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

Percentage Accurate: 99.4% → 99.4%

Time: 10.1s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. sub-neg 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. metadata-eval 99.3%

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

   3. sqrt-prod 99.5%

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

   4. pow1/2 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
7. Step-by-step derivation

   1. unpow1/2 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := 3 \cdot \left(y \cdot \sqrt{x}\right)\\ t_2 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-180}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0))
        (t_1 (* 3.0 (* y (sqrt x))))
        (t_2 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -1.6e+56)
     t_1
     (if (<= y -2.35e-180)
       t_2
       (if (<= y -1.1e-238)
         t_0
         (if (<= y 8e-148)
           t_2
           (if (<= y 9.5e-45) t_0 (if (<= y 2e+91) t_2 t_1))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = 3.0 * (y * sqrt(x));
	double t_2 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.6e+56) {
		tmp = t_1;
	} else if (y <= -2.35e-180) {
		tmp = t_2;
	} else if (y <= -1.1e-238) {
		tmp = t_0;
	} else if (y <= 8e-148) {
		tmp = t_2;
	} else if (y <= 9.5e-45) {
		tmp = t_0;
	} else if (y <= 2e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = 3.0d0 * (y * sqrt(x))
    t_2 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-1.6d+56)) then
        tmp = t_1
    else if (y <= (-2.35d-180)) then
        tmp = t_2
    else if (y <= (-1.1d-238)) then
        tmp = t_0
    else if (y <= 8d-148) then
        tmp = t_2
    else if (y <= 9.5d-45) then
        tmp = t_0
    else if (y <= 2d+91) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = 3.0 * (y * Math.sqrt(x));
	double t_2 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.6e+56) {
		tmp = t_1;
	} else if (y <= -2.35e-180) {
		tmp = t_2;
	} else if (y <= -1.1e-238) {
		tmp = t_0;
	} else if (y <= 8e-148) {
		tmp = t_2;
	} else if (y <= 9.5e-45) {
		tmp = t_0;
	} else if (y <= 2e+91) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = 3.0 * (y * math.sqrt(x))
	t_2 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -1.6e+56:
		tmp = t_1
	elif y <= -2.35e-180:
		tmp = t_2
	elif y <= -1.1e-238:
		tmp = t_0
	elif y <= 8e-148:
		tmp = t_2
	elif y <= 9.5e-45:
		tmp = t_0
	elif y <= 2e+91:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(3.0 * Float64(y * sqrt(x)))
	t_2 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -1.6e+56)
		tmp = t_1;
	elseif (y <= -2.35e-180)
		tmp = t_2;
	elseif (y <= -1.1e-238)
		tmp = t_0;
	elseif (y <= 8e-148)
		tmp = t_2;
	elseif (y <= 9.5e-45)
		tmp = t_0;
	elseif (y <= 2e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = 3.0 * (y * sqrt(x));
	t_2 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -1.6e+56)
		tmp = t_1;
	elseif (y <= -2.35e-180)
		tmp = t_2;
	elseif (y <= -1.1e-238)
		tmp = t_0;
	elseif (y <= 8e-148)
		tmp = t_2;
	elseif (y <= 9.5e-45)
		tmp = t_0;
	elseif (y <= 2e+91)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.6e+56], t$95$1, If[LessEqual[y, -2.35e-180], t$95$2, If[LessEqual[y, -1.1e-238], t$95$0, If[LessEqual[y, 8e-148], t$95$2, If[LessEqual[y, 9.5e-45], t$95$0, If[LessEqual[y, 2e+91], t$95$2, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000002e56 or 2.00000000000000016e91 < 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. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 82.1%

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

   if -1.60000000000000002e56 < y < -2.34999999999999988e-180 or -1.09999999999999996e-238 < y < 7.99999999999999949e-148 or 9.5000000000000002e-45 < y < 2.00000000000000016e91

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 68.7%

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

      2. sqrt-unprod 68.5%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. pow2 34.7%

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

      6. +-commutative 34.7%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in x around 0 61.0%

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

   if -2.34999999999999988e-180 < y < -1.09999999999999996e-238 or 7.99999999999999949e-148 < y < 9.5000000000000002e-45

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 70.9%

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

   10. Taylor expanded in y around 0 70.7%

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

       1. *-commutative 70.7%

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

   12. Simplified 70.7%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+56}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \mathbf{elif}\;y \leq -2.35 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-238}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-148}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-182}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-239}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -3.5e+63)
     (* (sqrt x) (* y 3.0))
     (if (<= y -2.8e-182)
       t_1
       (if (<= y -6.8e-239)
         t_0
         (if (<= y 1.18e-147)
           t_1
           (if (<= y 2.25e-45)
             t_0
             (if (<= y 3.3e+91) t_1 (* 3.0 (* y (sqrt x)))))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -3.5e+63) {
		tmp = sqrt(x) * (y * 3.0);
	} else if (y <= -2.8e-182) {
		tmp = t_1;
	} else if (y <= -6.8e-239) {
		tmp = t_0;
	} else if (y <= 1.18e-147) {
		tmp = t_1;
	} else if (y <= 2.25e-45) {
		tmp = t_0;
	} else if (y <= 3.3e+91) {
		tmp = t_1;
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-3.5d+63)) then
        tmp = sqrt(x) * (y * 3.0d0)
    else if (y <= (-2.8d-182)) then
        tmp = t_1
    else if (y <= (-6.8d-239)) then
        tmp = t_0
    else if (y <= 1.18d-147) then
        tmp = t_1
    else if (y <= 2.25d-45) then
        tmp = t_0
    else if (y <= 3.3d+91) then
        tmp = t_1
    else
        tmp = 3.0d0 * (y * sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -3.5e+63) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else if (y <= -2.8e-182) {
		tmp = t_1;
	} else if (y <= -6.8e-239) {
		tmp = t_0;
	} else if (y <= 1.18e-147) {
		tmp = t_1;
	} else if (y <= 2.25e-45) {
		tmp = t_0;
	} else if (y <= 3.3e+91) {
		tmp = t_1;
	} else {
		tmp = 3.0 * (y * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -3.5e+63:
		tmp = math.sqrt(x) * (y * 3.0)
	elif y <= -2.8e-182:
		tmp = t_1
	elif y <= -6.8e-239:
		tmp = t_0
	elif y <= 1.18e-147:
		tmp = t_1
	elif y <= 2.25e-45:
		tmp = t_0
	elif y <= 3.3e+91:
		tmp = t_1
	else:
		tmp = 3.0 * (y * math.sqrt(x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -3.5e+63)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	elseif (y <= -2.8e-182)
		tmp = t_1;
	elseif (y <= -6.8e-239)
		tmp = t_0;
	elseif (y <= 1.18e-147)
		tmp = t_1;
	elseif (y <= 2.25e-45)
		tmp = t_0;
	elseif (y <= 3.3e+91)
		tmp = t_1;
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -3.5e+63)
		tmp = sqrt(x) * (y * 3.0);
	elseif (y <= -2.8e-182)
		tmp = t_1;
	elseif (y <= -6.8e-239)
		tmp = t_0;
	elseif (y <= 1.18e-147)
		tmp = t_1;
	elseif (y <= 2.25e-45)
		tmp = t_0;
	elseif (y <= 3.3e+91)
		tmp = t_1;
	else
		tmp = 3.0 * (y * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -3.5e+63], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.8e-182], t$95$1, If[LessEqual[y, -6.8e-239], t$95$0, If[LessEqual[y, 1.18e-147], t$95$1, If[LessEqual[y, 2.25e-45], t$95$0, If[LessEqual[y, 3.3e+91], t$95$1, N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.50000000000000029e63

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 79.6%

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

      3. *-commutative 79.6%

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

   7. Simplified 79.6%

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

   if -3.50000000000000029e63 < y < -2.79999999999999993e-182 or -6.8e-239 < y < 1.18000000000000003e-147 or 2.2499999999999999e-45 < y < 3.30000000000000017e91

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 68.7%

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

      2. sqrt-unprod 68.5%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. pow2 34.7%

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

      6. +-commutative 34.7%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in x around 0 61.0%

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

   if -2.79999999999999993e-182 < y < -6.8e-239 or 1.18000000000000003e-147 < y < 2.2499999999999999e-45

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 70.9%

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

   10. Taylor expanded in y around 0 70.7%

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

       1. *-commutative 70.7%

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

   12. Simplified 70.7%

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

   if 3.30000000000000017e91 < 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* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 85.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq -2.8 \cdot 10^{-182}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-239}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\sqrt{x \cdot 9}\\ t_1 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-184}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-44}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (sqrt (* x 9.0)))) (t_1 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -5e+52)
     (* (sqrt x) (* y 3.0))
     (if (<= y -3.7e-184)
       t_1
       (if (<= y -4.4e-238)
         t_0
         (if (<= y 1.3e-147)
           t_1
           (if (<= y 1.9e-44)
             t_0
             (if (<= y 2e+91) t_1 (* 3.0 (* y (sqrt x)))))))))))

C

double code(double x, double y) {
	double t_0 = -sqrt((x * 9.0));
	double t_1 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -5e+52) {
		tmp = sqrt(x) * (y * 3.0);
	} else if (y <= -3.7e-184) {
		tmp = t_1;
	} else if (y <= -4.4e-238) {
		tmp = t_0;
	} else if (y <= 1.3e-147) {
		tmp = t_1;
	} else if (y <= 1.9e-44) {
		tmp = t_0;
	} else if (y <= 2e+91) {
		tmp = t_1;
	} else {
		tmp = 3.0 * (y * sqrt(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -sqrt((x * 9.0d0))
    t_1 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-5d+52)) then
        tmp = sqrt(x) * (y * 3.0d0)
    else if (y <= (-3.7d-184)) then
        tmp = t_1
    else if (y <= (-4.4d-238)) then
        tmp = t_0
    else if (y <= 1.3d-147) then
        tmp = t_1
    else if (y <= 1.9d-44) then
        tmp = t_0
    else if (y <= 2d+91) then
        tmp = t_1
    else
        tmp = 3.0d0 * (y * sqrt(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -Math.sqrt((x * 9.0));
	double t_1 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -5e+52) {
		tmp = Math.sqrt(x) * (y * 3.0);
	} else if (y <= -3.7e-184) {
		tmp = t_1;
	} else if (y <= -4.4e-238) {
		tmp = t_0;
	} else if (y <= 1.3e-147) {
		tmp = t_1;
	} else if (y <= 1.9e-44) {
		tmp = t_0;
	} else if (y <= 2e+91) {
		tmp = t_1;
	} else {
		tmp = 3.0 * (y * Math.sqrt(x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -math.sqrt((x * 9.0))
	t_1 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -5e+52:
		tmp = math.sqrt(x) * (y * 3.0)
	elif y <= -3.7e-184:
		tmp = t_1
	elif y <= -4.4e-238:
		tmp = t_0
	elif y <= 1.3e-147:
		tmp = t_1
	elif y <= 1.9e-44:
		tmp = t_0
	elif y <= 2e+91:
		tmp = t_1
	else:
		tmp = 3.0 * (y * math.sqrt(x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-sqrt(Float64(x * 9.0)))
	t_1 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -5e+52)
		tmp = Float64(sqrt(x) * Float64(y * 3.0));
	elseif (y <= -3.7e-184)
		tmp = t_1;
	elseif (y <= -4.4e-238)
		tmp = t_0;
	elseif (y <= 1.3e-147)
		tmp = t_1;
	elseif (y <= 1.9e-44)
		tmp = t_0;
	elseif (y <= 2e+91)
		tmp = t_1;
	else
		tmp = Float64(3.0 * Float64(y * sqrt(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -sqrt((x * 9.0));
	t_1 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -5e+52)
		tmp = sqrt(x) * (y * 3.0);
	elseif (y <= -3.7e-184)
		tmp = t_1;
	elseif (y <= -4.4e-238)
		tmp = t_0;
	elseif (y <= 1.3e-147)
		tmp = t_1;
	elseif (y <= 1.9e-44)
		tmp = t_0;
	elseif (y <= 2e+91)
		tmp = t_1;
	else
		tmp = 3.0 * (y * sqrt(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = (-N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -5e+52], N[(N[Sqrt[x], $MachinePrecision] * N[(y * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -3.7e-184], t$95$1, If[LessEqual[y, -4.4e-238], t$95$0, If[LessEqual[y, 1.3e-147], t$95$1, If[LessEqual[y, 1.9e-44], t$95$0, If[LessEqual[y, 2e+91], t$95$1, N[(3.0 * N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5e52

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*l* 79.6%

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

      3. *-commutative 79.6%

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

   7. Simplified 79.6%

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

   if -5e52 < y < -3.6999999999999999e-184 or -4.39999999999999982e-238 < y < 1.2999999999999999e-147 or 1.9e-44 < y < 2.00000000000000016e91

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 68.7%

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

      2. sqrt-unprod 68.5%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. pow2 34.7%

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

      6. +-commutative 34.7%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in x around 0 61.0%

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

   if -3.6999999999999999e-184 < y < -4.39999999999999982e-238 or 1.2999999999999999e-147 < y < 1.9e-44

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 99.7%

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

       1. sub-neg 99.7%

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

       2. associate-*r/ 99.7%

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

       3. metadata-eval 99.7%

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

       4. metadata-eval 99.7%

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

       5. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in x around inf 70.9%

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

   if 2.00000000000000016e91 < 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* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.4%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 85.5%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{x} \cdot \left(y \cdot 3\right)\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-184}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-238}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-44}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(y \cdot \sqrt{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ t_1 := -t\_0\\ t_2 := t\_0 \cdot y\\ t_3 := \sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{if}\;y \leq -1.72 \cdot 10^{+56}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-180}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-240}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-147}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0)))
        (t_1 (- t_0))
        (t_2 (* t_0 y))
        (t_3 (sqrt (/ 0.1111111111111111 x))))
   (if (<= y -1.72e+56)
     t_2
     (if (<= y -2.3e-180)
       t_3
       (if (<= y -8e-240)
         t_1
         (if (<= y 1.2e-147)
           t_3
           (if (<= y 9.5e-45) t_1 (if (<= y 2.9e+91) t_3 t_2))))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double t_3 = sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.72e+56) {
		tmp = t_2;
	} else if (y <= -2.3e-180) {
		tmp = t_3;
	} else if (y <= -8e-240) {
		tmp = t_1;
	} else if (y <= 1.2e-147) {
		tmp = t_3;
	} else if (y <= 9.5e-45) {
		tmp = t_1;
	} else if (y <= 2.9e+91) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sqrt((x * 9.0d0))
    t_1 = -t_0
    t_2 = t_0 * y
    t_3 = sqrt((0.1111111111111111d0 / x))
    if (y <= (-1.72d+56)) then
        tmp = t_2
    else if (y <= (-2.3d-180)) then
        tmp = t_3
    else if (y <= (-8d-240)) then
        tmp = t_1
    else if (y <= 1.2d-147) then
        tmp = t_3
    else if (y <= 9.5d-45) then
        tmp = t_1
    else if (y <= 2.9d+91) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double t_1 = -t_0;
	double t_2 = t_0 * y;
	double t_3 = Math.sqrt((0.1111111111111111 / x));
	double tmp;
	if (y <= -1.72e+56) {
		tmp = t_2;
	} else if (y <= -2.3e-180) {
		tmp = t_3;
	} else if (y <= -8e-240) {
		tmp = t_1;
	} else if (y <= 1.2e-147) {
		tmp = t_3;
	} else if (y <= 9.5e-45) {
		tmp = t_1;
	} else if (y <= 2.9e+91) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	t_1 = -t_0
	t_2 = t_0 * y
	t_3 = math.sqrt((0.1111111111111111 / x))
	tmp = 0
	if y <= -1.72e+56:
		tmp = t_2
	elif y <= -2.3e-180:
		tmp = t_3
	elif y <= -8e-240:
		tmp = t_1
	elif y <= 1.2e-147:
		tmp = t_3
	elif y <= 9.5e-45:
		tmp = t_1
	elif y <= 2.9e+91:
		tmp = t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	t_1 = Float64(-t_0)
	t_2 = Float64(t_0 * y)
	t_3 = sqrt(Float64(0.1111111111111111 / x))
	tmp = 0.0
	if (y <= -1.72e+56)
		tmp = t_2;
	elseif (y <= -2.3e-180)
		tmp = t_3;
	elseif (y <= -8e-240)
		tmp = t_1;
	elseif (y <= 1.2e-147)
		tmp = t_3;
	elseif (y <= 9.5e-45)
		tmp = t_1;
	elseif (y <= 2.9e+91)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	t_1 = -t_0;
	t_2 = t_0 * y;
	t_3 = sqrt((0.1111111111111111 / x));
	tmp = 0.0;
	if (y <= -1.72e+56)
		tmp = t_2;
	elseif (y <= -2.3e-180)
		tmp = t_3;
	elseif (y <= -8e-240)
		tmp = t_1;
	elseif (y <= 1.2e-147)
		tmp = t_3;
	elseif (y <= 9.5e-45)
		tmp = t_1;
	elseif (y <= 2.9e+91)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(t$95$0 * y), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -1.72e+56], t$95$2, If[LessEqual[y, -2.3e-180], t$95$3, If[LessEqual[y, -8e-240], t$95$1, If[LessEqual[y, 1.2e-147], t$95$3, If[LessEqual[y, 9.5e-45], t$95$1, If[LessEqual[y, 2.9e+91], t$95$3, t$95$2]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.72e56 or 2.90000000000000014e91 < 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%

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

      2. associate--l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf 82.3%

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

   if -1.72e56 < y < -2.29999999999999996e-180 or -7.9999999999999998e-240 < y < 1.19999999999999999e-147 or 9.5000000000000002e-45 < y < 2.90000000000000014e91

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. add-sqr-sqrt 68.7%

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

      2. sqrt-unprod 68.5%

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

      3. swap-sqr 34.7%

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

      4. add-sqr-sqrt 34.7%

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

      5. pow2 34.7%

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

      6. +-commutative 34.7%

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

   6. Applied egg-rr 34.7%

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

   7. Taylor expanded in x around 0 61.0%

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

   if -2.29999999999999996e-180 < y < -7.9999999999999998e-240 or 1.19999999999999999e-147 < y < 9.5000000000000002e-45

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 99.7%

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

       1. sub-neg 99.7%

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

       2. associate-*r/ 99.7%

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

       3. metadata-eval 99.7%

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

       4. metadata-eval 99.7%

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

       5. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in x around inf 70.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.72 \cdot 10^{+56}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{-240}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-147}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-45}:\\ \;\;\;\;-\sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+56} \lor \neg \left(y \leq 2 \cdot 10^{+91}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4.7e+56) (not (<= y 2e+91)))
   (* (sqrt (* x 9.0)) y)
   (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4.7e+56) || !(y <= 2e+91)) {
		tmp = sqrt((x * 9.0)) * y;
	} else {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4.7d+56)) .or. (.not. (y <= 2d+91))) then
        tmp = sqrt((x * 9.0d0)) * y
    else
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4.7e+56) || !(y <= 2e+91)) {
		tmp = Math.sqrt((x * 9.0)) * y;
	} else {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4.7e+56) or not (y <= 2e+91):
		tmp = math.sqrt((x * 9.0)) * y
	else:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4.7e+56) || !(y <= 2e+91))
		tmp = Float64(sqrt(Float64(x * 9.0)) * y);
	else
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4.7e+56) || ~((y <= 2e+91)))
		tmp = sqrt((x * 9.0)) * y;
	else
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4.7e+56], N[Not[LessEqual[y, 2e+91]], $MachinePrecision]], N[(N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision] * y), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7e56 or 2.00000000000000016e91 < 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%

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

      2. associate--l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.6%

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

      4. pow1/2 99.6%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.6%

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

   8. Simplified 99.6%

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

   9. Taylor expanded in y around inf 82.3%

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

   if -4.7e56 < y < 2.00000000000000016e91

   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. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 92.0%

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

      1. sub-neg 92.0%

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

      2. metadata-eval 92.0%

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

      3. associate-*r/ 92.0%

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

      4. metadata-eval 92.0%

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

      5. +-commutative 92.0%

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

   7. Simplified 92.0%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+56} \lor \neg \left(y \leq 2 \cdot 10^{+91}\right):\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x \cdot 9}\\ \mathbf{if}\;y \leq -3.1 \cdot 10^{+54}:\\ \;\;\;\;t\_0 \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (sqrt (* x 9.0))))
   (if (<= y -3.1e+54)
     (* t_0 y)
     (if (<= y 1.65e+91)
       (* (sqrt x) (+ (/ 0.3333333333333333 x) -3.0))
       (* t_0 (+ y -1.0))))))

C

double code(double x, double y) {
	double t_0 = sqrt((x * 9.0));
	double tmp;
	if (y <= -3.1e+54) {
		tmp = t_0 * y;
	} else if (y <= 1.65e+91) {
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_0 * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt((x * 9.0d0))
    if (y <= (-3.1d+54)) then
        tmp = t_0 * y
    else if (y <= 1.65d+91) then
        tmp = sqrt(x) * ((0.3333333333333333d0 / x) + (-3.0d0))
    else
        tmp = t_0 * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt((x * 9.0));
	double tmp;
	if (y <= -3.1e+54) {
		tmp = t_0 * y;
	} else if (y <= 1.65e+91) {
		tmp = Math.sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	} else {
		tmp = t_0 * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt((x * 9.0))
	tmp = 0
	if y <= -3.1e+54:
		tmp = t_0 * y
	elif y <= 1.65e+91:
		tmp = math.sqrt(x) * ((0.3333333333333333 / x) + -3.0)
	else:
		tmp = t_0 * (y + -1.0)
	return tmp

Julia

function code(x, y)
	t_0 = sqrt(Float64(x * 9.0))
	tmp = 0.0
	if (y <= -3.1e+54)
		tmp = Float64(t_0 * y);
	elseif (y <= 1.65e+91)
		tmp = Float64(sqrt(x) * Float64(Float64(0.3333333333333333 / x) + -3.0));
	else
		tmp = Float64(t_0 * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt((x * 9.0));
	tmp = 0.0;
	if (y <= -3.1e+54)
		tmp = t_0 * y;
	elseif (y <= 1.65e+91)
		tmp = sqrt(x) * ((0.3333333333333333 / x) + -3.0);
	else
		tmp = t_0 * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y, -3.1e+54], N[(t$95$0 * y), $MachinePrecision], If[LessEqual[y, 1.65e+91], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.0999999999999999e54

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around inf 79.7%

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

   if -3.0999999999999999e54 < y < 1.65000000000000009e91

   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. associate--l+ 99.3%

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

      4. distribute-lft-in 99.3%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 92.0%

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

      1. sub-neg 92.0%

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

      2. metadata-eval 92.0%

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

      3. associate-*r/ 92.0%

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

      4. metadata-eval 92.0%

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

      5. +-commutative 92.0%

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

   7. Simplified 92.0%

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

   if 1.65000000000000009e91 < 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%

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

      2. associate--l+ 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/r* 99.3%

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

      5. metadata-eval 99.3%

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

      6. sub-neg 99.3%

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

      7. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. *-commutative 99.3%

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

      2. metadata-eval 99.3%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in x around inf 85.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot y\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+91}:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{0.3333333333333333}{x} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot 9} \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.3%

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

   5. metadata-eval 99.3%

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

   6. sub-neg 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. metadata-eval 99.3%

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

   3. sqrt-prod 99.5%

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

   4. pow1/2 99.5%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
7. Step-by-step derivation

   1. unpow1/2 99.5%

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

8. Simplified 99.5%

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

9. Taylor expanded in y around 0 99.3%

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

    1. associate-*r* 99.3%

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

    2. associate-*r* 99.3%

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

    3. distribute-lft-out 99.3%

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

    4. sub-neg 99.3%

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

    5. associate-*r/ 99.3%

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

    6. metadata-eval 99.3%

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

    7. metadata-eval 99.3%

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

    8. associate-+l+ 99.3%

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

    9. +-commutative 99.3%

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

    10. *-commutative 99.3%

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

    11. associate-+r+ 99.3%

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

    12. associate-*l* 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return Float64(sqrt(x) * Float64(Float64(Float64(0.3333333333333333 / x) + -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[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] + N[(y * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. fma-undefine 99.4%

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

   2. +-commutative 99.4%

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

   3. +-commutative 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 10: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   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* 99.2%

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

      3. associate--l+ 99.2%

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

      4. distribute-lft-in 99.2%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

      1. add-sqr-sqrt 86.0%

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

      2. sqrt-unprod 81.8%

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

      3. swap-sqr 32.2%

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

      4. add-sqr-sqrt 32.2%

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

      5. pow2 32.2%

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

      6. +-commutative 32.2%

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

   6. Applied egg-rr 32.2%

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

   7. Taylor expanded in x around 0 70.2%

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

   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%

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

      2. associate--l+ 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-/r* 99.5%

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

      5. metadata-eval 99.5%

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

      6. sub-neg 99.5%

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

      7. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

      1. *-commutative 99.5%

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

      2. metadata-eval 99.5%

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

      3. sqrt-prod 99.7%

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

      4. pow1/2 99.7%

         \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(x \cdot 9\right)}^{0.5}} \cdot \left(\frac{0.1111111111111111}{x} + \left(y + -1\right)\right) \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 97.9%

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

   10. Taylor expanded in y around 0 48.9%

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

       1. *-commutative 48.9%

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

   12. Simplified 48.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 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. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.4%

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

   14. metadata-eval 99.4%

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

   15. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 58.4%

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

   2. sqrt-unprod 51.0%

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

   3. swap-sqr 23.1%

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

   4. add-sqr-sqrt 23.1%

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

   5. pow2 23.1%

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

   6. +-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 40.3%

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

8. Final simplification 40.3%

   \[\leadsto \sqrt{\frac{0.1111111111111111}{x}} \]
9. 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 2024095 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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