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

Percentage Accurate: 99.4% → 99.4%

Time: 8.9s

Alternatives: 11

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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. associate--l+ 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*r* 99.5%

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

   7. cancel-sign-sub 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-*r* 99.5%

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

   10. *-commutative 99.5%

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

   11. distribute-rgt-out-- 99.6%

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

   12. distribute-lft-neg-in 99.6%

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

   13. cancel-sign-sub 99.6%

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

   14. +-commutative 99.6%

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

   15. *-commutative 99.6%

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

   16. distribute-rgt-in 99.6%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ t_2 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-149}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-293}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (sqrt (/ 1.0 x))))
        (t_1 (* 3.0 (* (sqrt x) y)))
        (t_2 (* (sqrt x) -3.0)))
   (if (<= y -2.9e+19)
     t_1
     (if (<= y -3e-6)
       t_0
       (if (<= y -9.8e-149)
         t_2
         (if (<= y -2.2e-287)
           t_0
           (if (<= y 1.6e-293)
             t_2
             (if (<= y 6.4e-165) t_0 (if (<= y 0.0053) t_2 t_1)))))))))

C

double code(double x, double y) {
	double t_0 = 0.3333333333333333 * sqrt((1.0 / x));
	double t_1 = 3.0 * (sqrt(x) * y);
	double t_2 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -2.9e+19) {
		tmp = t_1;
	} else if (y <= -3e-6) {
		tmp = t_0;
	} else if (y <= -9.8e-149) {
		tmp = t_2;
	} else if (y <= -2.2e-287) {
		tmp = t_0;
	} else if (y <= 1.6e-293) {
		tmp = t_2;
	} else if (y <= 6.4e-165) {
		tmp = t_0;
	} else if (y <= 0.0053) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * sqrt((1.0d0 / x))
    t_1 = 3.0d0 * (sqrt(x) * y)
    t_2 = sqrt(x) * (-3.0d0)
    if (y <= (-2.9d+19)) then
        tmp = t_1
    else if (y <= (-3d-6)) then
        tmp = t_0
    else if (y <= (-9.8d-149)) then
        tmp = t_2
    else if (y <= (-2.2d-287)) then
        tmp = t_0
    else if (y <= 1.6d-293) then
        tmp = t_2
    else if (y <= 6.4d-165) then
        tmp = t_0
    else if (y <= 0.0053d0) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 0.3333333333333333 * Math.sqrt((1.0 / x));
	double t_1 = 3.0 * (Math.sqrt(x) * y);
	double t_2 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -2.9e+19) {
		tmp = t_1;
	} else if (y <= -3e-6) {
		tmp = t_0;
	} else if (y <= -9.8e-149) {
		tmp = t_2;
	} else if (y <= -2.2e-287) {
		tmp = t_0;
	} else if (y <= 1.6e-293) {
		tmp = t_2;
	} else if (y <= 6.4e-165) {
		tmp = t_0;
	} else if (y <= 0.0053) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 0.3333333333333333 * math.sqrt((1.0 / x))
	t_1 = 3.0 * (math.sqrt(x) * y)
	t_2 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -2.9e+19:
		tmp = t_1
	elif y <= -3e-6:
		tmp = t_0
	elif y <= -9.8e-149:
		tmp = t_2
	elif y <= -2.2e-287:
		tmp = t_0
	elif y <= 1.6e-293:
		tmp = t_2
	elif y <= 6.4e-165:
		tmp = t_0
	elif y <= 0.0053:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)))
	t_1 = Float64(3.0 * Float64(sqrt(x) * y))
	t_2 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -2.9e+19)
		tmp = t_1;
	elseif (y <= -3e-6)
		tmp = t_0;
	elseif (y <= -9.8e-149)
		tmp = t_2;
	elseif (y <= -2.2e-287)
		tmp = t_0;
	elseif (y <= 1.6e-293)
		tmp = t_2;
	elseif (y <= 6.4e-165)
		tmp = t_0;
	elseif (y <= 0.0053)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 0.3333333333333333 * sqrt((1.0 / x));
	t_1 = 3.0 * (sqrt(x) * y);
	t_2 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -2.9e+19)
		tmp = t_1;
	elseif (y <= -3e-6)
		tmp = t_0;
	elseif (y <= -9.8e-149)
		tmp = t_2;
	elseif (y <= -2.2e-287)
		tmp = t_0;
	elseif (y <= 1.6e-293)
		tmp = t_2;
	elseif (y <= 6.4e-165)
		tmp = t_0;
	elseif (y <= 0.0053)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -2.9e+19], t$95$1, If[LessEqual[y, -3e-6], t$95$0, If[LessEqual[y, -9.8e-149], t$95$2, If[LessEqual[y, -2.2e-287], t$95$0, If[LessEqual[y, 1.6e-293], t$95$2, If[LessEqual[y, 6.4e-165], t$95$0, If[LessEqual[y, 0.0053], t$95$2, t$95$1]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9e19 or 0.00530000000000000002 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out-- 99.7%

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

      12. distribute-lft-neg-in 99.7%

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

      13. cancel-sign-sub 99.7%

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

      14. +-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. distribute-rgt-in 99.7%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 78.8%

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

   if -2.9e19 < y < -3.0000000000000001e-6 or -9.8000000000000008e-149 < y < -2.2e-287 or 1.60000000000000003e-293 < y < 6.40000000000000026e-165

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   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 x around 0 70.6%

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

   5. Taylor expanded in y around 0 70.5%

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

   if -3.0000000000000001e-6 < y < -9.8000000000000008e-149 or -2.2e-287 < y < 1.60000000000000003e-293 or 6.40000000000000026e-165 < y < 0.00530000000000000002

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   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 0 97.0%

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

   5. Taylor expanded in x around inf 64.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+19}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq -9.8 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -2.2 \cdot 10^{-287}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-165}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 3: 60.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-291}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* 0.3333333333333333 (sqrt (/ 1.0 x)))))
   (if (<= y -1.8e+19)
     (* (sqrt x) (* 3.0 y))
     (if (<= y -3.2e-6)
       t_1
       (if (<= y -6e-149)
         t_0
         (if (<= y -1.12e-287)
           t_1
           (if (<= y 2.95e-291)
             t_0
             (if (<= y 8.6e-165)
               t_1
               (if (<= y 0.0053) t_0 (* 3.0 (* (sqrt x) y)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = 0.3333333333333333 * sqrt((1.0 / x));
	double tmp;
	if (y <= -1.8e+19) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= -3.2e-6) {
		tmp = t_1;
	} else if (y <= -6e-149) {
		tmp = t_0;
	} else if (y <= -1.12e-287) {
		tmp = t_1;
	} else if (y <= 2.95e-291) {
		tmp = t_0;
	} else if (y <= 8.6e-165) {
		tmp = t_1;
	} else if (y <= 0.0053) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = 0.3333333333333333d0 * sqrt((1.0d0 / x))
    if (y <= (-1.8d+19)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= (-3.2d-6)) then
        tmp = t_1
    else if (y <= (-6d-149)) then
        tmp = t_0
    else if (y <= (-1.12d-287)) then
        tmp = t_1
    else if (y <= 2.95d-291) then
        tmp = t_0
    else if (y <= 8.6d-165) then
        tmp = t_1
    else if (y <= 0.0053d0) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = 0.3333333333333333 * Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -1.8e+19) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= -3.2e-6) {
		tmp = t_1;
	} else if (y <= -6e-149) {
		tmp = t_0;
	} else if (y <= -1.12e-287) {
		tmp = t_1;
	} else if (y <= 2.95e-291) {
		tmp = t_0;
	} else if (y <= 8.6e-165) {
		tmp = t_1;
	} else if (y <= 0.0053) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = 0.3333333333333333 * math.sqrt((1.0 / x))
	tmp = 0
	if y <= -1.8e+19:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= -3.2e-6:
		tmp = t_1
	elif y <= -6e-149:
		tmp = t_0
	elif y <= -1.12e-287:
		tmp = t_1
	elif y <= 2.95e-291:
		tmp = t_0
	elif y <= 8.6e-165:
		tmp = t_1
	elif y <= 0.0053:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)))
	tmp = 0.0
	if (y <= -1.8e+19)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= -3.2e-6)
		tmp = t_1;
	elseif (y <= -6e-149)
		tmp = t_0;
	elseif (y <= -1.12e-287)
		tmp = t_1;
	elseif (y <= 2.95e-291)
		tmp = t_0;
	elseif (y <= 8.6e-165)
		tmp = t_1;
	elseif (y <= 0.0053)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = 0.3333333333333333 * sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -1.8e+19)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= -3.2e-6)
		tmp = t_1;
	elseif (y <= -6e-149)
		tmp = t_0;
	elseif (y <= -1.12e-287)
		tmp = t_1;
	elseif (y <= 2.95e-291)
		tmp = t_0;
	elseif (y <= 8.6e-165)
		tmp = t_1;
	elseif (y <= 0.0053)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.8e+19], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.2e-6], t$95$1, If[LessEqual[y, -6e-149], t$95$0, If[LessEqual[y, -1.12e-287], t$95$1, If[LessEqual[y, 2.95e-291], t$95$0, If[LessEqual[y, 8.6e-165], t$95$1, If[LessEqual[y, 0.0053], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.8e19

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out-- 99.7%

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

      12. distribute-lft-neg-in 99.7%

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

      13. cancel-sign-sub 99.7%

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

      14. +-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. distribute-rgt-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -1.8e19 < y < -3.1999999999999999e-6 or -6.0000000000000003e-149 < y < -1.12e-287 or 2.94999999999999986e-291 < y < 8.60000000000000013e-165

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   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 x around 0 70.6%

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

   5. Taylor expanded in y around 0 70.5%

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

   if -3.1999999999999999e-6 < y < -6.0000000000000003e-149 or -1.12e-287 < y < 2.94999999999999986e-291 or 8.60000000000000013e-165 < y < 0.00530000000000000002

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   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 0 97.0%

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

   5. Taylor expanded in x around inf 64.3%

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

   if 0.00530000000000000002 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. cancel-sign-sub 99.6%

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

      14. +-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. distribute-rgt-in 99.6%

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

   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 73.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -3.2 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq -6 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -1.12 \cdot 10^{-287}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-291}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{-165}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 4: 60.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := 0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{if}\;y \leq -1.2 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-293}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* 0.3333333333333333 (sqrt (/ 1.0 x)))))
   (if (<= y -1.2e+19)
     (* (sqrt x) (* 3.0 y))
     (if (<= y -2.7e-6)
       t_1
       (if (<= y -7.5e-149)
         t_0
         (if (<= y -3e-288)
           (* (sqrt x) (/ 0.3333333333333333 x))
           (if (<= y 2.25e-293)
             t_0
             (if (<= y 1.95e-162)
               t_1
               (if (<= y 0.0053) t_0 (* 3.0 (* (sqrt x) y)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = 0.3333333333333333 * sqrt((1.0 / x));
	double tmp;
	if (y <= -1.2e+19) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= -2.7e-6) {
		tmp = t_1;
	} else if (y <= -7.5e-149) {
		tmp = t_0;
	} else if (y <= -3e-288) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else if (y <= 2.25e-293) {
		tmp = t_0;
	} else if (y <= 1.95e-162) {
		tmp = t_1;
	} else if (y <= 0.0053) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = 0.3333333333333333d0 * sqrt((1.0d0 / x))
    if (y <= (-1.2d+19)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= (-2.7d-6)) then
        tmp = t_1
    else if (y <= (-7.5d-149)) then
        tmp = t_0
    else if (y <= (-3d-288)) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else if (y <= 2.25d-293) then
        tmp = t_0
    else if (y <= 1.95d-162) then
        tmp = t_1
    else if (y <= 0.0053d0) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = 0.3333333333333333 * Math.sqrt((1.0 / x));
	double tmp;
	if (y <= -1.2e+19) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= -2.7e-6) {
		tmp = t_1;
	} else if (y <= -7.5e-149) {
		tmp = t_0;
	} else if (y <= -3e-288) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else if (y <= 2.25e-293) {
		tmp = t_0;
	} else if (y <= 1.95e-162) {
		tmp = t_1;
	} else if (y <= 0.0053) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = 0.3333333333333333 * math.sqrt((1.0 / x))
	tmp = 0
	if y <= -1.2e+19:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= -2.7e-6:
		tmp = t_1
	elif y <= -7.5e-149:
		tmp = t_0
	elif y <= -3e-288:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	elif y <= 2.25e-293:
		tmp = t_0
	elif y <= 1.95e-162:
		tmp = t_1
	elif y <= 0.0053:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)))
	tmp = 0.0
	if (y <= -1.2e+19)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= -2.7e-6)
		tmp = t_1;
	elseif (y <= -7.5e-149)
		tmp = t_0;
	elseif (y <= -3e-288)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	elseif (y <= 2.25e-293)
		tmp = t_0;
	elseif (y <= 1.95e-162)
		tmp = t_1;
	elseif (y <= 0.0053)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = 0.3333333333333333 * sqrt((1.0 / x));
	tmp = 0.0;
	if (y <= -1.2e+19)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= -2.7e-6)
		tmp = t_1;
	elseif (y <= -7.5e-149)
		tmp = t_0;
	elseif (y <= -3e-288)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	elseif (y <= 2.25e-293)
		tmp = t_0;
	elseif (y <= 1.95e-162)
		tmp = t_1;
	elseif (y <= 0.0053)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.2e+19], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.7e-6], t$95$1, If[LessEqual[y, -7.5e-149], t$95$0, If[LessEqual[y, -3e-288], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-293], t$95$0, If[LessEqual[y, 1.95e-162], t$95$1, If[LessEqual[y, 0.0053], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.2e19

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out-- 99.7%

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

      12. distribute-lft-neg-in 99.7%

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

      13. cancel-sign-sub 99.7%

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

      14. +-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. distribute-rgt-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -1.2e19 < y < -2.69999999999999998e-6 or 2.2500000000000001e-293 < y < 1.95e-162

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   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 65.0%

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

   5. Taylor expanded in y around 0 65.1%

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

   if -2.69999999999999998e-6 < y < -7.49999999999999995e-149 or -2.99999999999999999e-288 < y < 2.2500000000000001e-293 or 1.95e-162 < y < 0.00530000000000000002

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   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 0 97.0%

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

   5. Taylor expanded in x around inf 64.3%

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

   if -7.49999999999999995e-149 < y < -2.99999999999999999e-288

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 99.3%

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

   5. Taylor expanded in x around 0 99.6%

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

   6. Taylor expanded in x around 0 79.0%

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

   if 0.00530000000000000002 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. cancel-sign-sub 99.6%

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

      14. +-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. distribute-rgt-in 99.6%

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

   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 73.2%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-6}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-288}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-293}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-162}:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{elif}\;y \leq 0.0053:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 5: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{if}\;x \leq 3 \cdot 10^{-176}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-158}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) (/ 0.3333333333333333 x))))
   (if (<= x 3e-176)
     t_0
     (if (<= x 4.6e-158)
       (* 3.0 (* (sqrt x) y))
       (if (<= x 4.2e-20) t_0 (* (sqrt x) (- (* 3.0 y) 3.0)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 3e-176) {
		tmp = t_0;
	} else if (x <= 4.6e-158) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 4.2e-20) {
		tmp = t_0;
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (0.3333333333333333d0 / x)
    if (x <= 3d-176) then
        tmp = t_0
    else if (x <= 4.6d-158) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 4.2d-20) then
        tmp = t_0
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * (0.3333333333333333 / x);
	double tmp;
	if (x <= 3e-176) {
		tmp = t_0;
	} else if (x <= 4.6e-158) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 4.2e-20) {
		tmp = t_0;
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * (0.3333333333333333 / x)
	tmp = 0
	if x <= 3e-176:
		tmp = t_0
	elif x <= 4.6e-158:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 4.2e-20:
		tmp = t_0
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * Float64(0.3333333333333333 / x))
	tmp = 0.0
	if (x <= 3e-176)
		tmp = t_0;
	elseif (x <= 4.6e-158)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 4.2e-20)
		tmp = t_0;
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 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 <= 3e-176)
		tmp = t_0;
	elseif (x <= 4.6e-158)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 4.2e-20)
		tmp = t_0;
	else
		tmp = sqrt(x) * ((3.0 * y) - 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, 3e-176], t$95$0, If[LessEqual[x, 4.6e-158], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-20], t$95$0, N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < 3e-176 or 4.5999999999999998e-158 < x < 4.1999999999999998e-20

   1. Initial program 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 99.4%

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

   5. Taylor expanded in x around 0 99.4%

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

   6. Taylor expanded in x around 0 74.4%

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

   if 3e-176 < x < 4.5999999999999998e-158

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 95.0%

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

   if 4.1999999999999998e-20 < x

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 96.1%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3 \cdot 10^{-176}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-158}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 6: 85.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -9.2e+18)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 0.8)
     (* (sqrt x) (- (/ 0.3333333333333333 x) 3.0))
     (* (sqrt x) (- (* 3.0 y) 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+18) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 0.8) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-9.2d+18)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 0.8d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) - 3.0d0)
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -9.2e+18) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 0.8) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -9.2e+18:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 0.8:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) - 3.0)
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -9.2e+18)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 0.8)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) - 3.0));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -9.2e+18)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 0.8)
		tmp = sqrt(x) * ((0.3333333333333333 / x) - 3.0);
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -9.2e+18], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.8], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.2e18

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.7%

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

      10. *-commutative 99.7%

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

      11. distribute-rgt-out-- 99.7%

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

      12. distribute-lft-neg-in 99.7%

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

      13. cancel-sign-sub 99.7%

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

      14. +-commutative 99.7%

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

      15. *-commutative 99.7%

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

      16. distribute-rgt-in 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around inf 83.8%

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

      1. associate-*r* 83.8%

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

      2. *-commutative 83.8%

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

   6. Simplified 83.8%

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

   if -9.2e18 < y < 0.80000000000000004

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   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 0 98.0%

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

      1. *-commutative 98.0%

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

      2. associate-*r/ 98.0%

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

      3. metadata-eval 98.0%

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

   6. Simplified 98.0%

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

   if 0.80000000000000004 < y

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around inf 77.7%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+18}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 0.8:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 7: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (3.0d0 * y))
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   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 x around 0 98.0%

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

      1. fma-udef 97.9%

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

      2. +-commutative 97.9%

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

   6. Applied egg-rr 97.9%

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

   if 0.110000000000000001 < x

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.9%

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

4. Final simplification 98.4%

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

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 99.5%

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

5. Taylor expanded in x around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 9: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (* 0.3333333333333333 (sqrt (/ 1.0 x))) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = 0.3333333333333333 * sqrt((1.0 / 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 <= 0.11d0) then
        tmp = 0.3333333333333333d0 * sqrt((1.0d0 / 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 <= 0.11) {
		tmp = 0.3333333333333333 * Math.sqrt((1.0 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = 0.3333333333333333 * math.sqrt((1.0 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = Float64(0.3333333333333333 * sqrt(Float64(1.0 / x)));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = 0.3333333333333333 * sqrt((1.0 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[(0.3333333333333333 * N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   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 x around 0 98.0%

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

   5. Taylor expanded in y around 0 68.0%

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

   if 0.110000000000000001 < x

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.5%

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

      7. cancel-sign-sub 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*r* 99.6%

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

      10. *-commutative 99.6%

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

      11. distribute-rgt-out-- 99.6%

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

      12. distribute-lft-neg-in 99.6%

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

      13. cancel-sign-sub 99.6%

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

      14. +-commutative 99.6%

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

      15. *-commutative 99.6%

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

      16. distribute-rgt-in 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around 0 45.4%

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

   5. Taylor expanded in x around inf 44.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;0.3333333333333333 \cdot \sqrt{\frac{1}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 10: 3.2% 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 99.5%

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

   1. +-commutative 99.5%

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

   2. associate--l+ 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*r* 99.5%

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

   7. cancel-sign-sub 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-*r* 99.5%

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

   10. *-commutative 99.5%

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

   11. distribute-rgt-out-- 99.6%

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

   12. distribute-lft-neg-in 99.6%

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

   13. cancel-sign-sub 99.6%

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

   14. +-commutative 99.6%

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

   15. *-commutative 99.6%

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

   16. distribute-rgt-in 99.6%

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

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 0 57.2%

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

5. Taylor expanded in x around inf 23.8%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.3%

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

   3. *-commutative 3.3%

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

   4. *-commutative 3.3%

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

   5. swap-sqr 3.3%

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

   6. add-sqr-sqrt 3.3%

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

   7. metadata-eval 3.3%

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

7. Applied egg-rr 3.3%

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

8. Final simplification 3.3%

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

Alternative 11: 25.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. +-commutative 99.5%

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

   2. associate--l+ 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. +-commutative 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*r* 99.5%

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

   7. cancel-sign-sub 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-*r* 99.5%

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

   10. *-commutative 99.5%

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

   11. distribute-rgt-out-- 99.6%

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

   12. distribute-lft-neg-in 99.6%

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

   13. cancel-sign-sub 99.6%

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

   14. +-commutative 99.6%

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

   15. *-commutative 99.6%

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

   16. distribute-rgt-in 99.6%

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

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 0 57.2%

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

5. Taylor expanded in x around inf 23.8%

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

6. Final simplification 23.8%

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

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 2023240 
(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)))