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

Percentage Accurate: 99.4% → 99.4%

Time: 11.4s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l* 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := 3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{if}\;y \leq -270000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-76}:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* 3.0 (* (sqrt x) y))))
   (if (<= y -270000000000.0)
     t_1
     (if (<= y -2.65e-76)
       (pow (* x 9.0) -0.5)
       (if (<= y 3.2e-278)
         t_0
         (if (<= y 3.2e-197)
           (sqrt (/ 0.1111111111111111 x))
           (if (<= y 1.0) t_0 t_1)))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = 3.0 * (sqrt(x) * y);
	double tmp;
	if (y <= -270000000000.0) {
		tmp = t_1;
	} else if (y <= -2.65e-76) {
		tmp = pow((x * 9.0), -0.5);
	} else if (y <= 3.2e-278) {
		tmp = t_0;
	} else if (y <= 3.2e-197) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = 3.0d0 * (sqrt(x) * y)
    if (y <= (-270000000000.0d0)) then
        tmp = t_1
    else if (y <= (-2.65d-76)) then
        tmp = (x * 9.0d0) ** (-0.5d0)
    else if (y <= 3.2d-278) then
        tmp = t_0
    else if (y <= 3.2d-197) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (y <= 1.0d0) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = 3.0 * (Math.sqrt(x) * y);
	double tmp;
	if (y <= -270000000000.0) {
		tmp = t_1;
	} else if (y <= -2.65e-76) {
		tmp = Math.pow((x * 9.0), -0.5);
	} else if (y <= 3.2e-278) {
		tmp = t_0;
	} else if (y <= 3.2e-197) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = 3.0 * (math.sqrt(x) * y)
	tmp = 0
	if y <= -270000000000.0:
		tmp = t_1
	elif y <= -2.65e-76:
		tmp = math.pow((x * 9.0), -0.5)
	elif y <= 3.2e-278:
		tmp = t_0
	elif y <= 3.2e-197:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif y <= 1.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(3.0 * Float64(sqrt(x) * y))
	tmp = 0.0
	if (y <= -270000000000.0)
		tmp = t_1;
	elseif (y <= -2.65e-76)
		tmp = Float64(x * 9.0) ^ -0.5;
	elseif (y <= 3.2e-278)
		tmp = t_0;
	elseif (y <= 3.2e-197)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = 3.0 * (sqrt(x) * y);
	tmp = 0.0;
	if (y <= -270000000000.0)
		tmp = t_1;
	elseif (y <= -2.65e-76)
		tmp = (x * 9.0) ^ -0.5;
	elseif (y <= 3.2e-278)
		tmp = t_0;
	elseif (y <= 3.2e-197)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -270000000000.0], t$95$1, If[LessEqual[y, -2.65e-76], N[Power[N[(x * 9.0), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[y, 3.2e-278], t$95$0, If[LessEqual[y, 3.2e-197], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.7e11 or 1 < y

   1. Initial program 99.6%

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

      1. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 80.9%

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

   if -2.7e11 < y < -2.65e-76

   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. associate-*l* 99.8%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 33.4%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around 0 72.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. clear-num 72.1%

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

       2. inv-pow 72.1%

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

       3. div-inv 72.4%

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

       4. metadata-eval 72.4%

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

       5. sqrt-pow1 72.3%

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

       6. metadata-eval 72.3%

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

   11. Applied egg-rr 72.3%

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

   if -2.65e-76 < y < 3.20000000000000018e-278 or 3.1999999999999997e-197 < y < 1

   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. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/r* 99.4%

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

      10. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 71.8%

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

   6. Taylor expanded in y around 0 71.4%

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

      1. *-commutative 71.4%

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

   8. Simplified 71.4%

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

   if 3.20000000000000018e-278 < y < 3.1999999999999997e-197

   1. Initial program 99.2%

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

      1. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 53.7%

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

   8. Simplified 54.0%

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

   9. Taylor expanded in x around 0 76.4%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -270000000000:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq -2.65 \cdot 10^{-76}:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-197}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ \mathbf{if}\;y \leq -470000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-76}:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-282}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)))
   (if (<= y -470000000.0)
     (* (sqrt x) (* 3.0 y))
     (if (<= y -2.8e-76)
       (pow (* x 9.0) -0.5)
       (if (<= y 2e-282)
         t_0
         (if (<= y 5.3e-196)
           (sqrt (/ 0.1111111111111111 x))
           (if (<= y 1.0) t_0 (* 3.0 (* (sqrt x) y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double tmp;
	if (y <= -470000000.0) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= -2.8e-76) {
		tmp = pow((x * 9.0), -0.5);
	} else if (y <= 2e-282) {
		tmp = t_0;
	} else if (y <= 5.3e-196) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    if (y <= (-470000000.0d0)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= (-2.8d-76)) then
        tmp = (x * 9.0d0) ** (-0.5d0)
    else if (y <= 2d-282) then
        tmp = t_0
    else if (y <= 5.3d-196) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (y <= 1.0d0) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double tmp;
	if (y <= -470000000.0) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= -2.8e-76) {
		tmp = Math.pow((x * 9.0), -0.5);
	} else if (y <= 2e-282) {
		tmp = t_0;
	} else if (y <= 5.3e-196) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (y <= 1.0) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	tmp = 0
	if y <= -470000000.0:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= -2.8e-76:
		tmp = math.pow((x * 9.0), -0.5)
	elif y <= 2e-282:
		tmp = t_0
	elif y <= 5.3e-196:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif y <= 1.0:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	tmp = 0.0
	if (y <= -470000000.0)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= -2.8e-76)
		tmp = Float64(x * 9.0) ^ -0.5;
	elseif (y <= 2e-282)
		tmp = t_0;
	elseif (y <= 5.3e-196)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	tmp = 0.0;
	if (y <= -470000000.0)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= -2.8e-76)
		tmp = (x * 9.0) ^ -0.5;
	elseif (y <= 2e-282)
		tmp = t_0;
	elseif (y <= 5.3e-196)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (y <= 1.0)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, If[LessEqual[y, -470000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.8e-76], N[Power[N[(x * 9.0), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[y, 2e-282], t$95$0, If[LessEqual[y, 5.3e-196], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[y, 1.0], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -4.7e8

   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. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.7%

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

   6. Taylor expanded in y around inf 84.7%

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

   if -4.7e8 < y < -2.8000000000000001e-76

   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. associate-*l* 99.8%

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

      2. associate--l+ 99.8%

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

      3. sub-neg 99.8%

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

      4. *-commutative 99.8%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 33.4%

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

   8. Simplified 33.3%

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

   9. Taylor expanded in x around 0 72.1%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. clear-num 72.1%

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

       2. inv-pow 72.1%

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

       3. div-inv 72.4%

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

       4. metadata-eval 72.4%

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

       5. sqrt-pow1 72.3%

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

       6. metadata-eval 72.3%

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

   11. Applied egg-rr 72.3%

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

   if -2.8000000000000001e-76 < y < 2e-282 or 5.3000000000000001e-196 < y < 1

   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. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/r* 99.4%

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

      10. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around inf 71.8%

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

   6. Taylor expanded in y around 0 71.4%

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

      1. *-commutative 71.4%

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

   8. Simplified 71.4%

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

   if 2e-282 < y < 5.3000000000000001e-196

   1. Initial program 99.2%

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

      1. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 53.7%

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

   8. Simplified 54.0%

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

   9. Taylor expanded in x around 0 76.4%

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

   if 1 < y

   1. Initial program 99.6%

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

      1. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 76.8%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -470000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-76}:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-282}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{-196}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -850000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -850000000000.0)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 4.4e+39)
     (* (sqrt x) (- (* 0.3333333333333333 (/ 1.0 x)) 3.0))
     (* (sqrt (* x 9.0)) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -850000000000.0) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 4.4e+39) {
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-850000000000.0d0)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 4.4d+39) then
        tmp = sqrt(x) * ((0.3333333333333333d0 * (1.0d0 / x)) - 3.0d0)
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -850000000000.0) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 4.4e+39) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -850000000000.0:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 4.4e+39:
		tmp = math.sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0)
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -850000000000.0)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 4.4e+39)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 * Float64(1.0 / x)) - 3.0));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -850000000000.0)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 4.4e+39)
		tmp = sqrt(x) * ((0.3333333333333333 * (1.0 / x)) - 3.0);
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -850000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+39], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 * N[(1.0 / x), $MachinePrecision]), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5e11

   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. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.7%

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

   6. Taylor expanded in y around inf 84.7%

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

   if -8.5e11 < y < 4.4000000000000003e39

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.5%

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

      3. sub-neg 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-lft-in 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-/r* 99.5%

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

      10. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.3%

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

   if 4.4000000000000003e39 < y

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

      2. div-inv 99.7%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 99.3%

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

      5. pow2 99.3%

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

      6. +-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around inf 83.3%

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

      1. *-commutative 4.7%

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

      2. metadata-eval 4.7%

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

      3. sqrt-prod 6.5%

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

      4. pow1/2 6.5%

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

   7. Applied egg-rr 83.3%

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

      1. unpow1/2 6.5%

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

   9. Simplified 83.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -850000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{x} \cdot \left(0.3333333333333333 \cdot \frac{1}{x} - 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -150000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -150000000.0)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 2.2e+41)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* 3.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -150000000.0) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 2.2e+41) {
		tmp = sqrt(x) * ((0.3333333333333333 / 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 <= (-150000000.0d0)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 2.2d+41) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / 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 <= -150000000.0) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 2.2e+41) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -150000000.0:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 2.2e+41:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -150000000.0)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 2.2e+41)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / 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 <= -150000000.0)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 2.2e+41)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -150000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+41], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $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 < -1.5e8

   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. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.7%

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

   6. Taylor expanded in y around inf 84.7%

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

   if -1.5e8 < y < 2.1999999999999999e41

   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. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. associate-*r* 98.2%

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

      2. sub-neg 98.2%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-rgt-in 98.2%

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

      7. associate-*r* 98.3%

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

      8. associate-*l/ 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r* 98.3%

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

      11. metadata-eval 98.3%

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

      12. distribute-rgt-in 98.3%

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

   7. Simplified 98.3%

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

   if 2.1999999999999999e41 < y

   1. Initial program 99.6%

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

      1. associate-*l* 99.5%

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

      2. associate--l+ 99.5%

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

      3. sub-neg 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-/r* 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.2%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -150000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+41}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -68000000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -68000000000000.0)
   (* (sqrt x) (* 3.0 y))
   (if (<= y 4.4e+39)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* (sqrt (* x 9.0)) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -68000000000000.0) {
		tmp = sqrt(x) * (3.0 * y);
	} else if (y <= 4.4e+39) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-68000000000000.0d0)) then
        tmp = sqrt(x) * (3.0d0 * y)
    else if (y <= 4.4d+39) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = sqrt((x * 9.0d0)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -68000000000000.0) {
		tmp = Math.sqrt(x) * (3.0 * y);
	} else if (y <= 4.4e+39) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -68000000000000.0:
		tmp = math.sqrt(x) * (3.0 * y)
	elif y <= 4.4e+39:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -68000000000000.0)
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	elseif (y <= 4.4e+39)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(sqrt(Float64(x * 9.0)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -68000000000000.0)
		tmp = sqrt(x) * (3.0 * y);
	elseif (y <= 4.4e+39)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -68000000000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+39], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.8e13

   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. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 84.7%

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

   6. Taylor expanded in y around inf 84.7%

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

   if -6.8e13 < y < 4.4000000000000003e39

   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. associate-*l* 99.6%

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

      2. associate--l+ 99.6%

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

      3. sub-neg 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-/r* 99.5%

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

      6. metadata-eval 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 98.3%

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

      1. associate-*r* 98.2%

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

      2. sub-neg 98.2%

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

      3. associate-*r/ 98.2%

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

      4. metadata-eval 98.2%

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

      5. metadata-eval 98.2%

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

      6. distribute-rgt-in 98.2%

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

      7. associate-*r* 98.3%

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

      8. associate-*l/ 98.3%

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

      9. metadata-eval 98.3%

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

      10. associate-*r* 98.3%

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

      11. metadata-eval 98.3%

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

      12. distribute-rgt-in 98.3%

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

   7. Simplified 98.3%

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

   if 4.4000000000000003e39 < y

   1. Initial program 99.6%

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

      1. metadata-eval 99.6%

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

      2. div-inv 99.7%

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

      3. clear-num 99.6%

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

      4. add-sqr-sqrt 99.3%

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

      5. pow2 99.3%

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

      6. +-commutative 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in x around inf 83.3%

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

      1. *-commutative 4.7%

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

      2. metadata-eval 4.7%

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

      3. sqrt-prod 6.5%

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

      4. pow1/2 6.5%

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

   7. Applied egg-rr 83.3%

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

      1. unpow1/2 6.5%

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

   9. Simplified 83.3%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -68000000000000:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+39}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0003) (pow (* x 9.0) -0.5) (* 3.0 (* (sqrt x) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0003) {
		tmp = pow((x * 9.0), -0.5);
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0003d0) then
        tmp = (x * 9.0d0) ** (-0.5d0)
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0003) {
		tmp = Math.pow((x * 9.0), -0.5);
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0003:
		tmp = math.pow((x * 9.0), -0.5)
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0003)
		tmp = Float64(x * 9.0) ^ -0.5;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0003)
		tmp = (x * 9.0) ^ -0.5;
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0003], N[Power[N[(x * 9.0), $MachinePrecision], -0.5], $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.99999999999999974e-4

   1. Initial program 99.3%

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

      1. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around 0 72.6%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. clear-num 72.6%

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

       2. inv-pow 72.6%

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

       3. div-inv 72.6%

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

       4. metadata-eval 72.6%

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

       5. sqrt-pow1 72.7%

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

       6. metadata-eval 72.7%

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

   11. Applied egg-rr 72.7%

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

   if 2.99999999999999974e-4 < x

   1. Initial program 99.6%

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

      1. associate-*l* 99.7%

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

      2. associate--l+ 99.7%

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

      3. sub-neg 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-/r* 99.7%

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

      6. metadata-eval 99.7%

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

      7. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 97.8%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0003:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.045:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.045) (pow (* x 9.0) -0.5) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.045) {
		tmp = pow((x * 9.0), -0.5);
	} 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 <= 0.045d0) then
        tmp = (x * 9.0d0) ** (-0.5d0)
    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 <= 0.045) {
		tmp = Math.pow((x * 9.0), -0.5);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.045:
		tmp = math.pow((x * 9.0), -0.5)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.045)
		tmp = Float64(x * 9.0) ^ -0.5;
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.045)
		tmp = (x * 9.0) ^ -0.5;
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.045], N[Power[N[(x * 9.0), $MachinePrecision], -0.5], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.044999999999999998

   1. Initial program 99.3%

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

      1. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around 0 72.6%

      \[\leadsto \sqrt{\color{blue}{\frac{0.1111111111111111}{x}}} \]
   10. Step-by-step derivation

       1. clear-num 72.6%

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

       2. inv-pow 72.6%

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

       3. div-inv 72.6%

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

       4. metadata-eval 72.6%

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

       5. sqrt-pow1 72.7%

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

       6. metadata-eval 72.7%

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

   11. Applied egg-rr 72.7%

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

   if 0.044999999999999998 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.8%

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

   6. Taylor expanded in y around 0 49.6%

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

      1. *-commutative 49.6%

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

   8. Simplified 49.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.045:\\ \;\;\;\;{\left(x \cdot 9\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.045:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.045) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.045) {
		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 <= 0.045d0) 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 <= 0.045) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.045:
		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 <= 0.045)
		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 <= 0.045)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.045], 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 < 0.044999999999999998

   1. Initial program 99.3%

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

      1. associate-*l* 99.4%

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

      2. associate--l+ 99.4%

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

      3. sub-neg 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-/r* 99.4%

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

      6. metadata-eval 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 38.1%

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

   8. Simplified 38.1%

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

   9. Taylor expanded in x around 0 72.6%

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

   if 0.044999999999999998 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. sub-neg 99.6%

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

      4. +-commutative 99.6%

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

      5. distribute-lft-in 99.6%

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

      6. metadata-eval 99.6%

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

      7. metadata-eval 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-/r* 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 97.8%

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

   6. Taylor expanded in y around 0 49.6%

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

      1. *-commutative 49.6%

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

   8. Simplified 49.6%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.045:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 3.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x \cdot 9} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* x 9.0)))

C

double code(double x, double y) {
	return sqrt((x * 9.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((x * 9.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((x * 9.0));
}

Python

def code(x, y):
	return math.sqrt((x * 9.0))

Julia

function code(x, y)
	return sqrt(Float64(x * 9.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((x * 9.0));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\left(3 \cdot \sqrt{x}\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
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. sub-neg 99.5%

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

   4. +-commutative 99.5%

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

   5. distribute-lft-in 99.5%

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

   6. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/r* 99.5%

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

   10. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 59.9%

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

   1. add-cube-cbrt 59.1%

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

   2. pow3 59.1%

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

   3. div-inv 59.1%

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

   4. sub-neg 59.1%

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

   5. metadata-eval 59.1%

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

7. Applied egg-rr 59.1%

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

8. Taylor expanded in x around -inf 0.0%

   \[\leadsto \color{blue}{-3 \cdot \left({1}^{0.16666666666666666} \cdot \left(\sqrt{x} \cdot {\left(\sqrt{-1}\right)}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. pow-base-1 0.0%

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

   2. *-lft-identity 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 2.9%

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

   5. *-commutative 2.9%

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

   6. neg-mul-1 2.9%

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

   7. distribute-rgt-neg-in 2.9%

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

   8. distribute-lft-neg-in 2.9%

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

   9. metadata-eval 2.9%

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

10. Simplified 2.9%

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

    1. *-commutative 2.9%

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

    2. metadata-eval 2.9%

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

    3. sqrt-prod 3.3%

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

    4. pow1/2 3.3%

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

12. Applied egg-rr 3.3%

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

    1. unpow1/2 3.3%

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

14. Simplified 3.3%

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

15. Final simplification 3.3%

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

Alternative 11: 38.3% 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 99.5%

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

   1. associate-*l* 99.6%

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

   2. associate--l+ 99.6%

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

   3. sub-neg 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. metadata-eval 99.6%

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

3. Simplified 99.6%

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

6. Applied egg-rr 24.2%

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

8. Simplified 24.2%

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

9. Taylor expanded in x around 0 29.2%

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

10. Final simplification 29.2%

    \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]
11. Add Preprocessing

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024013 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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