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

Percentage Accurate: 99.4% → 99.4%

Time: 9.9s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l* 99.3%

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

   2. associate--l+ 99.3%

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

   3. sub-neg 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. associate-*r* 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-commutative 99.3%

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

   4. metadata-eval 99.3%

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

   5. sqrt-prod 99.3%

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

   6. *-commutative 99.3%

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

   7. metadata-eval 99.3%

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

   8. sqrt-prod 99.5%

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

   9. +-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. associate-+r+ 99.5%

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

   3. metadata-eval 99.5%

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

   4. sub-neg 99.5%

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

   5. associate-+l- 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;x \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{3}{x} \cdot \frac{\sqrt{x}}{9}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;t_0 \cdot y\\ \mathbf{elif}\;x \leq 0.00062:\\ \;\;\;\;t_0 \cdot \left(-1 - \frac{-0.1111111111111111}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))))
   (if (<= x 2.5e-43)
     (* (/ 3.0 x) (/ (sqrt x) 9.0))
     (if (<= x 2.9e-24)
       (* t_0 y)
       (if (<= x 0.00062)
         (* t_0 (- -1.0 (/ -0.1111111111111111 x)))
         (* t_0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double tmp;
	if (x <= 2.5e-43) {
		tmp = (3.0 / x) * (sqrt(x) / 9.0);
	} else if (x <= 2.9e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.00062) {
		tmp = t_0 * (-1.0 - (-0.1111111111111111 / x));
	} else {
		tmp = t_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((x * 9.0d0))
    if (x <= 2.5d-43) then
        tmp = (3.0d0 / x) * (sqrt(x) / 9.0d0)
    else if (x <= 2.9d-24) then
        tmp = t_0 * y
    else if (x <= 0.00062d0) then
        tmp = t_0 * ((-1.0d0) - ((-0.1111111111111111d0) / x))
    else
        tmp = t_0 * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double tmp;
	if (x <= 2.5e-43) {
		tmp = (3.0 / x) * (Math.sqrt(x) / 9.0);
	} else if (x <= 2.9e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.00062) {
		tmp = t_0 * (-1.0 - (-0.1111111111111111 / x));
	} else {
		tmp = t_0 * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	tmp = 0
	if x <= 2.5e-43:
		tmp = (3.0 / x) * (math.sqrt(x) / 9.0)
	elif x <= 2.9e-24:
		tmp = t_0 * y
	elif x <= 0.00062:
		tmp = t_0 * (-1.0 - (-0.1111111111111111 / x))
	else:
		tmp = t_0 * (y + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	tmp = 0.0
	if (x <= 2.5e-43)
		tmp = Float64(Float64(3.0 / x) * Float64(sqrt(x) / 9.0));
	elseif (x <= 2.9e-24)
		tmp = Float64(t_0 * y);
	elseif (x <= 0.00062)
		tmp = Float64(t_0 * Float64(-1.0 - Float64(-0.1111111111111111 / x)));
	else
		tmp = Float64(t_0 * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	tmp = 0.0;
	if (x <= 2.5e-43)
		tmp = (3.0 / x) * (sqrt(x) / 9.0);
	elseif (x <= 2.9e-24)
		tmp = t_0 * y;
	elseif (x <= 0.00062)
		tmp = t_0 * (-1.0 - (-0.1111111111111111 / x));
	else
		tmp = t_0 * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 2.5e-43], N[(N[(3.0 / x), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.9e-24], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[x, 0.00062], N[(t$95$0 * N[(-1.0 - N[(-0.1111111111111111 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.50000000000000009e-43

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. associate-*r* 80.6%

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

      2. clear-num 80.6%

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

      3. un-div-inv 80.6%

         \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 80.7%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 80.7%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.7%

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

      1. times-frac 80.7%

         \[\leadsto \color{blue}{\frac{3}{x} \cdot \frac{\sqrt{x}}{9}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{3}{x} \cdot \frac{\sqrt{x}}{9}} \]

   if 2.50000000000000009e-43 < x < 2.8999999999999999e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 2.8999999999999999e-24 < x < 6.2e-4

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. sqrt-prod 99.6%

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

      9. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-+r+ 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-+l- 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 74.0%

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

      1. sub-neg 74.0%

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

      2. associate-*r/ 74.0%

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

      3. metadata-eval 74.0%

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

      4. metadata-eval 74.0%

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

      5. +-commutative 74.0%

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

      6. metadata-eval 74.0%

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

      7. distribute-neg-frac 74.0%

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

      8. unsub-neg 74.0%

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

   10. Simplified 74.0%

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

   if 6.2e-4 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.4%

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

      6. *-commutative 99.4%

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

      7. metadata-eval 99.4%

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

      8. sqrt-prod 99.6%

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

      9. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-+r+ 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-+l- 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.0%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{3}{x} \cdot \frac{\sqrt{x}}{9}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 0.00062:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(-1 - \frac{-0.1111111111111111}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 3: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{if}\;x \leq 2 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+259}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (sqrt x) x)))
        (t_1 (* 3.0 (* y (sqrt x)))))
   (if (<= x 2e-42)
     t_0
     (if (<= x 7e-24)
       t_1
       (if (<= x 2e-10) t_0 (if (<= x 4.5e+259) t_1 (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 * (sqrt(x) / x);
	double t_1 = 3.0 * (y * sqrt(x));
	double tmp;
	if (x <= 2e-42) {
		tmp = t_0;
	} else if (x <= 7e-24) {
		tmp = t_1;
	} else if (x <= 2e-10) {
		tmp = t_0;
	} else if (x <= 4.5e+259) {
		tmp = t_1;
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (sqrt(x) / x)
    t_1 = 3.0d0 * (y * sqrt(x))
    if (x <= 2d-42) then
        tmp = t_0
    else if (x <= 7d-24) then
        tmp = t_1
    else if (x <= 2d-10) then
        tmp = t_0
    else if (x <= 4.5d+259) then
        tmp = t_1
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.3333333333333333 * (Math.sqrt(x) / x);
	double t_1 = 3.0 * (y * Math.sqrt(x));
	double tmp;
	if (x <= 2e-42) {
		tmp = t_0;
	} else if (x <= 7e-24) {
		tmp = t_1;
	} else if (x <= 2e-10) {
		tmp = t_0;
	} else if (x <= 4.5e+259) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.3333333333333333 * (math.sqrt(x) / x)
	t_1 = 3.0 * (y * math.sqrt(x))
	tmp = 0
	if x <= 2e-42:
		tmp = t_0
	elif x <= 7e-24:
		tmp = t_1
	elif x <= 2e-10:
		tmp = t_0
	elif x <= 4.5e+259:
		tmp = t_1
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 * Float64(sqrt(x) / x))
	t_1 = Float64(3.0 * Float64(y * sqrt(x)))
	tmp = 0.0
	if (x <= 2e-42)
		tmp = t_0;
	elseif (x <= 7e-24)
		tmp = t_1;
	elseif (x <= 2e-10)
		tmp = t_0;
	elseif (x <= 4.5e+259)
		tmp = t_1;
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.3333333333333333 * (sqrt(x) / x);
	t_1 = 3.0 * (y * sqrt(x));
	tmp = 0.0;
	if (x <= 2e-42)
		tmp = t_0;
	elseif (x <= 7e-24)
		tmp = t_1;
	elseif (x <= 2e-10)
		tmp = t_0;
	elseif (x <= 4.5e+259)
		tmp = t_1;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2e-42], t$95$0, If[LessEqual[x, 7e-24], t$95$1, If[LessEqual[x, 2e-10], t$95$0, If[LessEqual[x, 4.5e+259], t$95$1, N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 2.00000000000000008e-42 or 6.9999999999999993e-24 < x < 2.00000000000000007e-10

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 80.2%

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

      1. associate-*r* 80.3%

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

      2. clear-num 80.3%

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

      3. un-div-inv 80.3%

         \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{9 \cdot x}} \]

      2. times-frac 80.3%

         \[\leadsto \color{blue}{\frac{3}{9} \cdot \frac{\sqrt{x}}{x}} \]

      3. metadata-eval 80.3%

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

   8. Simplified 80.3%

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

   if 2.00000000000000008e-42 < x < 6.9999999999999993e-24 or 2.00000000000000007e-10 < x < 4.4999999999999997e259

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-def 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 61.5%

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

   if 4.4999999999999997e259 < x

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.2%

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

      2. distribute-lft-in 99.2%

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

      3. *-commutative 99.2%

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

      4. metadata-eval 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. sqrt-prod 99.9%

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

      9. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-+r+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l- 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.9%

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

   9. Taylor expanded in y around 0 67.7%

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

       1. *-commutative 67.7%

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

   11. Simplified 67.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-42}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+259}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 4: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{if}\;x \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (sqrt x) x))))
   (if (<= x 4.2e-42)
     t_0
     (if (<= x 7.5e-24)
       (* 3.0 (* y (sqrt x)))
       (if (<= x 1.9e-10)
         t_0
         (if (<= x 1.35e+259) (* (sqrt x) (* y 3.0)) (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 * (sqrt(x) / x);
	double tmp;
	if (x <= 4.2e-42) {
		tmp = t_0;
	} else if (x <= 7.5e-24) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (x <= 1.9e-10) {
		tmp = t_0;
	} else if (x <= 1.35e+259) {
		tmp = sqrt(x) * (y * 3.0);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (sqrt(x) / x)
    if (x <= 4.2d-42) then
        tmp = t_0
    else if (x <= 7.5d-24) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (x <= 1.9d-10) then
        tmp = t_0
    else if (x <= 1.35d+259) then
        tmp = sqrt(x) * (y * 3.0d0)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.3333333333333333 * (Math.sqrt(x) / x);
	double tmp;
	if (x <= 4.2e-42) {
		tmp = t_0;
	} else if (x <= 7.5e-24) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (x <= 1.9e-10) {
		tmp = t_0;
	} else if (x <= 1.35e+259) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.3333333333333333 * (math.sqrt(x) / x)
	tmp = 0
	if x <= 4.2e-42:
		tmp = t_0
	elif x <= 7.5e-24:
		tmp = 3.0 * (y * math.sqrt(x))
	elif x <= 1.9e-10:
		tmp = t_0
	elif x <= 1.35e+259:
		tmp = math.sqrt(x) * (y * 3.0)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 * Float64(sqrt(x) / x))
	tmp = 0.0
	if (x <= 4.2e-42)
		tmp = t_0;
	elseif (x <= 7.5e-24)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (x <= 1.9e-10)
		tmp = t_0;
	elseif (x <= 1.35e+259)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.3333333333333333 * (sqrt(x) / x);
	tmp = 0.0;
	if (x <= 4.2e-42)
		tmp = t_0;
	elseif (x <= 7.5e-24)
		tmp = 3.0 * (y * sqrt(x));
	elseif (x <= 1.9e-10)
		tmp = t_0;
	elseif (x <= 1.35e+259)
		tmp = sqrt(x) * (y * 3.0);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4.2e-42], t$95$0, If[LessEqual[x, 7.5e-24], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9e-10], t$95$0, If[LessEqual[x, 1.35e+259], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 4.20000000000000013e-42 or 7.50000000000000007e-24 < x < 1.8999999999999999e-10

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in x around 0 80.2%

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

      1. associate-*r* 80.3%

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

      2. clear-num 80.3%

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

      3. un-div-inv 80.3%

         \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.4%

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

      1. *-commutative 80.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{9 \cdot x}} \]

      2. times-frac 80.3%

         \[\leadsto \color{blue}{\frac{3}{9} \cdot \frac{\sqrt{x}}{x}} \]

      3. metadata-eval 80.3%

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

   8. Simplified 80.3%

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

   if 4.20000000000000013e-42 < x < 7.50000000000000007e-24

   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* 98.8%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. fma-def 98.8%

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

      6. sub-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. distribute-lft-in 98.8%

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

      9. metadata-eval 98.8%

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

      10. metadata-eval 98.8%

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

      11. *-commutative 98.8%

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

      12. associate-/r* 99.0%

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

      13. associate-*r/ 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around inf 77.7%

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

   if 1.8999999999999999e-10 < x < 1.34999999999999994e259

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 60.0%

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

      3. *-commutative 60.0%

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

   6. Simplified 60.0%

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

   if 1.34999999999999994e259 < x

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.2%

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

      2. distribute-lft-in 99.2%

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

      3. *-commutative 99.2%

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

      4. metadata-eval 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. sqrt-prod 99.9%

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

      9. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-+r+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l- 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.9%

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

   9. Taylor expanded in y around 0 67.7%

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

       1. *-commutative 67.7%

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

   11. Simplified 67.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{-42}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 5: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;x \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= x 1.55e-43)
     t_0
     (if (<= x 6.8e-24)
       (* 3.0 (* y (sqrt x)))
       (if (<= x 2.5e-10)
         t_0
         (if (<= x 3e+259) (* (sqrt x) (* y 3.0)) (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 1.55e-43) {
		tmp = t_0;
	} else if (x <= 6.8e-24) {
		tmp = 3.0 * (y * sqrt(x));
	} else if (x <= 2.5e-10) {
		tmp = t_0;
	} else if (x <= 3e+259) {
		tmp = sqrt(x) * (y * 3.0);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (x <= 1.55d-43) then
        tmp = t_0
    else if (x <= 6.8d-24) then
        tmp = 3.0d0 * (y * sqrt(x))
    else if (x <= 2.5d-10) then
        tmp = t_0
    else if (x <= 3d+259) then
        tmp = sqrt(x) * (y * 3.0d0)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 1.55e-43) {
		tmp = t_0;
	} else if (x <= 6.8e-24) {
		tmp = 3.0 * (y * Math.sqrt(x));
	} else if (x <= 2.5e-10) {
		tmp = t_0;
	} else if (x <= 3e+259) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if x <= 1.55e-43:
		tmp = t_0
	elif x <= 6.8e-24:
		tmp = 3.0 * (y * math.sqrt(x))
	elif x <= 2.5e-10:
		tmp = t_0
	elif x <= 3e+259:
		tmp = math.sqrt(x) * (y * 3.0)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (x <= 1.55e-43)
		tmp = t_0;
	elseif (x <= 6.8e-24)
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	elseif (x <= 2.5e-10)
		tmp = t_0;
	elseif (x <= 3e+259)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (x <= 1.55e-43)
		tmp = t_0;
	elseif (x <= 6.8e-24)
		tmp = 3.0 * (y * sqrt(x));
	elseif (x <= 2.5e-10)
		tmp = t_0;
	elseif (x <= 3e+259)
		tmp = sqrt(x) * (y * 3.0);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.55e-43], t$95$0, If[LessEqual[x, 6.8e-24], N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-10], t$95$0, If[LessEqual[x, 3e+259], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 1.55e-43 or 6.79999999999999985e-24 < x < 2.50000000000000016e-10

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 80.3%

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

   if 1.55e-43 < x < 6.79999999999999985e-24

   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* 98.8%

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

      3. associate--l+ 98.8%

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

      4. distribute-lft-in 98.8%

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

      5. fma-def 98.8%

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

      6. sub-neg 98.8%

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

      7. +-commutative 98.8%

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

      8. distribute-lft-in 98.8%

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

      9. metadata-eval 98.8%

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

      10. metadata-eval 98.8%

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

      11. *-commutative 98.8%

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

      12. associate-/r* 99.0%

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

      13. associate-*r/ 99.0%

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

      14. metadata-eval 99.0%

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

      15. metadata-eval 99.0%

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

   3. Simplified 99.0%

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

   4. Taylor expanded in y around inf 77.7%

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

   if 2.50000000000000016e-10 < x < 3.00000000000000013e259

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 60.0%

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

      3. *-commutative 60.0%

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

   6. Simplified 60.0%

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

   if 3.00000000000000013e259 < x

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.2%

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

      2. distribute-lft-in 99.2%

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

      3. *-commutative 99.2%

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

      4. metadata-eval 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. sqrt-prod 99.9%

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

      9. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-+r+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l- 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.9%

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

   9. Taylor expanded in y around 0 67.7%

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

       1. *-commutative 67.7%

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

   11. Simplified 67.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 6: 62.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= x 3.3e-43)
     t_0
     (if (<= x 4e-24)
       (* (sqrt (* x 9.0)) y)
       (if (<= x 2e-10)
         t_0
         (if (<= x 3.4e+259) (* (sqrt x) (* y 3.0)) (* (sqrt x) -3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 3.3e-43) {
		tmp = t_0;
	} else if (x <= 4e-24) {
		tmp = sqrt((x * 9.0)) * y;
	} else if (x <= 2e-10) {
		tmp = t_0;
	} else if (x <= 3.4e+259) {
		tmp = sqrt(x) * (y * 3.0);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (x <= 3.3d-43) then
        tmp = t_0
    else if (x <= 4d-24) then
        tmp = sqrt((x * 9.0d0)) * y
    else if (x <= 2d-10) then
        tmp = t_0
    else if (x <= 3.4d+259) then
        tmp = sqrt(x) * (y * 3.0d0)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 3.3e-43) {
		tmp = t_0;
	} else if (x <= 4e-24) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else if (x <= 2e-10) {
		tmp = t_0;
	} else if (x <= 3.4e+259) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if x <= 3.3e-43:
		tmp = t_0
	elif x <= 4e-24:
		tmp = math.sqrt((x * 9.0)) * y
	elif x <= 2e-10:
		tmp = t_0
	elif x <= 3.4e+259:
		tmp = math.sqrt(x) * (y * 3.0)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (x <= 3.3e-43)
		tmp = t_0;
	elseif (x <= 4e-24)
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	elseif (x <= 2e-10)
		tmp = t_0;
	elseif (x <= 3.4e+259)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (x <= 3.3e-43)
		tmp = t_0;
	elseif (x <= 4e-24)
		tmp = sqrt((x * 9.0)) * y;
	elseif (x <= 2e-10)
		tmp = t_0;
	elseif (x <= 3.4e+259)
		tmp = sqrt(x) * (y * 3.0);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 3.3e-43], t$95$0, If[LessEqual[x, 4e-24], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2e-10], t$95$0, If[LessEqual[x, 3.4e+259], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.30000000000000016e-43 or 3.99999999999999969e-24 < x < 2.00000000000000007e-10

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 80.3%

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

   if 3.30000000000000016e-43 < x < 3.99999999999999969e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 2.00000000000000007e-10 < x < 3.39999999999999989e259

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 59.9%

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

      1. *-commutative 59.9%

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

      2. associate-*l* 60.0%

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

      3. *-commutative 60.0%

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

   6. Simplified 60.0%

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

   if 3.39999999999999989e259 < x

   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. associate-*l* 99.3%

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

      2. associate--l+ 99.3%

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

      3. sub-neg 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-/r* 99.3%

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

      6. metadata-eval 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.2%

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

      2. distribute-lft-in 99.2%

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

      3. *-commutative 99.2%

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

      4. metadata-eval 99.2%

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

      5. sqrt-prod 99.3%

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

      6. *-commutative 99.3%

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

      7. metadata-eval 99.3%

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

      8. sqrt-prod 99.9%

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

      9. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. distribute-lft-out 99.9%

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

      2. associate-+r+ 99.9%

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

      3. metadata-eval 99.9%

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

      4. sub-neg 99.9%

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

      5. associate-+l- 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in x around inf 99.9%

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

   9. Taylor expanded in y around 0 67.7%

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

       1. *-commutative 67.7%

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

   11. Simplified 67.7%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 4 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{+259}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 7: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;x \leq 7 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= x 7e-42)
     t_0
     (if (<= x 6.8e-24)
       (* (sqrt (* x 9.0)) y)
       (if (<= x 1.9e-10) t_0 (* 3.0 (* (sqrt x) (+ y -1.0))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 7e-42) {
		tmp = t_0;
	} else if (x <= 6.8e-24) {
		tmp = sqrt((x * 9.0)) * y;
	} else if (x <= 1.9e-10) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (x <= 7d-42) then
        tmp = t_0
    else if (x <= 6.8d-24) then
        tmp = sqrt((x * 9.0d0)) * y
    else if (x <= 1.9d-10) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 7e-42) {
		tmp = t_0;
	} else if (x <= 6.8e-24) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else if (x <= 1.9e-10) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if x <= 7e-42:
		tmp = t_0
	elif x <= 6.8e-24:
		tmp = math.sqrt((x * 9.0)) * y
	elif x <= 1.9e-10:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (x <= 7e-42)
		tmp = t_0;
	elseif (x <= 6.8e-24)
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	elseif (x <= 1.9e-10)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * (0.3333333333333333 / x);
	tmp = 0.0;
	if (x <= 7e-42)
		tmp = t_0;
	elseif (x <= 6.8e-24)
		tmp = sqrt((x * 9.0)) * y;
	elseif (x <= 1.9e-10)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 7e-42], t$95$0, If[LessEqual[x, 6.8e-24], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.9e-10], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 7.0000000000000004e-42 or 6.79999999999999985e-24 < x < 1.8999999999999999e-10

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in x around 0 80.3%

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

   if 7.0000000000000004e-42 < x < 6.79999999999999985e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 1.8999999999999999e-10 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 96.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-42}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 8: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.9e-42)
   (* 3.0 (/ (sqrt x) (* x 9.0)))
   (if (<= x 8.6e-24)
     (* (sqrt (* x 9.0)) y)
     (if (<= x 2.5e-10)
       (* (sqrt x) (/ 0.3333333333333333 x))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-42) {
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	} else if (x <= 8.6e-24) {
		tmp = sqrt((x * 9.0)) * y;
	} else if (x <= 2.5e-10) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.9d-42) then
        tmp = 3.0d0 * (sqrt(x) / (x * 9.0d0))
    else if (x <= 8.6d-24) then
        tmp = sqrt((x * 9.0d0)) * y
    else if (x <= 2.5d-10) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.9e-42) {
		tmp = 3.0 * (Math.sqrt(x) / (x * 9.0));
	} else if (x <= 8.6e-24) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else if (x <= 2.5e-10) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.9e-42:
		tmp = 3.0 * (math.sqrt(x) / (x * 9.0))
	elif x <= 8.6e-24:
		tmp = math.sqrt((x * 9.0)) * y
	elif x <= 2.5e-10:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.9e-42)
		tmp = Float64(3.0 * Float64(sqrt(x) / Float64(x * 9.0)));
	elseif (x <= 8.6e-24)
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	elseif (x <= 2.5e-10)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.9e-42)
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	elseif (x <= 8.6e-24)
		tmp = sqrt((x * 9.0)) * y;
	elseif (x <= 2.5e-10)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.9e-42], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.6e-24], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 2.5e-10], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.9000000000000003e-42

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. clear-num 80.5%

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

      2. un-div-inv 80.6%

         \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{x \cdot 9}} \]

   if 2.9000000000000003e-42 < x < 8.6000000000000006e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 8.6000000000000006e-24 < x < 2.50000000000000016e-10

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.0%

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

      3. associate--l+ 99.0%

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

      4. distribute-lft-in 99.2%

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

      5. fma-def 99.2%

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

      6. sub-neg 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.2%

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

      9. metadata-eval 99.2%

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

      10. metadata-eval 99.2%

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

      11. *-commutative 99.2%

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

      12. associate-/r* 99.1%

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

      13. associate-*r/ 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 76.2%

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

   if 2.50000000000000016e-10 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 96.3%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9 \cdot 10^{-42}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 9: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-42}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 0.0007:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.6e-42)
   (* 3.0 (/ (sqrt x) (* x 9.0)))
   (if (<= x 9.6e-24)
     (* (sqrt (* x 9.0)) y)
     (if (<= x 0.0007)
       (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
       (* 3.0 (* (sqrt x) (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.6e-42) {
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	} else if (x <= 9.6e-24) {
		tmp = sqrt((x * 9.0)) * y;
	} else if (x <= 0.0007) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.6d-42) then
        tmp = 3.0d0 * (sqrt(x) / (x * 9.0d0))
    else if (x <= 9.6d-24) then
        tmp = sqrt((x * 9.0d0)) * y
    else if (x <= 0.0007d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.6e-42) {
		tmp = 3.0 * (Math.sqrt(x) / (x * 9.0));
	} else if (x <= 9.6e-24) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else if (x <= 0.0007) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.6e-42:
		tmp = 3.0 * (math.sqrt(x) / (x * 9.0))
	elif x <= 9.6e-24:
		tmp = math.sqrt((x * 9.0)) * y
	elif x <= 0.0007:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.6e-42)
		tmp = Float64(3.0 * Float64(sqrt(x) / Float64(x * 9.0)));
	elseif (x <= 9.6e-24)
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	elseif (x <= 0.0007)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.6e-42)
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	elseif (x <= 9.6e-24)
		tmp = sqrt((x * 9.0)) * y;
	elseif (x <= 0.0007)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.6e-42], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.6e-24], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 0.0007], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 3.6000000000000002e-42

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. clear-num 80.5%

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

      2. un-div-inv 80.6%

         \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{x \cdot 9}} \]

   if 3.6000000000000002e-42 < x < 9.5999999999999993e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 9.5999999999999993e-24 < x < 6.99999999999999993e-4

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 73.7%

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

      1. sub-neg 73.7%

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

      2. associate-*r/ 73.9%

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

      3. metadata-eval 73.9%

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

      4. metadata-eval 73.9%

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

   6. Simplified 73.9%

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

   if 6.99999999999999993e-4 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 98.8%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{-42}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 9.6 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 0.0007:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 10: 85.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;x \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;t_0 \cdot y\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))))
   (if (<= x 7.6e-44)
     (* 3.0 (/ (sqrt x) (* x 9.0)))
     (if (<= x 6.4e-24)
       (* t_0 y)
       (if (<= x 0.0014)
         (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
         (* t_0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double tmp;
	if (x <= 7.6e-44) {
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	} else if (x <= 6.4e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.0014) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_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((x * 9.0d0))
    if (x <= 7.6d-44) then
        tmp = 3.0d0 * (sqrt(x) / (x * 9.0d0))
    else if (x <= 6.4d-24) then
        tmp = t_0 * y
    else if (x <= 0.0014d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = t_0 * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double tmp;
	if (x <= 7.6e-44) {
		tmp = 3.0 * (Math.sqrt(x) / (x * 9.0));
	} else if (x <= 6.4e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.0014) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_0 * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	tmp = 0
	if x <= 7.6e-44:
		tmp = 3.0 * (math.sqrt(x) / (x * 9.0))
	elif x <= 6.4e-24:
		tmp = t_0 * y
	elif x <= 0.0014:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = t_0 * (y + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	tmp = 0.0
	if (x <= 7.6e-44)
		tmp = Float64(3.0 * Float64(sqrt(x) / Float64(x * 9.0)));
	elseif (x <= 6.4e-24)
		tmp = Float64(t_0 * y);
	elseif (x <= 0.0014)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(t_0 * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	tmp = 0.0;
	if (x <= 7.6e-44)
		tmp = 3.0 * (sqrt(x) / (x * 9.0));
	elseif (x <= 6.4e-24)
		tmp = t_0 * y;
	elseif (x <= 0.0014)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = t_0 * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 7.6e-44], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] / N[(x * 9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e-24], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[x, 0.0014], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 7.6000000000000002e-44

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. clear-num 80.5%

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

      2. un-div-inv 80.6%

         \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      3. div-inv 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      4. metadata-eval 80.7%

         \[\leadsto 3 \cdot \frac{\sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.7%

      \[\leadsto 3 \cdot \color{blue}{\frac{\sqrt{x}}{x \cdot 9}} \]

   if 7.6000000000000002e-44 < x < 6.40000000000000025e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 6.40000000000000025e-24 < x < 0.00139999999999999999

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 73.7%

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

      1. sub-neg 73.7%

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

      2. associate-*r/ 73.9%

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

      3. metadata-eval 73.9%

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

      4. metadata-eval 73.9%

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

   6. Simplified 73.9%

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

   if 0.00139999999999999999 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.4%

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

      6. *-commutative 99.4%

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

      7. metadata-eval 99.4%

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

      8. sqrt-prod 99.6%

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

      9. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-+r+ 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-+l- 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.0%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{-44}:\\ \;\;\;\;3 \cdot \frac{\sqrt{x}}{x \cdot 9}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 0.0014:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 11: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;x \leq 6 \cdot 10^{-42}:\\ \;\;\;\;\frac{3}{x} \cdot \frac{\sqrt{x}}{9}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-24}:\\ \;\;\;\;t_0 \cdot y\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))))
   (if (<= x 6e-42)
     (* (/ 3.0 x) (/ (sqrt x) 9.0))
     (if (<= x 6.1e-24)
       (* t_0 y)
       (if (<= x 0.00092)
         (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
         (* t_0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double tmp;
	if (x <= 6e-42) {
		tmp = (3.0 / x) * (sqrt(x) / 9.0);
	} else if (x <= 6.1e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.00092) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_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((x * 9.0d0))
    if (x <= 6d-42) then
        tmp = (3.0d0 / x) * (sqrt(x) / 9.0d0)
    else if (x <= 6.1d-24) then
        tmp = t_0 * y
    else if (x <= 0.00092d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = t_0 * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double tmp;
	if (x <= 6e-42) {
		tmp = (3.0 / x) * (Math.sqrt(x) / 9.0);
	} else if (x <= 6.1e-24) {
		tmp = t_0 * y;
	} else if (x <= 0.00092) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_0 * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	tmp = 0
	if x <= 6e-42:
		tmp = (3.0 / x) * (math.sqrt(x) / 9.0)
	elif x <= 6.1e-24:
		tmp = t_0 * y
	elif x <= 0.00092:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = t_0 * (y + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	tmp = 0.0
	if (x <= 6e-42)
		tmp = Float64(Float64(3.0 / x) * Float64(sqrt(x) / 9.0));
	elseif (x <= 6.1e-24)
		tmp = Float64(t_0 * y);
	elseif (x <= 0.00092)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(t_0 * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	tmp = 0.0;
	if (x <= 6e-42)
		tmp = (3.0 / x) * (sqrt(x) / 9.0);
	elseif (x <= 6.1e-24)
		tmp = t_0 * y;
	elseif (x <= 0.00092)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = t_0 * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 6e-42], N[(N[(3.0 / x), $MachinePrecision] * N[(N[Sqrt[x], $MachinePrecision] / 9.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.1e-24], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[x, 0.00092], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 6.00000000000000054e-42

   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. associate-*l* 99.2%

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

      2. associate--l+ 99.2%

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

      3. sub-neg 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in x around 0 80.4%

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

      1. associate-*r* 80.6%

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

      2. clear-num 80.6%

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

      3. un-div-inv 80.6%

         \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 80.7%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 80.7%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 80.7%

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

      1. times-frac 80.7%

         \[\leadsto \color{blue}{\frac{3}{x} \cdot \frac{\sqrt{x}}{9}} \]

   8. Simplified 80.7%

      \[\leadsto \color{blue}{\frac{3}{x} \cdot \frac{\sqrt{x}}{9}} \]

   if 6.00000000000000054e-42 < x < 6.10000000000000036e-24

   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. associate-*l* 99.1%

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

      2. associate--l+ 99.1%

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

      3. sub-neg 99.1%

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

      4. *-commutative 99.1%

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

      5. associate-/r* 99.1%

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

      6. metadata-eval 99.1%

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

      7. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.7%

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

      6. *-commutative 99.7%

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

      7. metadata-eval 99.7%

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

      8. sqrt-prod 99.7%

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

      9. +-commutative 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-+r+ 99.7%

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

      3. metadata-eval 99.7%

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

      4. sub-neg 99.7%

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

      5. associate-+l- 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 78.3%

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

   if 6.10000000000000036e-24 < x < 9.2000000000000003e-4

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.3%

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

      5. fma-def 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 73.7%

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

      1. sub-neg 73.7%

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

      2. associate-*r/ 73.9%

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

      3. metadata-eval 73.9%

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

      4. metadata-eval 73.9%

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

   6. Simplified 73.9%

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

   if 9.2000000000000003e-4 < x

   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. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. associate-*r* 99.3%

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

      2. distribute-lft-in 99.3%

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

      3. *-commutative 99.3%

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

      4. metadata-eval 99.3%

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

      5. sqrt-prod 99.4%

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

      6. *-commutative 99.4%

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

      7. metadata-eval 99.4%

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

      8. sqrt-prod 99.6%

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

      9. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. distribute-lft-out 99.6%

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

      2. associate-+r+ 99.6%

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

      3. metadata-eval 99.6%

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

      4. sub-neg 99.6%

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

      5. associate-+l- 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around inf 99.0%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6 \cdot 10^{-42}:\\ \;\;\;\;\frac{3}{x} \cdot \frac{\sqrt{x}}{9}\\ \mathbf{elif}\;x \leq 6.1 \cdot 10^{-24}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;x \leq 0.00092:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 12: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* 3.0 (* (+ y (+ (/ 0.1111111111111111 x) -1.0)) (sqrt x))))

C

double code(double x, double y) {
	return 3.0 * ((y + ((0.1111111111111111 / x) + -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 + ((0.1111111111111111d0 / x) + (-1.0d0))) * sqrt(x))
end function

Java

public static double code(double x, double y) {
	return 3.0 * ((y + ((0.1111111111111111 / x) + -1.0)) * Math.sqrt(x));
}

Python

def code(x, y):
	return 3.0 * ((y + ((0.1111111111111111 / x) + -1.0)) * math.sqrt(x))

Julia

function code(x, y)
	return Float64(3.0 * Float64(Float64(y + Float64(Float64(0.1111111111111111 / x) + -1.0)) * sqrt(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 3.0 * ((y + ((0.1111111111111111 / x) + -1.0)) * sqrt(x));
end

Wolfram

code[x_, y_] := N[(3.0 * N[(N[(y + N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l* 99.3%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   2. associate--l+ 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

   3. sub-neg 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

   4. *-commutative 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

   5. associate-/r* 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

   6. metadata-eval 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

   7. metadata-eval 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]

4. Final simplification 99.3%

   \[\leadsto 3 \cdot \left(\left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right) \cdot \sqrt{x}\right) \]

Alternative 13: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \left(y \cdot 3 + \left(\frac{0.3333333333333333}{x} + -3\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (+ (* y 3.0) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	return sqrt(x) * ((y * 3.0) + ((0.3333333333333333 / x) + -3.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * ((y * 3.0d0) + ((0.3333333333333333d0 / x) + (-3.0d0)))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(x) * ((y * 3.0) + ((0.3333333333333333 / x) + -3.0));
}

Python

def code(x, y):
	return math.sqrt(x) * ((y * 3.0) + ((0.3333333333333333 / x) + -3.0))

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(y * 3.0) + Float64(Float64(0.3333333333333333 / x) + -3.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * ((y * 3.0) + ((0.3333333333333333 / x) + -3.0));
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(y * 3.0), $MachinePrecision] + N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   2. associate-*l* 99.3%

      \[\leadsto \color{blue}{\sqrt{x} \cdot \left(3 \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   3. associate--l+ 99.3%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

   4. distribute-lft-in 99.3%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot y + 3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)\right)} \]

   5. fma-def 99.3%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{fma}\left(3, y, 3 \cdot \left(\frac{1}{x \cdot 9} - 1\right)\right)} \]

   6. sub-neg 99.3%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, 3 \cdot \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right) \]

   7. +-commutative 99.3%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, 3 \cdot \color{blue}{\left(\left(-1\right) + \frac{1}{x \cdot 9}\right)}\right) \]

   8. distribute-lft-in 99.3%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, \color{blue}{3 \cdot \left(-1\right) + 3 \cdot \frac{1}{x \cdot 9}}\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, 3 \cdot \color{blue}{-1} + 3 \cdot \frac{1}{x \cdot 9}\right) \]

   10. metadata-eval 99.3%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, \color{blue}{-3} + 3 \cdot \frac{1}{x \cdot 9}\right) \]

   11. *-commutative 99.3%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + 3 \cdot \frac{1}{\color{blue}{9 \cdot x}}\right) \]

   12. associate-/r* 99.3%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + 3 \cdot \color{blue}{\frac{\frac{1}{9}}{x}}\right) \]

   13. associate-*r/ 99.4%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \color{blue}{\frac{3 \cdot \frac{1}{9}}{x}}\right) \]

   14. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \frac{3 \cdot \color{blue}{0.1111111111111111}}{x}\right) \]

   15. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \frac{\color{blue}{0.3333333333333333}}{x}\right) \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.3%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot y + \left(-3 + \frac{0.3333333333333333}{x}\right)\right)} \]

   2. +-commutative 99.3%

      \[\leadsto \sqrt{x} \cdot \left(3 \cdot y + \color{blue}{\left(\frac{0.3333333333333333}{x} + -3\right)}\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\left(3 \cdot y + \left(\frac{0.3333333333333333}{x} + -3\right)\right)} \]

6. Final simplification 99.3%

   \[\leadsto \sqrt{x} \cdot \left(y \cdot 3 + \left(\frac{0.3333333333333333}{x} + -3\right)\right) \]

Alternative 14: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (* 0.3333333333333333 (/ (sqrt x) x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 0.3333333333333333 * (sqrt(x) / x);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = 0.3333333333333333d0 * (sqrt(x) / x)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 0.3333333333333333 * (Math.sqrt(x) / x);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = 0.3333333333333333 * (math.sqrt(x) / x)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(x) / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = 0.3333333333333333 * (sqrt(x) / x);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(0.3333333333333333 * N[(N[Sqrt[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.2%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.2%

         \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. associate--l+ 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

      3. sub-neg 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

      4. *-commutative 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

      5. associate-/r* 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

      6. metadata-eval 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

      7. metadata-eval 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]

   4. Taylor expanded in x around 0 74.2%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\frac{0.1111111111111111}{x}}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 74.4%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \frac{0.1111111111111111}{x}} \]

      2. clear-num 74.4%

         \[\leadsto \left(3 \cdot \sqrt{x}\right) \cdot \color{blue}{\frac{1}{\frac{x}{0.1111111111111111}}} \]

      3. un-div-inv 74.4%

         \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{\frac{x}{0.1111111111111111}}} \]

      4. div-inv 74.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{x \cdot \frac{1}{0.1111111111111111}}} \]

      5. metadata-eval 74.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{x \cdot \color{blue}{9}} \]

   6. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{\frac{3 \cdot \sqrt{x}}{x \cdot 9}} \]
   7. Step-by-step derivation

      1. *-commutative 74.4%

         \[\leadsto \frac{3 \cdot \sqrt{x}}{\color{blue}{9 \cdot x}} \]

      2. times-frac 74.4%

         \[\leadsto \color{blue}{\frac{3}{9} \cdot \frac{\sqrt{x}}{x}} \]

      3. metadata-eval 74.4%

         \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\sqrt{x}}{x} \]

   8. Simplified 74.4%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\sqrt{x}}{x}} \]

   if 0.112000000000000002 < x

   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. associate-*l* 99.4%

         \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. associate--l+ 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

      3. sub-neg 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

      4. *-commutative 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

      5. associate-/r* 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

      6. metadata-eval 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.3%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)} \]

      2. distribute-lft-in 99.3%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]

      3. *-commutative 99.3%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      4. metadata-eval 99.3%

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      5. sqrt-prod 99.4%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      6. *-commutative 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      8. sqrt-prod 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      9. +-commutative 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \color{blue}{\left(-1 + \frac{0.1111111111111111}{x}\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \]

      2. associate-+r+ 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(y + \color{blue}{\left(-1\right)}\right) + \frac{0.1111111111111111}{x}\right) \]

      4. sub-neg 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(y - 1\right)} + \frac{0.1111111111111111}{x}\right) \]

      5. associate-+l- 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   8. Taylor expanded in x around inf 99.0%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]

   9. Taylor expanded in y around 0 43.6%

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. *-commutative 43.6%

          \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 15: 26.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (sqrt (* x 9.0)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt((x * 9.0));
	} 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 <= 0.112d0) then
        tmp = sqrt((x * 9.0d0))
    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 <= 0.112) {
		tmp = Math.sqrt((x * 9.0));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt((x * 9.0))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = sqrt(Float64(x * 9.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = sqrt((x * 9.0));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.2%

      \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   2. Step-by-step derivation

      1. associate-*l* 99.2%

         \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. associate--l+ 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

      3. sub-neg 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

      4. *-commutative 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

      5. associate-/r* 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

      6. metadata-eval 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

      7. metadata-eval 99.2%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.2%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)} \]

      2. distribute-lft-in 99.2%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]

      3. *-commutative 99.2%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      4. metadata-eval 99.2%

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      5. sqrt-prod 99.3%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      6. *-commutative 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      7. metadata-eval 99.3%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      8. sqrt-prod 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      9. +-commutative 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \color{blue}{\left(-1 + \frac{0.1111111111111111}{x}\right)} \]

   5. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 99.4%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \]

      2. associate-+r+ 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

      3. metadata-eval 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(y + \color{blue}{\left(-1\right)}\right) + \frac{0.1111111111111111}{x}\right) \]

      4. sub-neg 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(y - 1\right)} + \frac{0.1111111111111111}{x}\right) \]

      5. associate-+l- 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   7. Simplified 99.4%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   8. Taylor expanded in x around inf 23.7%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]

   9. Taylor expanded in y around 0 1.7%

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. *-commutative 1.7%

          \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   11. Simplified 1.7%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
   12. 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 4.7%

          \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \]

       3. swap-sqr 4.7%

          \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-3 \cdot -3\right)}} \]

       4. add-sqr-sqrt 4.7%

          \[\leadsto \sqrt{\color{blue}{x} \cdot \left(-3 \cdot -3\right)} \]

       5. metadata-eval 4.7%

          \[\leadsto \sqrt{x \cdot \color{blue}{9}} \]

   13. Applied egg-rr 4.7%

       \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

   if 0.112000000000000002 < x

   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. associate-*l* 99.4%

         \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

      2. associate--l+ 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

      3. sub-neg 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

      4. *-commutative 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

      5. associate-/r* 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

      6. metadata-eval 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

      7. metadata-eval 99.4%

         \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 99.3%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)} \]

      2. distribute-lft-in 99.3%

         \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]

      3. *-commutative 99.3%

         \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      4. metadata-eval 99.3%

         \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      5. sqrt-prod 99.4%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      6. *-commutative 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      7. metadata-eval 99.4%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      8. sqrt-prod 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

      9. +-commutative 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \color{blue}{\left(-1 + \frac{0.1111111111111111}{x}\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-out 99.6%

         \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \]

      2. associate-+r+ 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

      3. metadata-eval 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(y + \color{blue}{\left(-1\right)}\right) + \frac{0.1111111111111111}{x}\right) \]

      4. sub-neg 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(y - 1\right)} + \frac{0.1111111111111111}{x}\right) \]

      5. associate-+l- 99.6%

         \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

   8. Taylor expanded in x around inf 99.0%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]

   9. Taylor expanded in y around 0 43.6%

      \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
   10. Step-by-step derivation

       1. *-commutative 43.6%

          \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

   11. Simplified 43.6%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 16: 3.4% 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 99.3%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
2. Step-by-step derivation

   1. associate-*l* 99.3%

      \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)\right)} \]

   2. associate--l+ 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \color{blue}{\left(y + \left(\frac{1}{x \cdot 9} - 1\right)\right)}\right) \]

   3. sub-neg 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \color{blue}{\left(\frac{1}{x \cdot 9} + \left(-1\right)\right)}\right)\right) \]

   4. *-commutative 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{1}{\color{blue}{9 \cdot x}} + \left(-1\right)\right)\right)\right) \]

   5. associate-/r* 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\color{blue}{\frac{\frac{1}{9}}{x}} + \left(-1\right)\right)\right)\right) \]

   6. metadata-eval 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{\color{blue}{0.1111111111111111}}{x} + \left(-1\right)\right)\right)\right) \]

   7. metadata-eval 99.3%

      \[\leadsto 3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + \color{blue}{-1}\right)\right)\right) \]

3. Simplified 99.3%

   \[\leadsto \color{blue}{3 \cdot \left(\sqrt{x} \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 99.3%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot \left(y + \left(\frac{0.1111111111111111}{x} + -1\right)\right)} \]

   2. distribute-lft-in 99.3%

      \[\leadsto \color{blue}{\left(3 \cdot \sqrt{x}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right)} \]

   3. *-commutative 99.3%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   4. metadata-eval 99.3%

      \[\leadsto \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   5. sqrt-prod 99.3%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   6. *-commutative 99.3%

      \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\left(\sqrt{x} \cdot 3\right)} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   7. metadata-eval 99.3%

      \[\leadsto \sqrt{x \cdot 9} \cdot y + \left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   8. sqrt-prod 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot y + \color{blue}{\sqrt{x \cdot 9}} \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   9. +-commutative 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \color{blue}{\left(-1 + \frac{0.1111111111111111}{x}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot y + \sqrt{x \cdot 9} \cdot \left(-1 + \frac{0.1111111111111111}{x}\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y + \left(-1 + \frac{0.1111111111111111}{x}\right)\right)} \]

   2. associate-+r+ 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(\left(y + -1\right) + \frac{0.1111111111111111}{x}\right)} \]

   3. metadata-eval 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\left(y + \color{blue}{\left(-1\right)}\right) + \frac{0.1111111111111111}{x}\right) \]

   4. sub-neg 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \left(\color{blue}{\left(y - 1\right)} + \frac{0.1111111111111111}{x}\right) \]

   5. associate-+l- 99.5%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

7. Simplified 99.5%

   \[\leadsto \color{blue}{\sqrt{x \cdot 9} \cdot \left(y - \left(1 - \frac{0.1111111111111111}{x}\right)\right)} \]

8. Taylor expanded in x around inf 56.7%

   \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]

9. Taylor expanded in y around 0 20.1%

   \[\leadsto \color{blue}{-3 \cdot \sqrt{x}} \]
10. Step-by-step derivation

    1. *-commutative 20.1%

       \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]

11. Simplified 20.1%

    \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
12. 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 3.7%

       \[\leadsto \color{blue}{\sqrt{\left(\sqrt{x} \cdot -3\right) \cdot \left(\sqrt{x} \cdot -3\right)}} \]

    3. swap-sqr 3.7%

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right) \cdot \left(-3 \cdot -3\right)}} \]

    4. add-sqr-sqrt 3.7%

       \[\leadsto \sqrt{\color{blue}{x} \cdot \left(-3 \cdot -3\right)} \]

    5. metadata-eval 3.7%

       \[\leadsto \sqrt{x \cdot \color{blue}{9}} \]

13. Applied egg-rr 3.7%

    \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]

14. Final simplification 3.7%

    \[\leadsto \sqrt{x \cdot 9} \]

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 2023271 
(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)))