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

Percentage Accurate: 99.4% → 99.3%

Time: 12.8s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

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(0.3333333333333333 / x) + Float64(-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[(0.3333333333333333 / x), $MachinePrecision] + N[(-3.0 + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. distribute-lft-in 99.5%

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

   2. div-inv 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. +-commutative 99.5%

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

   7. distribute-rgt-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x} + -2}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-272}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-131}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-64}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1950000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (+ (/ 0.1111111111111111 x) -2.0)))
        (t_1 (* 3.0 (* (sqrt x) y)))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -5.2)
     t_1
     (if (<= y -4e-272)
       t_2
       (if (<= y 1.2e-131)
         t_0
         (if (<= y 2.85e-64) t_2 (if (<= y 1950000.0) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	double t_1 = 3.0 * (sqrt(x) * y);
	double t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = t_1;
	} else if (y <= -4e-272) {
		tmp = t_2;
	} else if (y <= 1.2e-131) {
		tmp = t_0;
	} else if (y <= 2.85e-64) {
		tmp = t_2;
	} else if (y <= 1950000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    t_1 = 3.0d0 * (sqrt(x) * y)
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-5.2d0)) then
        tmp = t_1
    else if (y <= (-4d-272)) then
        tmp = t_2
    else if (y <= 1.2d-131) then
        tmp = t_0
    else if (y <= 2.85d-64) then
        tmp = t_2
    else if (y <= 1950000.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	double t_1 = 3.0 * (Math.sqrt(x) * y);
	double t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = t_1;
	} else if (y <= -4e-272) {
		tmp = t_2;
	} else if (y <= 1.2e-131) {
		tmp = t_0;
	} else if (y <= 2.85e-64) {
		tmp = t_2;
	} else if (y <= 1950000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(((0.1111111111111111 / x) + -2.0))
	t_1 = 3.0 * (math.sqrt(x) * y)
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -5.2:
		tmp = t_1
	elif y <= -4e-272:
		tmp = t_2
	elif y <= 1.2e-131:
		tmp = t_0
	elif y <= 2.85e-64:
		tmp = t_2
	elif y <= 1950000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0))
	t_1 = Float64(3.0 * Float64(sqrt(x) * y))
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -5.2)
		tmp = t_1;
	elseif (y <= -4e-272)
		tmp = t_2;
	elseif (y <= 1.2e-131)
		tmp = t_0;
	elseif (y <= 2.85e-64)
		tmp = t_2;
	elseif (y <= 1950000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	t_1 = 3.0 * (sqrt(x) * y);
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -5.2)
		tmp = t_1;
	elseif (y <= -4e-272)
		tmp = t_2;
	elseif (y <= 1.2e-131)
		tmp = t_0;
	elseif (y <= 2.85e-64)
		tmp = t_2;
	elseif (y <= 1950000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -5.2], t$95$1, If[LessEqual[y, -4e-272], t$95$2, If[LessEqual[y, 1.2e-131], t$95$0, If[LessEqual[y, 2.85e-64], t$95$2, If[LessEqual[y, 1950000.0], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.20000000000000018 or 1.95e6 < y

   1. Initial program 99.4%

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

      1. *-commutative 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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 81.0%

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

   if -5.20000000000000018 < y < -3.99999999999999972e-272 or 1.2e-131 < y < 2.8500000000000001e-64

   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* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. div-inv 99.5%

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

      3. associate-*r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 67.4%

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

   8. Taylor expanded in y around 0 66.4%

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

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -3.99999999999999972e-272 < y < 1.2e-131 or 2.8500000000000001e-64 < y < 1.95e6

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 68.3%

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

      12. sqrt-unprod 69.1%

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

      13. swap-sqr 26.2%

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

      14. add-sqr-sqrt 26.3%

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

      15. pow2 26.3%

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

      16. +-commutative 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in x around 0 68.4%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 68.4%

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

      1. sub-neg 68.4%

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

      2. associate-*r/ 68.4%

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

      3. metadata-eval 68.4%

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

      4. metadata-eval 68.4%

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

   10. Simplified 68.4%

       \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x} + -2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -4 \cdot 10^{-272}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-131}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1950000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x} + -2}\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-271}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 66000000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (+ (/ 0.1111111111111111 x) -2.0))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -5.2)
     (* 3.0 (* (sqrt x) y))
     (if (<= y -5.2e-271)
       t_1
       (if (<= y 3.1e-133)
         t_0
         (if (<= y 3.7e-64)
           t_1
           (if (<= y 66000000000.0) t_0 (* (sqrt x) (* y 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= -5.2e-271) {
		tmp = t_1;
	} else if (y <= 3.1e-133) {
		tmp = t_0;
	} else if (y <= 3.7e-64) {
		tmp = t_1;
	} else if (y <= 66000000000.0) {
		tmp = t_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) :: tmp
    t_0 = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-5.2d0)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= (-5.2d-271)) then
        tmp = t_1
    else if (y <= 3.1d-133) then
        tmp = t_0
    else if (y <= 3.7d-64) then
        tmp = t_1
    else if (y <= 66000000000.0d0) then
        tmp = t_0
    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(((0.1111111111111111 / x) + -2.0));
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= -5.2e-271) {
		tmp = t_1;
	} else if (y <= 3.1e-133) {
		tmp = t_0;
	} else if (y <= 3.7e-64) {
		tmp = t_1;
	} else if (y <= 66000000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(((0.1111111111111111 / x) + -2.0))
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -5.2:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= -5.2e-271:
		tmp = t_1
	elif y <= 3.1e-133:
		tmp = t_0
	elif y <= 3.7e-64:
		tmp = t_1
	elif y <= 66000000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -5.2)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= -5.2e-271)
		tmp = t_1;
	elseif (y <= 3.1e-133)
		tmp = t_0;
	elseif (y <= 3.7e-64)
		tmp = t_1;
	elseif (y <= 66000000000.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -5.2)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= -5.2e-271)
		tmp = t_1;
	elseif (y <= 3.1e-133)
		tmp = t_0;
	elseif (y <= 3.7e-64)
		tmp = t_1;
	elseif (y <= 66000000000.0)
		tmp = t_0;
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -5.2], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.2e-271], t$95$1, If[LessEqual[y, 3.1e-133], t$95$0, If[LessEqual[y, 3.7e-64], t$95$1, If[LessEqual[y, 66000000000.0], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.20000000000000018

   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. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 78.3%

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

   if -5.20000000000000018 < y < -5.2e-271 or 3.10000000000000016e-133 < y < 3.69999999999999999e-64

   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* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. div-inv 99.5%

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

      3. associate-*r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 67.4%

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

   8. Taylor expanded in y around 0 66.4%

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

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -5.2e-271 < y < 3.10000000000000016e-133 or 3.69999999999999999e-64 < y < 6.6e10

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 68.3%

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

      12. sqrt-unprod 69.1%

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

      13. swap-sqr 26.2%

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

      14. add-sqr-sqrt 26.3%

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

      15. pow2 26.3%

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

      16. +-commutative 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in x around 0 68.4%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 68.4%

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

      1. sub-neg 68.4%

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

      2. associate-*r/ 68.4%

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

      3. metadata-eval 68.4%

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

      4. metadata-eval 68.4%

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

   10. Simplified 68.4%

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

   if 6.6e10 < y

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 66000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x} + -2}\\ t_1 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (+ (/ 0.1111111111111111 x) -2.0))) (t_1 (* (sqrt x) -3.0)))
   (if (<= y -5.2)
     (* y (sqrt (* x 9.0)))
     (if (<= y -2.25e-273)
       t_1
       (if (<= y 7.2e-124)
         t_0
         (if (<= y 4.1e-65)
           t_1
           (if (<= y 3100000.0) t_0 (* (sqrt x) (* y 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	double t_1 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= -2.25e-273) {
		tmp = t_1;
	} else if (y <= 7.2e-124) {
		tmp = t_0;
	} else if (y <= 4.1e-65) {
		tmp = t_1;
	} else if (y <= 3100000.0) {
		tmp = t_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) :: tmp
    t_0 = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    t_1 = sqrt(x) * (-3.0d0)
    if (y <= (-5.2d0)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= (-2.25d-273)) then
        tmp = t_1
    else if (y <= 7.2d-124) then
        tmp = t_0
    else if (y <= 4.1d-65) then
        tmp = t_1
    else if (y <= 3100000.0d0) then
        tmp = t_0
    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(((0.1111111111111111 / x) + -2.0));
	double t_1 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -5.2) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= -2.25e-273) {
		tmp = t_1;
	} else if (y <= 7.2e-124) {
		tmp = t_0;
	} else if (y <= 4.1e-65) {
		tmp = t_1;
	} else if (y <= 3100000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(((0.1111111111111111 / x) + -2.0))
	t_1 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -5.2:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= -2.25e-273:
		tmp = t_1
	elif y <= 7.2e-124:
		tmp = t_0
	elif y <= 4.1e-65:
		tmp = t_1
	elif y <= 3100000.0:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0))
	t_1 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -5.2)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= -2.25e-273)
		tmp = t_1;
	elseif (y <= 7.2e-124)
		tmp = t_0;
	elseif (y <= 4.1e-65)
		tmp = t_1;
	elseif (y <= 3100000.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	t_1 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -5.2)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= -2.25e-273)
		tmp = t_1;
	elseif (y <= 7.2e-124)
		tmp = t_0;
	elseif (y <= 4.1e-65)
		tmp = t_1;
	elseif (y <= 3100000.0)
		tmp = t_0;
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -5.2], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.25e-273], t$95$1, If[LessEqual[y, 7.2e-124], t$95$0, If[LessEqual[y, 4.1e-65], t$95$1, If[LessEqual[y, 3100000.0], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.20000000000000018

   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. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. clear-num 99.4%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.5%

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

      15. *-commutative 99.5%

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

      16. metadata-eval 99.5%

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

      17. sqrt-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around inf 78.4%

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

   if -5.20000000000000018 < y < -2.2499999999999998e-273 or 7.20000000000000019e-124 < y < 4.09999999999999987e-65

   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* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. div-inv 99.5%

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

      3. associate-*r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 67.4%

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

   8. Taylor expanded in y around 0 66.4%

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

      1. *-commutative 66.4%

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

   10. Simplified 66.4%

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

   if -2.2499999999999998e-273 < y < 7.20000000000000019e-124 or 4.09999999999999987e-65 < y < 3.1e6

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 68.3%

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

      12. sqrt-unprod 69.1%

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

      13. swap-sqr 26.2%

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

      14. add-sqr-sqrt 26.3%

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

      15. pow2 26.3%

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

      16. +-commutative 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in x around 0 68.4%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 68.4%

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

      1. sub-neg 68.4%

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

      2. associate-*r/ 68.4%

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

      3. metadata-eval 68.4%

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

      4. metadata-eval 68.4%

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

   10. Simplified 68.4%

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

   if 3.1e6 < y

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-273}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-124}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3100000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}\\ \mathbf{elif}\;y \leq 920000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (+ (/ 0.1111111111111111 x) -2.0))))
   (if (<= y -5.2)
     (* y (sqrt (* x 9.0)))
     (if (<= y -8.8e-274)
       (* (sqrt x) -3.0)
       (if (<= y 2.55e-129)
         t_0
         (if (<= y 2.8e-64)
           (/ 1.0 (* -0.3333333333333333 (pow x -0.5)))
           (if (<= y 920000000.0) t_0 (* (sqrt x) (* y 3.0)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	double tmp;
	if (y <= -5.2) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= -8.8e-274) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 2.55e-129) {
		tmp = t_0;
	} else if (y <= 2.8e-64) {
		tmp = 1.0 / (-0.3333333333333333 * pow(x, -0.5));
	} else if (y <= 920000000.0) {
		tmp = t_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) :: tmp
    t_0 = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    if (y <= (-5.2d0)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= (-8.8d-274)) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 2.55d-129) then
        tmp = t_0
    else if (y <= 2.8d-64) then
        tmp = 1.0d0 / ((-0.3333333333333333d0) * (x ** (-0.5d0)))
    else if (y <= 920000000.0d0) then
        tmp = t_0
    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(((0.1111111111111111 / x) + -2.0));
	double tmp;
	if (y <= -5.2) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= -8.8e-274) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 2.55e-129) {
		tmp = t_0;
	} else if (y <= 2.8e-64) {
		tmp = 1.0 / (-0.3333333333333333 * Math.pow(x, -0.5));
	} else if (y <= 920000000.0) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(((0.1111111111111111 / x) + -2.0))
	tmp = 0
	if y <= -5.2:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= -8.8e-274:
		tmp = math.sqrt(x) * -3.0
	elif y <= 2.55e-129:
		tmp = t_0
	elif y <= 2.8e-64:
		tmp = 1.0 / (-0.3333333333333333 * math.pow(x, -0.5))
	elif y <= 920000000.0:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0))
	tmp = 0.0
	if (y <= -5.2)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= -8.8e-274)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 2.55e-129)
		tmp = t_0;
	elseif (y <= 2.8e-64)
		tmp = Float64(1.0 / Float64(-0.3333333333333333 * (x ^ -0.5)));
	elseif (y <= 920000000.0)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	tmp = 0.0;
	if (y <= -5.2)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= -8.8e-274)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 2.55e-129)
		tmp = t_0;
	elseif (y <= 2.8e-64)
		tmp = 1.0 / (-0.3333333333333333 * (x ^ -0.5));
	elseif (y <= 920000000.0)
		tmp = t_0;
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -5.2], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -8.8e-274], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 2.55e-129], t$95$0, If[LessEqual[y, 2.8e-64], N[(1.0 / N[(-0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 920000000.0], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5.20000000000000018

   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. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. clear-num 99.4%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.5%

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

      15. *-commutative 99.5%

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

      16. metadata-eval 99.5%

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

      17. sqrt-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in y around inf 78.4%

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

   if -5.20000000000000018 < y < -8.7999999999999998e-274

   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. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. div-inv 99.5%

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

      3. associate-*r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 65.4%

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

   8. Taylor expanded in y around 0 64.3%

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

      1. *-commutative 64.3%

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

   10. Simplified 64.3%

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

   if -8.7999999999999998e-274 < y < 2.5499999999999999e-129 or 2.80000000000000004e-64 < y < 9.2e8

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 68.3%

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

      12. sqrt-unprod 69.1%

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

      13. swap-sqr 26.2%

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

      14. add-sqr-sqrt 26.3%

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

      15. pow2 26.3%

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

      16. +-commutative 26.3%

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

   6. Applied egg-rr 26.3%

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

   7. Taylor expanded in x around 0 68.4%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 68.4%

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

      1. sub-neg 68.4%

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

      2. associate-*r/ 68.4%

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

      3. metadata-eval 68.4%

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

      4. metadata-eval 68.4%

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

   10. Simplified 68.4%

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

   if 2.5499999999999999e-129 < y < 2.80000000000000004e-64

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.1%

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

      3. +-commutative 99.1%

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

      4. associate--l+ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. sub-neg 99.2%

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

      9. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around 0 99.1%

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

      1. *-commutative 99.1%

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

      2. sub-neg 99.1%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

      5. metadata-eval 99.2%

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

      6. associate-*r* 99.2%

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

      7. distribute-rgt-in 99.2%

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

      8. associate-*l/ 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

   7. Simplified 99.2%

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

      1. flip-+ 80.3%

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

      2. associate-*r/ 80.3%

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

      3. sub-neg 80.3%

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

      4. frac-times 80.3%

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

      5. metadata-eval 80.3%

         \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{\color{blue}{0.1111111111111111}}{x \cdot x} + \left(--3 \cdot -3\right)\right)}{\frac{0.3333333333333333}{x} - -3} \]

      6. pow2 80.3%

         \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{\color{blue}{{x}^{2}}} + \left(--3 \cdot -3\right)\right)}{\frac{0.3333333333333333}{x} - -3} \]

      7. metadata-eval 80.3%

         \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + \left(-\color{blue}{9}\right)\right)}{\frac{0.3333333333333333}{x} - -3} \]

      8. metadata-eval 80.3%

         \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + \color{blue}{-9}\right)}{\frac{0.3333333333333333}{x} - -3} \]

      9. sub-neg 80.3%

         \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + -9\right)}{\color{blue}{\frac{0.3333333333333333}{x} + \left(--3\right)}} \]

      10. metadata-eval 80.3%

          \[\leadsto \frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + -9\right)}{\frac{0.3333333333333333}{x} + \color{blue}{3}} \]

   9. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\frac{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + -9\right)}{\frac{0.3333333333333333}{x} + 3}} \]
   10. Step-by-step derivation

       1. clear-num 80.3%

          \[\leadsto \color{blue}{\frac{1}{\frac{\frac{0.3333333333333333}{x} + 3}{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + -9\right)}}} \]

       2. inv-pow 80.3%

          \[\leadsto \color{blue}{{\left(\frac{\frac{0.3333333333333333}{x} + 3}{\sqrt{x} \cdot \left(\frac{0.1111111111111111}{{x}^{2}} + -9\right)}\right)}^{-1}} \]

   11. Applied egg-rr 99.4%

       \[\leadsto \color{blue}{{\left({x}^{-0.5} \cdot \frac{1}{\frac{0.3333333333333333}{x} + -3}\right)}^{-1}} \]
   12. Step-by-step derivation

       1. unpow-1 99.4%

          \[\leadsto \color{blue}{\frac{1}{{x}^{-0.5} \cdot \frac{1}{\frac{0.3333333333333333}{x} + -3}}} \]

       2. *-commutative 99.4%

          \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.3333333333333333}{x} + -3} \cdot {x}^{-0.5}}} \]

       3. +-commutative 99.4%

          \[\leadsto \frac{1}{\frac{1}{\color{blue}{-3 + \frac{0.3333333333333333}{x}}} \cdot {x}^{-0.5}} \]

   13. Simplified 99.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{1}{-3 + \frac{0.3333333333333333}{x}} \cdot {x}^{-0.5}}} \]

   14. Taylor expanded in x around inf 79.8%

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

   if 9.2e8 < y

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot y\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq -8.8 \cdot 10^{-274}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-129}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-64}:\\ \;\;\;\;\frac{1}{-0.3333333333333333 \cdot {x}^{-0.5}}\\ \mathbf{elif}\;y \leq 920000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{if}\;x \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-23}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (+ (/ 0.1111111111111111 x) -2.0))))
   (if (<= x 2.7e-53)
     t_0
     (if (<= x 6e-23)
       (* 3.0 (* (sqrt x) y))
       (if (<= x 4.6e-8) t_0 (* (sqrt x) (* 3.0 (+ y -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	double tmp;
	if (x <= 2.7e-53) {
		tmp = t_0;
	} else if (x <= 6e-23) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 4.6e-8) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    if (x <= 2.7d-53) then
        tmp = t_0
    else if (x <= 6d-23) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 4.6d-8) then
        tmp = t_0
    else
        tmp = sqrt(x) * (3.0d0 * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	double tmp;
	if (x <= 2.7e-53) {
		tmp = t_0;
	} else if (x <= 6e-23) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 4.6e-8) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(((0.1111111111111111 / x) + -2.0))
	tmp = 0
	if x <= 2.7e-53:
		tmp = t_0
	elif x <= 6e-23:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 4.6e-8:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0))
	tmp = 0.0
	if (x <= 2.7e-53)
		tmp = t_0;
	elseif (x <= 6e-23)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 4.6e-8)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(((0.1111111111111111 / x) + -2.0));
	tmp = 0.0;
	if (x <= 2.7e-53)
		tmp = t_0;
	elseif (x <= 6e-23)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 4.6e-8)
		tmp = t_0;
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.7e-53], t$95$0, If[LessEqual[x, 6e-23], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-8], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.6999999999999999e-53 or 6.00000000000000006e-23 < x < 4.6000000000000002e-8

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.4%

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

      11. add-sqr-sqrt 91.5%

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

      12. sqrt-unprod 88.4%

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

      13. swap-sqr 31.8%

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

      14. add-sqr-sqrt 31.9%

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

      15. pow2 31.9%

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

      16. +-commutative 31.9%

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

   6. Applied egg-rr 31.9%

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

   7. Taylor expanded in x around 0 79.1%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 78.4%

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

      1. sub-neg 78.4%

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

      2. associate-*r/ 78.5%

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

      3. metadata-eval 78.5%

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

      4. metadata-eval 78.5%

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

   10. Simplified 78.5%

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

   if 2.6999999999999999e-53 < x < 6.00000000000000006e-23

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. +-commutative 99.1%

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

      4. associate--l+ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-/r* 99.1%

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

      7. metadata-eval 99.1%

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

      8. sub-neg 99.1%

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

      9. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around inf 73.1%

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

   if 4.6000000000000002e-8 < 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. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 98.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.7 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-23}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 16000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -4e+46)
   (* y (sqrt (* x 9.0)))
   (if (<= y 16000000000.0)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -4e+46) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= 16000000000.0) {
		tmp = sqrt(x) * ((0.3333333333333333 / 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) :: tmp
    if (y <= (-4d+46)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= 16000000000.0d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / 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 tmp;
	if (y <= -4e+46) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= 16000000000.0) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -4e+46:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= 16000000000.0:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -4e+46)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= 16000000000.0)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -4e+46)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= 16000000000.0)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -4e+46], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 16000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4e46

   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* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. clear-num 99.4%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.5%

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

      15. *-commutative 99.5%

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

      16. metadata-eval 99.5%

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

      17. sqrt-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around inf 82.5%

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

   if -4e46 < y < 1.6e10

   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. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. sub-neg 95.3%

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

      3. associate-*r/ 95.2%

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

      4. metadata-eval 95.2%

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

      5. metadata-eval 95.2%

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

      6. associate-*r* 95.3%

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

      7. distribute-rgt-in 95.3%

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

      8. associate-*l/ 95.4%

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

      9. metadata-eval 95.4%

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

      10. metadata-eval 95.4%

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

      11. *-commutative 95.4%

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

   7. Simplified 95.4%

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

   if 1.6e10 < y

   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* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 16000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return sqrt(x) * (3.0 * ((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) * (3.0d0 * ((0.1111111111111111d0 / x) + (y + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   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. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 9: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e-5) (sqrt (+ (/ 0.1111111111111111 x) -2.0)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-5) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.8d-5) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.8e-5) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.8e-5:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.8e-5)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.8e-5)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.8e-5], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.8000000000000002e-5

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      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. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.4%

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

      11. add-sqr-sqrt 88.0%

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

      12. sqrt-unprod 83.7%

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

      13. swap-sqr 32.9%

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

      14. add-sqr-sqrt 33.0%

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

      15. pow2 33.0%

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

      16. +-commutative 33.0%

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

   6. Applied egg-rr 33.0%

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

   7. Taylor expanded in x around 0 73.9%

      \[\leadsto \sqrt{\color{blue}{0.6666666666666666 \cdot \left(3 \cdot y - 3\right) + 0.1111111111111111 \cdot \frac{1}{x}}} \]

   8. Taylor expanded in y around 0 73.3%

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

      1. sub-neg 73.3%

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

      2. associate-*r/ 73.3%

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

      3. metadata-eval 73.3%

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

      4. metadata-eval 73.3%

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

   10. Simplified 73.3%

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

   if 3.8000000000000002e-5 < 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. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. distribute-lft-in 99.5%

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

      2. div-inv 99.5%

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

      3. associate-*r* 99.5%

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

      4. metadata-eval 99.5%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 98.8%

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

   8. Taylor expanded in y around 0 47.9%

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

      1. *-commutative 47.9%

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

   10. Simplified 47.9%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. distribute-lft-in 99.5%

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

   2. div-inv 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. +-commutative 99.5%

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

   7. distribute-rgt-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around inf 65.7%

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

8. Taylor expanded in y around 0 26.7%

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

   1. *-commutative 26.7%

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

10. Simplified 26.7%

    \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
11. 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 3.1%

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

    3. swap-sqr 3.1%

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

    4. add-sqr-sqrt 3.1%

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

    5. metadata-eval 3.1%

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

    6. pow1/2 3.1%

       \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \]

12. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \]
13. Step-by-step derivation

    1. unpow1/2 3.1%

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

14. Simplified 3.1%

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

15. Final simplification 3.1%

    \[\leadsto \sqrt{x \cdot 9} \]
16. Add Preprocessing

Alternative 11: 26.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. distribute-lft-in 99.5%

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

   2. div-inv 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. +-commutative 99.5%

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

   7. distribute-rgt-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around inf 65.7%

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

8. Taylor expanded in y around 0 26.7%

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

   1. *-commutative 26.7%

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

10. Simplified 26.7%

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

11. Final simplification 26.7%

    \[\leadsto \sqrt{x} \cdot -3 \]
12. 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 2024031 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (* 3.0 (+ (* y (sqrt x)) (* (- (/ 1.0 (* x 9.0)) 1.0) (sqrt x))))

  (* (* 3.0 (sqrt x)) (- (+ y (/ 1.0 (* x 9.0))) 1.0)))