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

Percentage Accurate: 99.4% → 99.3%

Time: 10.8s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. distribute-lft-in 99.5%

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

   2. div-inv 99.5%

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

   3. associate-*r* 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. +-commutative 99.5%

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

   7. distribute-rgt-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 85.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.8e-53)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 3.3e-21)
     (* 3.0 (* (sqrt x) y))
     (if (<= x 5.7e-8)
       (sqrt (+ (/ 0.1111111111111111 x) -2.0))
       (* (sqrt x) (* 3.0 (+ y -1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.8e-53) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 3.3e-21) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (x <= 5.7e-8) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.8d-53) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 3.3d-21) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (x <= 5.7d-8) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = sqrt(x) * (3.0d0 * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.8e-53) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 3.3e-21) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (x <= 5.7e-8) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * (3.0 * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.8e-53:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 3.3e-21:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif x <= 5.7e-8:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * (3.0 * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.8e-53)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 3.3e-21)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (x <= 5.7e-8)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.8e-53)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 3.3e-21)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (x <= 5.7e-8)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * (3.0 * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.8e-53], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.3e-21], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-8], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2.79999999999999985e-53

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.4%

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

      11. add-sqr-sqrt 91.9%

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

      12. sqrt-unprod 88.5%

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

      13. swap-sqr 27.3%

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

      14. add-sqr-sqrt 27.4%

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

      15. pow2 27.4%

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

      16. +-commutative 27.4%

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

   6. Applied egg-rr 27.4%

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

   7. Taylor expanded in x around 0 78.8%

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

   8. Taylor expanded in y around inf 78.8%

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

      1. *-commutative 78.8%

         \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

   10. Simplified 78.8%

       \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

   11. Taylor expanded in y around 0 78.1%

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

   if 2.79999999999999985e-53 < x < 3.30000000000000009e-21

   1. Initial program 99.1%

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

      1. *-commutative 99.1%

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

      2. associate-*l* 99.1%

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

      3. +-commutative 99.1%

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

      4. associate--l+ 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-/r* 99.1%

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

      7. metadata-eval 99.1%

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

      8. sub-neg 99.1%

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

      9. metadata-eval 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in y around inf 73.1%

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

   if 3.30000000000000009e-21 < x < 5.70000000000000009e-8

   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* 99.2%

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

      3. +-commutative 99.2%

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

      4. associate--l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.4%

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

      6. associate-*r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. div-inv 99.4%

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

      9. associate-+r+ 99.4%

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

      10. fma-udef 99.4%

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

      11. add-sqr-sqrt 85.9%

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

      12. sqrt-unprod 86.9%

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

      13. swap-sqr 86.9%

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

      14. add-sqr-sqrt 87.1%

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

      15. pow2 87.1%

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

      16. +-commutative 87.1%

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

   6. Applied egg-rr 87.1%

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

   7. Taylor expanded in x around 0 82.9%

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

   8. Taylor expanded in y around 0 83.0%

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

      1. sub-neg 83.0%

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

      2. associate-*r/ 83.0%

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

      3. metadata-eval 83.0%

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

      4. metadata-eval 83.0%

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

   10. Simplified 83.0%

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

   if 5.70000000000000009e-8 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 98.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-53}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot \left(y + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 5400000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.1e+45)
   (* y (sqrt (* x 9.0)))
   (if (<= y 5400000000.0)
     (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+45) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= 5400000000.0) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.1d+45)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= 5400000000.0d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.1e+45) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= 5400000000.0) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.1e+45:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= 5400000000.0:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.1e+45)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= 5400000000.0)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.1e+45)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= 5400000000.0)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.1e+45], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5400000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.09999999999999988e45

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. clear-num 99.4%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.5%

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

      15. *-commutative 99.5%

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

      16. metadata-eval 99.5%

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

      17. sqrt-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around inf 82.5%

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

   if -3.09999999999999988e45 < y < 5.4e9

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 95.3%

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

      1. *-commutative 95.3%

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

      2. sub-neg 95.3%

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

      3. associate-*r/ 95.2%

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

      4. metadata-eval 95.2%

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

      5. metadata-eval 95.2%

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

      6. associate-*r* 95.3%

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

      7. distribute-rgt-in 95.3%

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

      8. associate-*l/ 95.4%

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

      9. metadata-eval 95.4%

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

      10. metadata-eval 95.4%

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

      11. *-commutative 95.4%

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

   7. Simplified 95.4%

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

   if 5.4e9 < y

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+45}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 5400000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+27} \lor \neg \left(y \leq 30000000\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -2.35e+27) (not (<= y 30000000.0)))
   (* 3.0 (* (sqrt x) y))
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -2.35e+27) || !(y <= 30000000.0)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-2.35d+27)) .or. (.not. (y <= 30000000.0d0))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -2.35e+27) || !(y <= 30000000.0)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -2.35e+27) or not (y <= 30000000.0):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -2.35e+27) || !(y <= 30000000.0))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -2.35e+27) || ~((y <= 30000000.0)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -2.35e+27], N[Not[LessEqual[y, 30000000.0]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.34999999999999988e27 or 3e7 < y

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 82.1%

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

   if -2.34999999999999988e27 < y < 3e7

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 48.4%

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

      12. sqrt-unprod 49.2%

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

      13. swap-sqr 23.1%

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

      14. add-sqr-sqrt 23.1%

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

      15. pow2 23.1%

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

      16. +-commutative 23.1%

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

   6. Applied egg-rr 23.1%

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

   7. Taylor expanded in x around 0 48.5%

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

   8. Taylor expanded in y around 0 48.5%

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

      1. sub-neg 48.5%

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

      2. associate-*r/ 48.5%

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

      3. metadata-eval 48.5%

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

      4. metadata-eval 48.5%

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

   10. Simplified 48.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.35 \cdot 10^{+27} \lor \neg \left(y \leq 30000000\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+24}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 27000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.15e+24)
   (* 3.0 (* (sqrt x) y))
   (if (<= y 27000000000.0)
     (sqrt (+ (/ 0.1111111111111111 x) -2.0))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+24) {
		tmp = 3.0 * (sqrt(x) * y);
	} else if (y <= 27000000000.0) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-2.15d+24)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else if (y <= 27000000000.0d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.15e+24) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else if (y <= 27000000000.0) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.15e+24:
		tmp = 3.0 * (math.sqrt(x) * y)
	elif y <= 27000000000.0:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.15e+24)
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	elseif (y <= 27000000000.0)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.15e+24)
		tmp = 3.0 * (sqrt(x) * y);
	elseif (y <= 27000000000.0)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.15e+24], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 27000000000.0], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.14999999999999994e24

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 80.4%

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

   if -2.14999999999999994e24 < y < 2.7e10

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 48.4%

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

      12. sqrt-unprod 49.2%

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

      13. swap-sqr 23.1%

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

      14. add-sqr-sqrt 23.1%

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

      15. pow2 23.1%

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

      16. +-commutative 23.1%

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

   6. Applied egg-rr 23.1%

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

   7. Taylor expanded in x around 0 48.5%

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

   8. Taylor expanded in y around 0 48.5%

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

      1. sub-neg 48.5%

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

      2. associate-*r/ 48.5%

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

      3. metadata-eval 48.5%

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

      4. metadata-eval 48.5%

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

   10. Simplified 48.5%

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

   if 2.7e10 < y

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.15 \cdot 10^{+24}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{elif}\;y \leq 27000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 22000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -3.2e+24)
   (* y (sqrt (* x 9.0)))
   (if (<= y 22000000000.0)
     (sqrt (+ (/ 0.1111111111111111 x) -2.0))
     (* (sqrt x) (* y 3.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+24) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= 22000000000.0) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-3.2d+24)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= 22000000000.0d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else
        tmp = sqrt(x) * (y * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -3.2e+24) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= 22000000000.0) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * (y * 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -3.2e+24:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= 22000000000.0:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * (y * 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -3.2e+24)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= 22000000000.0)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -3.2e+24)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= 22000000000.0)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * (y * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -3.2e+24], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 22000000000.0], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1999999999999997e24

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. associate-*r* 99.4%

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

      2. *-commutative 99.4%

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

      3. +-commutative 99.4%

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

      4. associate-+r+ 99.4%

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

      5. +-commutative 99.4%

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

      6. distribute-lft-in 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. clear-num 99.4%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.5%

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

      15. *-commutative 99.5%

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

      16. metadata-eval 99.5%

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

      17. sqrt-prod 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around inf 80.5%

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

   if -3.1999999999999997e24 < y < 2.2e10

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-in 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. associate-*r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. div-inv 99.5%

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

      9. associate-+r+ 99.5%

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

      10. fma-udef 99.5%

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

      11. add-sqr-sqrt 48.4%

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

      12. sqrt-unprod 49.2%

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

      13. swap-sqr 23.1%

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

      14. add-sqr-sqrt 23.1%

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

      15. pow2 23.1%

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

      16. +-commutative 23.1%

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

   6. Applied egg-rr 23.1%

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

   7. Taylor expanded in x around 0 48.5%

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

   8. Taylor expanded in y around 0 48.5%

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

      1. sub-neg 48.5%

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

      2. associate-*r/ 48.5%

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

      3. metadata-eval 48.5%

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

      4. metadata-eval 48.5%

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

   10. Simplified 48.5%

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

   if 2.2e10 < y

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.6%

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

      3. +-commutative 99.6%

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

      4. associate--l+ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/r* 99.6%

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

      7. metadata-eval 99.6%

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

      8. sub-neg 99.6%

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

      9. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l* 84.0%

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

      3. *-commutative 84.0%

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

   7. Simplified 84.0%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+24}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 22000000000:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) (* y 3.0)))
   (* (sqrt (* x 9.0)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 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 (x <= 0.112d0) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (y * 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 (x <= 0.112) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	} else {
		tmp = Math.sqrt((x * 9.0)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0))
	else:
		tmp = math.sqrt((x * 9.0)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + Float64(y * 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 (x <= 0.112)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + (y * 3.0));
	else
		tmp = sqrt((x * 9.0)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. distribute-lft-in 99.4%

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

      2. div-inv 99.4%

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

      3. associate-*r* 99.4%

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

      4. metadata-eval 99.4%

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

      5. div-inv 99.5%

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

      6. +-commutative 99.5%

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

      7. distribute-rgt-in 99.5%

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

      8. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in y around inf 97.5%

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

   if 0.112000000000000002 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+r+ 99.5%

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

      5. +-commutative 99.5%

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

      6. distribute-lft-in 99.5%

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

      7. metadata-eval 99.5%

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

      8. sub-neg 99.5%

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

      9. clear-num 99.5%

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

      10. div-inv 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. metadata-eval 99.5%

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

      14. sqrt-prod 99.6%

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

      15. *-commutative 99.6%

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

      16. metadata-eval 99.6%

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

      17. sqrt-prod 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. distribute-lft-out 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 99.1%

      \[\leadsto \sqrt{x \cdot 9} \cdot \color{blue}{\left(y - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + y \cdot 3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt(x) * (3.0d0 * ((0.1111111111111111d0 / x) + (y + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 9: 2.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt (* y 2.0)))

C

double code(double x, double y) {
	return sqrt((y * 2.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((y * 2.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt((y * 2.0));
}

Python

def code(x, y):
	return math.sqrt((y * 2.0))

Julia

function code(x, y)
	return sqrt(Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt((y * 2.0));
end

Wolfram

code[x_, y_] := N[Sqrt[N[(y * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. distribute-lft-in 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. associate-*r* 99.5%

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

   7. metadata-eval 99.5%

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

   8. div-inv 99.5%

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

   9. associate-+r+ 99.5%

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

   10. fma-udef 99.5%

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

   11. add-sqr-sqrt 53.0%

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

   12. sqrt-unprod 45.6%

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

   13. swap-sqr 22.2%

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

   14. add-sqr-sqrt 22.2%

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

   15. pow2 22.2%

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

   16. +-commutative 22.2%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in x around 0 34.7%

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

8. Taylor expanded in y around inf 34.2%

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

   1. *-commutative 34.2%

      \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

10. Simplified 34.2%

    \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

11. Taylor expanded in y around inf 2.5%

    \[\leadsto \sqrt{\color{blue}{2 \cdot y}} \]
12. Step-by-step derivation

    1. *-commutative 2.5%

       \[\leadsto \sqrt{\color{blue}{y \cdot 2}} \]

13. Simplified 2.5%

    \[\leadsto \sqrt{\color{blue}{y \cdot 2}} \]

14. Final simplification 2.5%

    \[\leadsto \sqrt{y \cdot 2} \]
15. Add Preprocessing

Alternative 10: 37.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. distribute-lft-in 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. associate-*r* 99.5%

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

   7. metadata-eval 99.5%

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

   8. div-inv 99.5%

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

   9. associate-+r+ 99.5%

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

   10. fma-udef 99.5%

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

   11. add-sqr-sqrt 53.0%

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

   12. sqrt-unprod 45.6%

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

   13. swap-sqr 22.2%

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

   14. add-sqr-sqrt 22.2%

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

   15. pow2 22.2%

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

   16. +-commutative 22.2%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in x around 0 34.7%

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

8. Taylor expanded in y around inf 34.2%

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

   1. *-commutative 34.2%

      \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

10. Simplified 34.2%

    \[\leadsto \sqrt{\color{blue}{y \cdot 2} + 0.1111111111111111 \cdot \frac{1}{x}} \]

11. Taylor expanded in y around 0 34.1%

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

12. Final simplification 34.1%

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

Alternative 11: 0.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{-2} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (sqrt -2.0))

C

double code(double x, double y) {
	return sqrt(-2.0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sqrt((-2.0d0))
end function

Java

public static double code(double x, double y) {
	return Math.sqrt(-2.0);
}

Python

def code(x, y):
	return math.sqrt(-2.0)

Julia

function code(x, y)
	return sqrt(-2.0)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(-2.0);
end

Wolfram

code[x_, y_] := N[Sqrt[-2.0], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. distribute-lft-in 99.5%

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

   3. distribute-lft-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. associate-*r* 99.5%

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

   7. metadata-eval 99.5%

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

   8. div-inv 99.5%

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

   9. associate-+r+ 99.5%

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

   10. fma-udef 99.5%

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

   11. add-sqr-sqrt 53.0%

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

   12. sqrt-unprod 45.6%

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

   13. swap-sqr 22.2%

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

   14. add-sqr-sqrt 22.2%

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

   15. pow2 22.2%

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

   16. +-commutative 22.2%

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

6. Applied egg-rr 22.2%

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

7. Taylor expanded in x around 0 34.7%

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

8. Taylor expanded in y around 0 33.8%

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

   1. sub-neg 33.8%

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

   2. associate-*r/ 33.8%

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

   3. metadata-eval 33.8%

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

   4. metadata-eval 33.8%

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

10. Simplified 33.8%

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

11. Taylor expanded in x around inf 0.0%

    \[\leadsto \color{blue}{\sqrt{-2}} \]

12. Final simplification 0.0%

    \[\leadsto \sqrt{-2} \]
13. Add Preprocessing

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024031 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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