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, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + {\left(x \cdot 3\right)}^{-1}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sqrt x) (fma 3.0 y (+ -3.0 (pow (* x 3.0) -1.0)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.5%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

   12. associate-/r* 99.5%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

   3. div-inv 99.5%

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

   4. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \color{blue}{{\left(x \cdot 3\right)}^{-1}}\right) \]
7. Add Preprocessing

Alternative 2: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -15000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.08 \cdot 10^{-60}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-38}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot 9}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))))
   (if (<= y -15000000.0)
     t_0
     (if (<= y 1.08e-60)
       (sqrt (/ 0.1111111111111111 x))
       (if (<= y 1.9e-38)
         (* (sqrt x) -3.0)
         (if (<= y 4.9e+22) (/ 1.0 (sqrt (* x 9.0))) t_0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double tmp;
	if (y <= -15000000.0) {
		tmp = t_0;
	} else if (y <= 1.08e-60) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (y <= 1.9e-38) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 4.9e+22) {
		tmp = 1.0 / sqrt((x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (sqrt(x) * y)
    if (y <= (-15000000.0d0)) then
        tmp = t_0
    else if (y <= 1.08d-60) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (y <= 1.9d-38) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 4.9d+22) then
        tmp = 1.0d0 / sqrt((x * 9.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double tmp;
	if (y <= -15000000.0) {
		tmp = t_0;
	} else if (y <= 1.08e-60) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (y <= 1.9e-38) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 4.9e+22) {
		tmp = 1.0 / Math.sqrt((x * 9.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	tmp = 0
	if y <= -15000000.0:
		tmp = t_0
	elif y <= 1.08e-60:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif y <= 1.9e-38:
		tmp = math.sqrt(x) * -3.0
	elif y <= 4.9e+22:
		tmp = 1.0 / math.sqrt((x * 9.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -15000000.0)
		tmp = t_0;
	elseif (y <= 1.08e-60)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (y <= 1.9e-38)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 4.9e+22)
		tmp = Float64(1.0 / sqrt(Float64(x * 9.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -15000000.0)
		tmp = t_0;
	elseif (y <= 1.08e-60)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (y <= 1.9e-38)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 4.9e+22)
		tmp = 1.0 / sqrt((x * 9.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -15000000.0], t$95$0, If[LessEqual[y, 1.08e-60], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.9e-38], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 4.9e+22], N[(1.0 / N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.5e7 or 4.89999999999999979e22 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.7%

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

   if -1.5e7 < y < 1.07999999999999997e-60

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 56.7%

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

      1. add-sqr-sqrt 56.6%

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

      2. sqrt-unprod 56.7%

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

      3. associate-*l* 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-*l* 56.7%

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

      6. add-sqr-sqrt 56.8%

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

      7. inv-pow 56.8%

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

      8. metadata-eval 56.8%

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

      9. unpow-prod-down 56.9%

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

      10. unpow-1 56.9%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{1}{x \cdot 3}}} \]

      11. *-commutative 56.9%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \frac{1}{\color{blue}{3 \cdot x}}} \]

      12. associate-/r* 56.8%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{3}}{x}}} \]

      13. metadata-eval 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. associate-*r/ 56.9%

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

      2. metadata-eval 56.9%

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

   9. Simplified 56.9%

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

   if 1.07999999999999997e-60 < y < 1.9e-38

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. associate--l+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. fma-define 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-/r* 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-neg-frac 99.7%

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

      10. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 99.7%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-/l* 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if 1.9e-38 < y < 4.89999999999999979e22

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.1%

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

      3. associate--l+ 99.1%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 65.1%

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

   6. Taylor expanded in x around -inf 0.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\sqrt{\frac{1}{x}} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \]
   7. Step-by-step derivation

      1. associate-*r* 0.0%

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

      2. unpow2 0.0%

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

      3. *-commutative 0.0%

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

      4. rem-square-sqrt 1.8%

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

      5. associate-*l* 1.8%

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

      6. metadata-eval 1.8%

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

   8. Simplified 1.8%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 65.1%

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

      3. swap-sqr 65.2%

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

      4. add-sqr-sqrt 65.3%

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

      5. metadata-eval 65.3%

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

      6. *-commutative 65.3%

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

      7. div-inv 65.3%

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

      8. clear-num 65.4%

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

      9. sqrt-div 65.4%

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

      10. metadata-eval 65.4%

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

      11. div-inv 65.4%

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

      12. metadata-eval 65.4%

          \[\leadsto \frac{1}{\sqrt{x \cdot \color{blue}{9}}} \]

   10. Applied egg-rr 65.4%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{x \cdot 9}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 62.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -1.95:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+23}:\\ \;\;\;\;0.3333333333333333 \cdot {x}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* 3.0 (* (sqrt x) y))))
   (if (<= y -1.95)
     t_0
     (if (<= y 1.95e-61)
       (sqrt (/ 0.1111111111111111 x))
       (if (<= y 1.15e-34)
         (* (sqrt x) -3.0)
         (if (<= y 6.2e+23) (* 0.3333333333333333 (pow x -0.5)) t_0))))))

C

double code(double x, double y) {
	double t_0 = 3.0 * (sqrt(x) * y);
	double tmp;
	if (y <= -1.95) {
		tmp = t_0;
	} else if (y <= 1.95e-61) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (y <= 1.15e-34) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 6.2e+23) {
		tmp = 0.3333333333333333 * pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 3.0d0 * (sqrt(x) * y)
    if (y <= (-1.95d0)) then
        tmp = t_0
    else if (y <= 1.95d-61) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (y <= 1.15d-34) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 6.2d+23) then
        tmp = 0.3333333333333333d0 * (x ** (-0.5d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 3.0 * (Math.sqrt(x) * y);
	double tmp;
	if (y <= -1.95) {
		tmp = t_0;
	} else if (y <= 1.95e-61) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (y <= 1.15e-34) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 6.2e+23) {
		tmp = 0.3333333333333333 * Math.pow(x, -0.5);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 3.0 * (math.sqrt(x) * y)
	tmp = 0
	if y <= -1.95:
		tmp = t_0
	elif y <= 1.95e-61:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif y <= 1.15e-34:
		tmp = math.sqrt(x) * -3.0
	elif y <= 6.2e+23:
		tmp = 0.3333333333333333 * math.pow(x, -0.5)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(3.0 * Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -1.95)
		tmp = t_0;
	elseif (y <= 1.95e-61)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (y <= 1.15e-34)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 6.2e+23)
		tmp = Float64(0.3333333333333333 * (x ^ -0.5));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 3.0 * (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -1.95)
		tmp = t_0;
	elseif (y <= 1.95e-61)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (y <= 1.15e-34)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 6.2e+23)
		tmp = 0.3333333333333333 * (x ^ -0.5);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.95], t$95$0, If[LessEqual[y, 1.95e-61], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.15e-34], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 6.2e+23], N[(0.3333333333333333 * N[Power[x, -0.5], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.94999999999999996 or 6.19999999999999941e23 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.7%

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

   if -1.94999999999999996 < y < 1.95000000000000016e-61

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 56.7%

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

      1. add-sqr-sqrt 56.6%

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

      2. sqrt-unprod 56.7%

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

      3. associate-*l* 56.7%

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

      4. *-commutative 56.7%

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

      5. associate-*l* 56.7%

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

      6. add-sqr-sqrt 56.8%

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

      7. inv-pow 56.8%

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

      8. metadata-eval 56.8%

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

      9. unpow-prod-down 56.9%

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

      10. unpow-1 56.9%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{1}{x \cdot 3}}} \]

      11. *-commutative 56.9%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \frac{1}{\color{blue}{3 \cdot x}}} \]

      12. associate-/r* 56.8%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{3}}{x}}} \]

      13. metadata-eval 56.8%

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

   7. Applied egg-rr 56.8%

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

      1. associate-*r/ 56.9%

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

      2. metadata-eval 56.9%

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

   9. Simplified 56.9%

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

   if 1.95000000000000016e-61 < y < 1.15000000000000006e-34

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. associate--l+ 100.0%

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

      4. distribute-lft-in 100.0%

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

      5. fma-define 100.0%

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

      6. sub-neg 100.0%

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

      7. +-commutative 100.0%

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

      8. distribute-lft-in 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

      11. *-commutative 100.0%

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

      12. associate-/r* 100.0%

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

      13. associate-*r/ 100.0%

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

      14. metadata-eval 100.0%

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

      15. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf 99.7%

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

      1. *-commutative 99.7%

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

      2. distribute-lft-out 99.7%

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

      3. sub-neg 99.7%

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

      4. associate-*r/ 99.7%

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

      5. metadata-eval 99.7%

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

      6. metadata-eval 99.7%

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

      7. +-commutative 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-neg-frac 99.7%

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

      10. unsub-neg 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around 0 99.7%

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

      1. associate-*r/ 100.0%

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

      2. *-commutative 100.0%

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

      3. sub-neg 100.0%

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

      4. associate-*r/ 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. +-commutative 100.0%

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

      8. associate-/l* 99.4%

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

   10. Simplified 99.4%

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

   11. Taylor expanded in x around inf 100.0%

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

       1. *-commutative 100.0%

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

   13. Simplified 100.0%

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

   if 1.15000000000000006e-34 < y < 6.19999999999999941e23

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.1%

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

      3. associate--l+ 99.1%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.2%

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

      14. metadata-eval 99.2%

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

      15. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 99.2%

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

      1. *-commutative 99.2%

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

      2. distribute-lft-out 99.2%

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

      3. sub-neg 99.2%

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

      4. associate-*r/ 99.0%

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

      5. metadata-eval 99.0%

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

      6. metadata-eval 99.0%

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

      7. +-commutative 99.0%

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

      8. metadata-eval 99.0%

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

      9. distribute-neg-frac 99.0%

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

      10. unsub-neg 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in x around 0 65.2%

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

      1. associate-*r/ 65.2%

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

      2. *-rgt-identity 65.2%

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

   10. Simplified 65.2%

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

       1. associate-*r* 65.1%

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

       2. clear-num 65.1%

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

       3. un-div-inv 65.0%

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

       4. inv-pow 65.0%

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

       5. sqrt-pow1 65.1%

          \[\leadsto \frac{y \cdot 0.3333333333333333}{\frac{y}{\color{blue}{{x}^{\left(\frac{-1}{2}\right)}}}} \]

       6. metadata-eval 65.1%

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

   12. Applied egg-rr 65.1%

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

       1. associate-/r/ 65.3%

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

       2. *-commutative 65.3%

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

       3. associate-/l* 65.3%

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

       4. *-inverses 65.3%

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

       5. metadata-eval 65.3%

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

   14. Simplified 65.3%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot {x}^{-0.5}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot y\\ \mathbf{if}\;y \leq -120000000:\\ \;\;\;\;t\_0 \cdot \left(3 - \frac{3}{y}\right)\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) y)))
   (if (<= y -120000000.0)
     (* t_0 (- 3.0 (/ 3.0 y)))
     (if (<= y 3.1e+23)
       (* (sqrt x) (- (* 0.3333333333333333 (/ 1.0 x)) 3.0))
       (* 3.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * y;
	double tmp;
	if (y <= -120000000.0) {
		tmp = t_0 * (3.0 - (3.0 / y));
	} else if (y <= 3.1e+23) {
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = 3.0 * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * y
    if (y <= (-120000000.0d0)) then
        tmp = t_0 * (3.0d0 - (3.0d0 / y))
    else if (y <= 3.1d+23) then
        tmp = sqrt(x) * ((0.3333333333333333d0 * (1.0d0 / x)) - 3.0d0)
    else
        tmp = 3.0d0 * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * y;
	double tmp;
	if (y <= -120000000.0) {
		tmp = t_0 * (3.0 - (3.0 / y));
	} else if (y <= 3.1e+23) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = 3.0 * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * y
	tmp = 0
	if y <= -120000000.0:
		tmp = t_0 * (3.0 - (3.0 / y))
	elif y <= 3.1e+23:
		tmp = math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0)
	else:
		tmp = 3.0 * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * y)
	tmp = 0.0
	if (y <= -120000000.0)
		tmp = Float64(t_0 * Float64(3.0 - Float64(3.0 / y)));
	elseif (y <= 3.1e+23)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 3.0));
	else
		tmp = Float64(3.0 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * y;
	tmp = 0.0;
	if (y <= -120000000.0)
		tmp = t_0 * (3.0 - (3.0 / y));
	elseif (y <= 3.1e+23)
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	else
		tmp = 3.0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -120000000.0], N[(t$95$0 * N[(3.0 - N[(3.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+23], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(3.0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.2e8

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 99.3%

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

      1. *-commutative 99.3%

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

      2. distribute-lft-out 99.3%

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

      3. sub-neg 99.3%

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

      4. associate-*r/ 99.3%

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

      5. metadata-eval 99.3%

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

      6. metadata-eval 99.3%

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

      7. +-commutative 99.3%

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

      8. metadata-eval 99.3%

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

      9. distribute-neg-frac 99.3%

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

      10. unsub-neg 99.3%

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

   7. Simplified 99.3%

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

   8. Taylor expanded in x around inf 84.9%

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

      1. associate-*r* 85.0%

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

      2. associate-*r/ 85.0%

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

      3. metadata-eval 85.0%

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

   10. Simplified 85.0%

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

   if -1.2e8 < y < 3.09999999999999971e23

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 96.6%

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

   if 3.09999999999999971e23 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.0%

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

Alternative 5: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8200:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \mathbf{elif}\;y \leq 10^{+25}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8200.0)
   (* (sqrt x) (- (* 3.0 y) 3.0))
   (if (<= y 1e+25)
     (* (sqrt x) (- (* 0.3333333333333333 (/ 1.0 x)) 3.0))
     (* 3.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8200.0) {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 1e+25) {
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} 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) :: tmp
    if (y <= (-8200.0d0)) then
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    else if (y <= 1d+25) then
        tmp = sqrt(x) * ((0.3333333333333333d0 * (1.0d0 / x)) - 3.0d0)
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8200.0) {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 1e+25) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8200.0:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	elif y <= 1e+25:
		tmp = math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0)
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8200.0)
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	elseif (y <= 1e+25)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 3.0));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8200.0)
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	elseif (y <= 1e+25)
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8200.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+25], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8200

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.9%

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

   if -8200 < y < 1.00000000000000009e25

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 96.6%

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

   if 1.00000000000000009e25 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.0%

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

Alternative 6: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -170000000 \lor \neg \left(y \leq 2.4 \cdot 10^{+23}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -170000000.0) (not (<= y 2.4e+23)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -170000000.0) || !(y <= 2.4e+23)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-170000000.0d0)) .or. (.not. (y <= 2.4d+23))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -170000000.0) || !(y <= 2.4e+23)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -170000000.0) or not (y <= 2.4e+23):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -170000000.0) || !(y <= 2.4e+23))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -170000000.0) || ~((y <= 2.4e+23)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -170000000.0], N[Not[LessEqual[y, 2.4e+23]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7e8 or 2.4e23 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.7%

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

   if -1.7e8 < y < 2.4e23

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. associate-*r/ 96.6%

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

      3. metadata-eval 96.6%

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

      4. metadata-eval 96.6%

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

      5. +-commutative 96.6%

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

      6. metadata-eval 96.6%

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

      7. distribute-neg-frac 96.6%

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

      8. unsub-neg 96.6%

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

   7. Simplified 96.6%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -170000000 \lor \neg \left(y \leq 2.4 \cdot 10^{+23}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+24}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.4e-9)
   (* (sqrt x) (- (* 3.0 y) 3.0))
   (if (<= y 4.1e+24)
     (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))
     (* 3.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-9) {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 4.1e+24) {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} 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) :: tmp
    if (y <= (-2.4d-9)) then
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    else if (y <= 4.1d+24) then
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.4e-9) {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	} else if (y <= 4.1e+24) {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.4e-9:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	elif y <= 4.1e+24:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.4e-9)
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	elseif (y <= 4.1e+24)
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.4e-9)
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	elseif (y <= 4.1e+24)
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.4e-9], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+24], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e-9

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.9%

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

   if -2.4e-9 < y < 4.1000000000000001e24

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 96.6%

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

      1. sub-neg 96.6%

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

      2. associate-*r/ 96.6%

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

      3. metadata-eval 96.6%

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

      4. metadata-eval 96.6%

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

      5. +-commutative 96.6%

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

      6. metadata-eval 96.6%

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

      7. distribute-neg-frac 96.6%

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

      8. unsub-neg 96.6%

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

   7. Simplified 96.6%

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

   if 4.1000000000000001e24 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.0%

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

Alternative 8: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. sub-neg 99.4%

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

   2. +-commutative 99.4%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. metadata-eval 99.3%

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

   3. sqrt-prod 99.5%

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

   4. pow1 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow1 99.5%

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

8. Simplified 99.5%

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

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.5%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

   12. associate-/r* 99.5%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

   3. div-inv 99.5%

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

   4. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

   \[\leadsto \sqrt{x} \cdot \mathsf{fma}\left(3, y, -3 + \color{blue}{{\left(x \cdot 3\right)}^{-1}}\right) \]
7. Step-by-step derivation

   1. fma-undefine 99.5%

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

   2. +-commutative 99.5%

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

   3. associate-+r+ 99.5%

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

   4. *-commutative 99.5%

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

   5. unpow-1 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-/r* 99.4%

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

   8. metadata-eval 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 10: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00049:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.00049) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.00049) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.00049d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.00049) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.00049:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.00049)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.00049)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.00049], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.8999999999999998e-4

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. add-sqr-sqrt 74.1%

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

      2. sqrt-unprod 74.3%

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

      3. associate-*l* 74.3%

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

      4. *-commutative 74.3%

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

      5. associate-*l* 74.3%

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

      6. add-sqr-sqrt 74.5%

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

      7. inv-pow 74.5%

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

      8. metadata-eval 74.5%

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

      9. unpow-prod-down 74.6%

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

      10. unpow-1 74.6%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{1}{x \cdot 3}}} \]

      11. *-commutative 74.6%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \frac{1}{\color{blue}{3 \cdot x}}} \]

      12. associate-/r* 74.5%

          \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{3}}{x}}} \]

      13. metadata-eval 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*r/ 74.6%

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

      2. metadata-eval 74.6%

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

   9. Simplified 74.6%

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

   if 4.8999999999999998e-4 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 90.9%

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

      1. *-commutative 90.9%

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

      2. distribute-lft-out 90.9%

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

      3. sub-neg 90.9%

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

      4. associate-*r/ 90.9%

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

      5. metadata-eval 90.9%

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

      6. metadata-eval 90.9%

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

      7. +-commutative 90.9%

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

      8. metadata-eval 90.9%

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

      9. distribute-neg-frac 90.9%

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

      10. unsub-neg 90.9%

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

   7. Simplified 90.9%

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

   8. Taylor expanded in y around 0 32.1%

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

      1. associate-*r/ 32.1%

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

      2. *-commutative 32.1%

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

      3. sub-neg 32.1%

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

      4. associate-*r/ 32.1%

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

      5. metadata-eval 32.1%

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

      6. metadata-eval 32.1%

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

      7. +-commutative 32.1%

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

      8. associate-/l* 32.1%

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

   10. Simplified 32.1%

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

   11. Taylor expanded in x around inf 39.8%

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

       1. *-commutative 39.8%

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

   13. Simplified 39.8%

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

Alternative 11: 38.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.1111111111111111}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (/ 0.1111111111111111 x)))

C

double code(double x, double y) {
	return sqrt((0.1111111111111111 / x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((0.1111111111111111d0 / x))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((0.1111111111111111 / x));
}

Python

def code(x, y):
	return math.sqrt((0.1111111111111111 / x))

Julia

function code(x, y)
	return sqrt(Float64(0.1111111111111111 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((0.1111111111111111 / x));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.5%

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

   5. fma-define 99.5%

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

   6. sub-neg 99.5%

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

   7. +-commutative 99.5%

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

   8. distribute-lft-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. metadata-eval 99.5%

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

   11. *-commutative 99.5%

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

   12. associate-/r* 99.5%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 36.5%

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

   1. add-sqr-sqrt 36.4%

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

   2. sqrt-unprod 36.5%

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

   3. associate-*l* 36.5%

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

   4. *-commutative 36.5%

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

   5. associate-*l* 36.5%

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

   6. add-sqr-sqrt 36.6%

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

   7. inv-pow 36.6%

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

   8. metadata-eval 36.6%

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

   9. unpow-prod-down 36.6%

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

   10. unpow-1 36.6%

       \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{1}{x \cdot 3}}} \]

   11. *-commutative 36.6%

       \[\leadsto \sqrt{0.3333333333333333 \cdot \frac{1}{\color{blue}{3 \cdot x}}} \]

   12. associate-/r* 36.5%

       \[\leadsto \sqrt{0.3333333333333333 \cdot \color{blue}{\frac{\frac{1}{3}}{x}}} \]

   13. metadata-eval 36.5%

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

7. Applied egg-rr 36.5%

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

   1. associate-*r/ 36.6%

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

   2. metadata-eval 36.6%

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

9. Simplified 36.6%

   \[\leadsto \color{blue}{\sqrt{\frac{0.1111111111111111}{x}}} \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024111 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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