (x <= 1.9e+212) {
		tmp = t_0;
	} else if (x <= 2.05e+219) {
		tmp = t_1;
	} else if (x <= 8.2e+254) {
		tmp = t_2;
	} else if (x <= 9.5e+267) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (y * 3.0)
	t_1 = -math.sqrt((x * 9.0))
	t_2 = 3.0 * (y * math.sqrt(x))
	tmp = 0
	if x <= 1.55e-17:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 5.3e+91:
		tmp = t_2
	elif x <= 7.2e+110:
		tmp = t_1
	elif x <= 2.85e+138:
		tmp = t_2
	elif x <= 1.56e+193:
		tmp = t_1
	elif x <= 1.9e+212:
		tmp = t_0
	elif x <= 2.05e+219:
		tmp = t_1
	elif x <= 8.2e+254:
		tmp = t_2
	elif x <= 9.5e+267:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(y * 3.0))
	t_1 = Float64(-sqrt(Float64(x * 9.0)))
	t_2 = Float64(3.0 * Float64(y * sqrt(x)))
	tmp = 0.0
	if (x <= 1.55e-17)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 5.3e+91)
		tmp = t_2;
	elseif (x <= 7.2e+110)
		tmp = t_1;
	elseif (x <= 2.85e+138)
		tmp = t_2;
	elseif (x <= 1.56e+193)
		tmp = t_1;
	elseif (x <= 1.9e+212)
		tmp = t_0;
	elseif (x <= 2.05e+219)
		tmp = t_1;
	elseif (x <= 8.2e+254)
		tmp = t_2;
	elseif (x <= 9.5e+267)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (y * 3.0);
	t_1 = -sqrt((x * 9.0));
	t_2 = 3.0 * (y * sqrt(x));
	tmp = 0.0;
	if (x <= 1.55e-17)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 5.3e+91)
		tmp = t_2;
	elseif (x <= 7.2e+110)
		tmp = t_1;
	elseif (x <= 2.85e+138)
		tmp = t_2;
	elseif (x <= 1.56e+193)
		tmp = t_1;
	elseif (x <= 1.9e+212)
		tmp = t_0;
	elseif (x <= 2.05e+219)
		tmp = t_1;
	elseif (x <= 8.2e+254)
		tmp = t_2;
	elseif (x <= 9.5e+267)
		tmp = sqrt(x) * -3.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.55e-17], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 5.3e+91], t$95$2, If[LessEqual[x, 7.2e+110], t$95$1, If[LessEqual[x, 2.85e+138], t$95$2, If[LessEqual[x, 1.56e+193], t$95$1, If[LessEqual[x, 1.9e+212], t$95$0, If[LessEqual[x, 2.05e+219], t$95$1, If[LessEqual[x, 8.2e+254], t$95$2, If[LessEqual[x, 9.5e+267], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], t$95$0]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x < 1.5499999999999999e-17

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. metadata-eval 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.8%

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

      4. pow1/2 78.8%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.8%

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

   9. Simplified 78.8%

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

   if 1.5499999999999999e-17 < x < 5.29999999999999997e91 or 7.1999999999999994e110 < x < 2.84999999999999993e138 or 2.0499999999999999e219 < x < 8.19999999999999974e254

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 65.7%

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

   if 5.29999999999999997e91 < x < 7.1999999999999994e110 or 2.84999999999999993e138 < x < 1.5599999999999999e193 or 1.89999999999999994e212 < x < 2.0499999999999999e219

   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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 87.4%

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

   6. Taylor expanded in y around 0 65.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(0.3333333333333333 + -3 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 65.3%

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

      2. *-commutative 65.3%

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

   8. Simplified 65.3%

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

   9. Taylor expanded in x around inf 65.3%

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

       1. *-commutative 65.3%

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

   11. Simplified 65.3%

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

       1. sqrt-div 65.2%

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

       2. metadata-eval 65.2%

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

       3. un-div-inv 65.2%

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

       4. *-commutative 65.2%

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

   13. Applied egg-rr 65.2%

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

       1. *-commutative 65.2%

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

   15. Simplified 65.2%

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

       1. *-un-lft-identity 65.2%

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

       2. *-commutative 65.2%

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

       3. clear-num 65.3%

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

       4. *-un-lft-identity 65.3%

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

       5. *-commutative 65.3%

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

       6. times-frac 65.3%

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

       7. metadata-eval 65.3%

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

       8. pow1/2 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot 1 \]

       9. pow1 65.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot 1 \]

       10. pow-div 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot \color{blue}{{x}^{\left(0.5 - 1\right)}}} \cdot 1 \]

       11. metadata-eval 65.4%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}} \cdot 1 \]

   17. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}} \cdot 1} \]
   18. Step-by-step derivation

       1. *-rgt-identity 65.4%

          \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}} \]

       2. associate-/r* 65.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333}}{{x}^{-0.5}}} \]

       3. metadata-eval 65.6%

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

   19. Simplified 65.6%

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

   21. Applied egg-rr 65.6%

       \[\leadsto \color{blue}{-1 \cdot \sqrt{x \cdot 9}} \]
   22. Step-by-step derivation

       1. neg-mul-1 65.6%

          \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   23. Simplified 65.6%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   if 1.5599999999999999e193 < x < 1.89999999999999994e212 or 9.50000000000000066e267 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.8%

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

      5. fma-define 99.8%

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

      6. sub-neg 99.8%

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

      7. +-commutative 99.8%

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

      8. distribute-lft-in 99.8%

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

      9. metadata-eval 99.8%

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

      10. metadata-eval 99.8%

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

      11. *-commutative 99.8%

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

      12. associate-/r* 99.8%

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

      13. associate-*r/ 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 61.8%

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

      1. *-commutative 61.8%

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

      2. associate-*l* 61.9%

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

      3. *-commutative 61.9%

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

   7. Simplified 61.9%

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

   if 8.19999999999999974e254 < x < 9.50000000000000066e267

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-*l* 100.0%

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

      3. associate--l+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. fma-define 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-/r* 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 100.0%

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

   6. Taylor expanded in y around 0 67.6%

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

      1. *-commutative 67.6%

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

   8. Simplified 67.6%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 5.3 \cdot 10^{+91}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{+110}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 2.85 \cdot 10^{+138}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.56 \cdot 10^{+193}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{+212}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{+219}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+254}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := -t\_0\\ t_2 := t\_0 \cdot y\\ \mathbf{if}\;x \leq 1.65 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+110}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+195} \lor \neg \left(x \leq 1.4 \cdot 10^{+214} \lor \neg \left(x \leq 6.5 \cdot 10^{+218}\right) \land x \leq 2.1 \cdot 10^{+258}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))) (t_1 (- t_0)) (t_2 (* t_0 y)))
   (if (<= x 1.65e-17)
     (sqrt (/ 0.1111111111111111 x))
     (if (<= x 1.1e+91)
       t_2
       (if (<= x 8.2e+110)
         t_1
         (if (<= x 3.5e+139)
           (* y (* (sqrt x) 3.0))
           (if (or (<= x 4.1e+195)
                   (not
                    (or (<= x 1.4e+214)
                        (and (not (<= x 6.5e+218)) (<= x 2.1e+258)))))
             t_1
             t_2)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double tmp;
	if (x <= 1.65e-17) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 1.1e+91) {
		tmp = t_2;
	} else if (x <= 8.2e+110) {
		tmp = t_1;
	} else if (x <= 3.5e+139) {
		tmp = y * (sqrt(x) * 3.0);
	} else if ((x <= 4.1e+195) || !((x <= 1.4e+214) || (!(x <= 6.5e+218) && (x <= 2.1e+258)))) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = sqrt((x * 9.0d0))
    t_1 = -t_0
    t_2 = t_0 * y
    if (x <= 1.65d-17) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 1.1d+91) then
        tmp = t_2
    else if (x <= 8.2d+110) then
        tmp = t_1
    else if (x <= 3.5d+139) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if ((x <= 4.1d+195) .or. (.not. (x <= 1.4d+214) .or. (.not. (x <= 6.5d+218)) .and. (x <= 2.1d+258))) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double tmp;
	if (x <= 1.65e-17) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 1.1e+91) {
		tmp = t_2;
	} else if (x <= 8.2e+110) {
		tmp = t_1;
	} else if (x <= 3.5e+139) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if ((x <= 4.1e+195) || !((x <= 1.4e+214) || (!(x <= 6.5e+218) && (x <= 2.1e+258)))) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	t_1 = -t_0
	t_2 = t_0 * y
	tmp = 0
	if x <= 1.65e-17:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 1.1e+91:
		tmp = t_2
	elif x <= 8.2e+110:
		tmp = t_1
	elif x <= 3.5e+139:
		tmp = y * (math.sqrt(x) * 3.0)
	elif (x <= 4.1e+195) or not ((x <= 1.4e+214) or (not (x <= 6.5e+218) and (x <= 2.1e+258))):
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	t_1 = Float64(-t_0)
	t_2 = Float64(t_0 * y)
	tmp = 0.0
	if (x <= 1.65e-17)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 1.1e+91)
		tmp = t_2;
	elseif (x <= 8.2e+110)
		tmp = t_1;
	elseif (x <= 3.5e+139)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif ((x <= 4.1e+195) || !((x <= 1.4e+214) || (!(x <= 6.5e+218) && (x <= 2.1e+258))))
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	t_1 = -t_0;
	t_2 = t_0 * y;
	tmp = 0.0;
	if (x <= 1.65e-17)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 1.1e+91)
		tmp = t_2;
	elseif (x <= 8.2e+110)
		tmp = t_1;
	elseif (x <= 3.5e+139)
		tmp = y * (sqrt(x) * 3.0);
	elseif ((x <= 4.1e+195) || ~(((x <= 1.4e+214) || (~((x <= 6.5e+218)) && (x <= 2.1e+258)))))
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(t$95$0 * y), $MachinePrecision]}, If[LessEqual[x, 1.65e-17], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.1e+91], t$95$2, If[LessEqual[x, 8.2e+110], t$95$1, If[LessEqual[x, 3.5e+139], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 4.1e+195], N[Not[Or[LessEqual[x, 1.4e+214], And[N[Not[LessEqual[x, 6.5e+218]], $MachinePrecision], LessEqual[x, 2.1e+258]]]], $MachinePrecision]], t$95$1, t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.65e-17

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. metadata-eval 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.8%

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

      4. pow1/2 78.8%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.8%

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

   9. Simplified 78.8%

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

   if 1.65e-17 < x < 1.1e91 or 4.1e195 < x < 1.3999999999999999e214 or 6.4999999999999999e218 < x < 2.09999999999999997e258

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. associate-+l+ 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.8%

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

      4. pow1/2 99.8%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around inf 64.1%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{y} \]

   if 1.1e91 < x < 8.1999999999999997e110 or 3.49999999999999978e139 < x < 4.1e195 or 1.3999999999999999e214 < x < 6.4999999999999999e218 or 2.09999999999999997e258 < 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%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 82.5%

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

   6. Taylor expanded in y around 0 61.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(0.3333333333333333 + -3 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 61.3%

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

      2. *-commutative 61.3%

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

   8. Simplified 61.3%

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

   9. Taylor expanded in x around inf 61.3%

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

       1. *-commutative 61.3%

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

   11. Simplified 61.3%

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

       1. sqrt-div 61.2%

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

       2. metadata-eval 61.2%

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

       3. un-div-inv 61.2%

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

       4. *-commutative 61.2%

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

   13. Applied egg-rr 61.2%

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

       1. *-commutative 61.2%

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

   15. Simplified 61.2%

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

       1. *-un-lft-identity 61.2%

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

       2. *-commutative 61.2%

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

       3. clear-num 61.3%

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

       4. *-un-lft-identity 61.3%

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

       5. *-commutative 61.3%

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

       6. times-frac 61.3%

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

       7. metadata-eval 61.3%

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

       8. pow1/2 61.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot 1 \]

       9. pow1 61.3%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot 1 \]

       10. pow-div 61.3%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot \color{blue}{{x}^{\left(0.5 - 1\right)}}} \cdot 1 \]

       11. metadata-eval 61.3%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}} \cdot 1 \]

   17. Applied egg-rr 61.3%

       \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}} \cdot 1} \]
   18. Step-by-step derivation

       1. *-rgt-identity 61.3%

          \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}} \]

       2. associate-/r* 61.5%

          \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333}}{{x}^{-0.5}}} \]

       3. metadata-eval 61.5%

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

   19. Simplified 61.5%

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

   21. Applied egg-rr 63.2%

       \[\leadsto \color{blue}{-1 \cdot \sqrt{x \cdot 9}} \]
   22. Step-by-step derivation

       1. neg-mul-1 63.2%

          \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   23. Simplified 63.2%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   if 8.1999999999999997e110 < x < 3.49999999999999978e139

   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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   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. Add Preprocessing

   5. Taylor expanded in y around inf 75.8%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.65 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+110}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{+139}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+195} \lor \neg \left(x \leq 1.4 \cdot 10^{+214} \lor \neg \left(x \leq 6.5 \cdot 10^{+218}\right) \land x \leq 2.1 \cdot 10^{+258}\right):\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -340000 \lor \neg \left(y \leq 1.25 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot \left(3 + \frac{\frac{0.3333333333333333}{x}}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -340000.0) (not (<= y 1.25e-10)))
   (* y (* (sqrt x) (+ 3.0 (/ (/ 0.3333333333333333 x) y))))
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -340000.0) || !(y <= 1.25e-10)) {
		tmp = y * (sqrt(x) * (3.0 + ((0.3333333333333333 / x) / y)));
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / 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 ((y <= (-340000.0d0)) .or. (.not. (y <= 1.25d-10))) then
        tmp = y * (sqrt(x) * (3.0d0 + ((0.3333333333333333d0 / x) / y)))
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -340000.0) || !(y <= 1.25e-10)) {
		tmp = y * (Math.sqrt(x) * (3.0 + ((0.3333333333333333 / x) / y)));
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -340000.0) or not (y <= 1.25e-10):
		tmp = y * (math.sqrt(x) * (3.0 + ((0.3333333333333333 / x) / y)))
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -340000.0) || !(y <= 1.25e-10))
		tmp = Float64(y * Float64(sqrt(x) * Float64(3.0 + Float64(Float64(0.3333333333333333 / x) / y))));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -340000.0) || ~((y <= 1.25e-10)))
		tmp = y * (sqrt(x) * (3.0 + ((0.3333333333333333 / x) / y)));
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -340000.0], N[Not[LessEqual[y, 1.25e-10]], $MachinePrecision]], N[(y * N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 + N[(N[(0.3333333333333333 / x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.4e5 or 1.25000000000000008e-10 < 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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 99.4%

      \[\leadsto \color{blue}{y \cdot \left(3 \cdot \sqrt{x} + \sqrt{x} \cdot \frac{0.3333333333333333 \cdot \frac{1}{x} - 3}{y}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 99.4%

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

      2. distribute-rgt-out 99.4%

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

      3. sub-neg 99.4%

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

      4. metadata-eval 99.4%

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

      5. associate-*r/ 99.4%

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

      6. metadata-eval 99.4%

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

      7. +-commutative 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in x around 0 97.9%

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

      1. associate-/r* 97.9%

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

   10. Simplified 97.9%

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

   if -3.4e5 < y < 1.25000000000000008e-10

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)} \]
   6. Step-by-step derivation

      1. sub-neg 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-*r/ 98.9%

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

      4. metadata-eval 98.9%

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

      5. +-commutative 98.9%

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

   7. Simplified 98.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -340000 \lor \neg \left(y \leq 1.25 \cdot 10^{-10}\right):\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot \left(3 + \frac{\frac{0.3333333333333333}{x}}{y}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.4e-25)
   (sqrt (/ 0.1111111111111111 x))
   (* (sqrt (* x 9.0)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.4e-25) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.4d-25) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.4e-25) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.4e-25:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.4e-25)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.4e-25)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.4e-25], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.40000000000000002e-25

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 79.5%

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

      1. metadata-eval 79.5%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 79.7%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 79.7%

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

      4. pow1/2 79.7%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 79.7%

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

   9. Simplified 79.7%

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

   if 3.40000000000000002e-25 < 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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

      1. *-commutative 99.6%

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

      2. metadata-eval 99.6%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 93.9%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.6e-25)
   (sqrt (/ 0.1111111111111111 x))
   (* 3.0 (* (sqrt x) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.6e-25) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 5.6d-25) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.6e-25) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.6e-25:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.6e-25)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.6e-25)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.6e-25], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.59999999999999976e-25

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 79.5%

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

      1. metadata-eval 79.5%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 79.7%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 79.7%

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

      4. pow1/2 79.7%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 79.7%

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

   9. Simplified 79.7%

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

   if 5.59999999999999976e-25 < 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. sub-neg 99.5%

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

      2. +-commutative 99.5%

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

      3. associate-+l+ 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 93.8%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{-25}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. fma-undefine 99.4%

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

   2. +-commutative 99.4%

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

   3. +-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 10: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.5e-18) (sqrt (/ 0.1111111111111111 x)) (- (sqrt (* x 9.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.5e-18) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = -sqrt((x * 9.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7.5d-18) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = -sqrt((x * 9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.5e-18) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = -Math.sqrt((x * 9.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.5e-18:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = -math.sqrt((x * 9.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.5e-18)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(-sqrt(Float64(x * 9.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.5e-18)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = -sqrt((x * 9.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.5e-18], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 7.50000000000000015e-18

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.4%

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

      1. metadata-eval 78.4%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.7%

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

      4. pow1/2 78.7%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.7%

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

   9. Simplified 78.7%

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

   if 7.50000000000000015e-18 < 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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 87.0%

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

   6. Taylor expanded in y around 0 46.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{x}} \cdot \left(0.3333333333333333 + -3 \cdot x\right)} \]
   7. Step-by-step derivation

      1. *-commutative 46.1%

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

      2. *-commutative 46.1%

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

   8. Simplified 46.1%

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

   9. Taylor expanded in x around inf 41.5%

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

       1. *-commutative 41.5%

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

   11. Simplified 41.5%

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

       1. sqrt-div 41.5%

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

       2. metadata-eval 41.5%

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

       3. un-div-inv 41.5%

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

       4. *-commutative 41.5%

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

   13. Applied egg-rr 41.5%

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

       1. *-commutative 41.5%

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

   15. Simplified 41.5%

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

       1. *-un-lft-identity 41.5%

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

       2. *-commutative 41.5%

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

       3. clear-num 41.5%

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

       4. *-un-lft-identity 41.5%

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

       5. *-commutative 41.5%

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

       6. times-frac 41.5%

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

       7. metadata-eval 41.5%

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

       8. pow1/2 41.5%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{\color{blue}{{x}^{0.5}}}{x}} \cdot 1 \]

       9. pow1 41.5%

          \[\leadsto \frac{1}{-0.3333333333333333 \cdot \frac{{x}^{0.5}}{\color{blue}{{x}^{1}}}} \cdot 1 \]

       10. pow-div 41.5%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot \color{blue}{{x}^{\left(0.5 - 1\right)}}} \cdot 1 \]

       11. metadata-eval 41.5%

           \[\leadsto \frac{1}{-0.3333333333333333 \cdot {x}^{\color{blue}{-0.5}}} \cdot 1 \]

   17. Applied egg-rr 41.5%

       \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}} \cdot 1} \]
   18. Step-by-step derivation

       1. *-rgt-identity 41.5%

          \[\leadsto \color{blue}{\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}} \]

       2. associate-/r* 41.6%

          \[\leadsto \color{blue}{\frac{\frac{1}{-0.3333333333333333}}{{x}^{-0.5}}} \]

       3. metadata-eval 41.6%

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

   19. Simplified 41.6%

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

   21. Applied egg-rr 42.3%

       \[\leadsto \color{blue}{-1 \cdot \sqrt{x \cdot 9}} \]
   22. Step-by-step derivation

       1. neg-mul-1 42.3%

          \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]

   23. Simplified 42.3%

       \[\leadsto \color{blue}{-\sqrt{x \cdot 9}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.85e-17) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.85e-17) {
		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 <= 1.85d-17) 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 <= 1.85e-17) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.85e-17:
		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 <= 1.85e-17)
		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 <= 1.85e-17)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.85e-17], 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 < 1.8499999999999999e-17

   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%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. metadata-eval 78.6%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

      2. sqrt-prod 78.8%

         \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. div-inv 78.8%

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

      4. pow1/2 78.8%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 78.8%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 78.8%

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

   9. Simplified 78.8%

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

   if 1.8499999999999999e-17 < 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%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 95.6%

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

   6. Taylor expanded in y around 0 41.8%

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

      1. *-commutative 41.8%

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

   8. Simplified 41.8%

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

Alternative 12: 37.2% 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%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 38.2%

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

   1. metadata-eval 38.2%

      \[\leadsto \color{blue}{\sqrt{0.1111111111111111}} \cdot \sqrt{\frac{1}{x}} \]

   2. sqrt-prod 38.3%

      \[\leadsto \color{blue}{\sqrt{0.1111111111111111 \cdot \frac{1}{x}}} \]

   3. div-inv 38.3%

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

   4. pow1/2 38.3%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

7. Applied egg-rr 38.3%

   \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 38.3%

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

9. Simplified 38.3%

   \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]
10. Add Preprocessing

Alternative 13: 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%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around inf 60.6%

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

6. Taylor expanded in y around 0 23.4%

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

   1. *-commutative 23.4%

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

8. Simplified 23.4%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.4%

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

   3. swap-sqr 3.4%

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

   4. add-sqr-sqrt 3.4%

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

   5. metadata-eval 3.4%

      \[\leadsto \sqrt{x \cdot \color{blue}{9}} \]

10. Applied egg-rr 3.4%

    \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]
11. Add Preprocessing

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 2024107 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (* 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)))