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

Percentage Accurate: 99.4% → 99.4%

Time: 11.0s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.4%

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

   3. 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-def 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/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.4%

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

   2. inv-pow 99.4%

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

   3. div-inv 99.5%

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

   4. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-/r* 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   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-def 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/ 99.5%

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

   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. Final simplification 99.5%

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

Alternative 3: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0015:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{+239} \lor \neg \left(x \leq 4.7 \cdot 10^{+257}\right) \land x \leq 5.4 \cdot 10^{+293}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0015)
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))
   (if (or (<= x 1.76e+239) (and (not (<= x 4.7e+257)) (<= x 5.4e+293)))
     (* 3.0 (* (sqrt x) y))
     (* (sqrt x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0015) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else if ((x <= 1.76e+239) || (!(x <= 4.7e+257) && (x <= 5.4e+293))) {
		tmp = 3.0 * (sqrt(x) * y);
	} 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 <= 0.0015d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else if ((x <= 1.76d+239) .or. (.not. (x <= 4.7d+257)) .and. (x <= 5.4d+293)) then
        tmp = 3.0d0 * (sqrt(x) * y)
    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 <= 0.0015) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else if ((x <= 1.76e+239) || (!(x <= 4.7e+257) && (x <= 5.4e+293))) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0015:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	elif (x <= 1.76e+239) or (not (x <= 4.7e+257) and (x <= 5.4e+293)):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0015)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	elseif ((x <= 1.76e+239) || (!(x <= 4.7e+257) && (x <= 5.4e+293)))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0015)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	elseif ((x <= 1.76e+239) || (~((x <= 4.7e+257)) && (x <= 5.4e+293)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0015], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[x, 1.76e+239], And[N[Not[LessEqual[x, 4.7e+257]], $MachinePrecision], LessEqual[x, 5.4e+293]]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0015

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. add-sqr-sqrt 75.5%

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

      2. sqrt-unprod 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. swap-sqr 75.7%

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

      6. swap-sqr 33.2%

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

      7. add-sqr-sqrt 33.1%

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

      8. pow2 33.1%

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

      9. sub-neg 33.1%

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

      10. metadata-eval 33.1%

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

      11. +-commutative 33.1%

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

      12. un-div-inv 33.1%

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

      13. metadata-eval 33.1%

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

   7. Applied egg-rr 33.1%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. associate-*r/ 75.2%

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

      3. metadata-eval 75.2%

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

      4. metadata-eval 75.2%

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

   10. Simplified 75.2%

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

   if 0.0015 < x < 1.76e239 or 4.7e257 < x < 5.4000000000000002e293

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 63.0%

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

   if 1.76e239 < x < 4.7e257 or 5.4000000000000002e293 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in y around 0 90.2%

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

      1. *-commutative 90.2%

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

   8. Simplified 90.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0015:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;x \leq 1.76 \cdot 10^{+239} \lor \neg \left(x \leq 4.7 \cdot 10^{+257}\right) \land x \leq 5.4 \cdot 10^{+293}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+238}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+257} \lor \neg \left(x \leq 5.5 \cdot 10^{+293}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.002)
   (sqrt (+ (/ 0.1111111111111111 x) -2.0))
   (if (<= x 3.1e+238)
     (* y (* (sqrt x) 3.0))
     (if (or (<= x 3.8e+257) (not (<= x 5.5e+293)))
       (* (sqrt x) -3.0)
       (* 3.0 (* (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.002) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (x <= 3.1e+238) {
		tmp = y * (sqrt(x) * 3.0);
	} else if ((x <= 3.8e+257) || !(x <= 5.5e+293)) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.002d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    else if (x <= 3.1d+238) then
        tmp = y * (sqrt(x) * 3.0d0)
    else if ((x <= 3.8d+257) .or. (.not. (x <= 5.5d+293))) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.002) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else if (x <= 3.1e+238) {
		tmp = y * (Math.sqrt(x) * 3.0);
	} else if ((x <= 3.8e+257) || !(x <= 5.5e+293)) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.002:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	elif x <= 3.1e+238:
		tmp = y * (math.sqrt(x) * 3.0)
	elif (x <= 3.8e+257) or not (x <= 5.5e+293):
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.002)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	elseif (x <= 3.1e+238)
		tmp = Float64(y * Float64(sqrt(x) * 3.0));
	elseif ((x <= 3.8e+257) || !(x <= 5.5e+293))
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.002)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	elseif (x <= 3.1e+238)
		tmp = y * (sqrt(x) * 3.0);
	elseif ((x <= 3.8e+257) || ~((x <= 5.5e+293)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.002], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 3.1e+238], N[(y * N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 3.8e+257], N[Not[LessEqual[x, 5.5e+293]], $MachinePrecision]], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < 2e-3

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. add-sqr-sqrt 75.5%

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

      2. sqrt-unprod 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. swap-sqr 75.7%

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

      6. swap-sqr 33.2%

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

      7. add-sqr-sqrt 33.1%

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

      8. pow2 33.1%

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

      9. sub-neg 33.1%

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

      10. metadata-eval 33.1%

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

      11. +-commutative 33.1%

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

      12. un-div-inv 33.1%

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

      13. metadata-eval 33.1%

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

   7. Applied egg-rr 33.1%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. associate-*r/ 75.2%

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

      3. metadata-eval 75.2%

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

      4. metadata-eval 75.2%

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

   10. Simplified 75.2%

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

   if 2e-3 < x < 3.10000000000000012e238

   1. Initial program 99.5%

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

      1. expm1-log1p-u 93.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-udef 93.8%

         \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      3. *-commutative 93.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x} \cdot 3}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      4. metadata-eval 93.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{x} \cdot \color{blue}{\sqrt{9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      5. sqrt-prod 93.8%

         \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{x \cdot 9}}\right)} - 1\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

   4. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)} - 1\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]
   5. Step-by-step derivation

      1. expm1-def 93.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x \cdot 9}\right)\right)} \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right) \]

      2. expm1-log1p 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around inf 60.6%

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

      1. *-commutative 60.6%

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

      2. *-commutative 60.6%

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

      3. associate-*r* 60.7%

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

      4. *-commutative 60.7%

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

   9. Simplified 60.7%

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

   if 3.10000000000000012e238 < x < 3.79999999999999998e257 or 5.5000000000000003e293 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-def 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.6%

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

   6. Taylor expanded in y around 0 90.2%

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

      1. *-commutative 90.2%

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

   8. Simplified 90.2%

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

   if 3.79999999999999998e257 < x < 5.5000000000000003e293

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. +-commutative 99.4%

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

      4. associate--l+ 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 79.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+238}:\\ \;\;\;\;y \cdot \left(\sqrt{x} \cdot 3\right)\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+257} \lor \neg \left(x \leq 5.5 \cdot 10^{+293}\right):\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+23} \lor \neg \left(y \leq 8.2 \cdot 10^{+21}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -4e+23) (not (<= y 8.2e+21)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -4e+23) || !(y <= 8.2e+21)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-4d+23)) .or. (.not. (y <= 8.2d+21))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -4e+23) || !(y <= 8.2e+21)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -4e+23) or not (y <= 8.2e+21):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -4e+23) || !(y <= 8.2e+21))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -4e+23) || ~((y <= 8.2e+21)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -4e+23], N[Not[LessEqual[y, 8.2e+21]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999997e23 or 8.2e21 < y

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around inf 81.1%

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

   if -3.9999999999999997e23 < y < 8.2e21

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.5%

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

      3. +-commutative 99.5%

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

      4. associate--l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.4%

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

      7. metadata-eval 99.4%

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

      8. sub-neg 99.4%

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

      9. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 96.4%

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

      1. *-commutative 96.4%

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

      2. sub-neg 96.4%

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

      3. associate-*r/ 96.4%

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

      4. metadata-eval 96.4%

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

      5. metadata-eval 96.4%

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

      6. associate-*r* 96.5%

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

      7. distribute-rgt-in 96.5%

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

      8. associate-*l/ 96.6%

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

      9. metadata-eval 96.6%

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

      10. metadata-eval 96.6%

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

      11. *-commutative 96.6%

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

   7. Simplified 96.6%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+23} \lor \neg \left(y \leq 8.2 \cdot 10^{+21}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	return (sqrt(x) * 3.0) * ((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 = (sqrt(x) * 3.0d0) * ((y + (1.0d0 / (x * 9.0d0))) + (-1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

   \[\leadsto \left(\sqrt{x} \cdot 3\right) \cdot \left(\left(y + \frac{1}{x \cdot 9}\right) + -1\right) \]
4. Add Preprocessing

Alternative 7: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.026)
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.026d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = sqrt(x) * ((3.0d0 * y) - 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.026) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.026:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.026)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(sqrt(x) * Float64(Float64(3.0 * y) - 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.026)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.026], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(3.0 * y), $MachinePrecision] - 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0259999999999999988

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. sub-neg 75.7%

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

      3. associate-*r/ 75.8%

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

      4. metadata-eval 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-*r* 75.9%

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

      7. distribute-rgt-in 75.9%

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

      8. associate-*l/ 76.0%

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

      9. metadata-eval 76.0%

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

      10. metadata-eval 76.0%

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

      11. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   if 0.0259999999999999988 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.0045)
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
   (* (* (sqrt x) 3.0) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.0045) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.0045d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = (sqrt(x) * 3.0d0) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.0045) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = (Math.sqrt(x) * 3.0) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.0045:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = (math.sqrt(x) * 3.0) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.0045)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(sqrt(x) * 3.0) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.0045)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = (sqrt(x) * 3.0) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.0045], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00449999999999999966

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. *-commutative 75.7%

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

      2. sub-neg 75.7%

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

      3. associate-*r/ 75.8%

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

      4. metadata-eval 75.8%

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

      5. metadata-eval 75.8%

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

      6. associate-*r* 75.9%

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

      7. distribute-rgt-in 75.9%

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

      8. associate-*l/ 76.0%

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

      9. metadata-eval 76.0%

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

      10. metadata-eval 76.0%

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

      11. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   if 0.00449999999999999966 < 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. Add Preprocessing

   3. Taylor expanded in y around inf 99.4%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0045:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{x} \cdot 3\right) \cdot \left(y + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 10: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate--l+ 99.5%

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

   2. sub-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 11: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.056) (sqrt (+ (/ 0.1111111111111111 x) -2.0)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.056) {
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	} 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 <= 0.056d0) then
        tmp = sqrt(((0.1111111111111111d0 / x) + (-2.0d0)))
    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 <= 0.056) {
		tmp = Math.sqrt(((0.1111111111111111 / x) + -2.0));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.056:
		tmp = math.sqrt(((0.1111111111111111 / x) + -2.0))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.056)
		tmp = sqrt(Float64(Float64(0.1111111111111111 / x) + -2.0));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.056)
		tmp = sqrt(((0.1111111111111111 / x) + -2.0));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.056], N[Sqrt[N[(N[(0.1111111111111111 / x), $MachinePrecision] + -2.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0560000000000000012

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. add-sqr-sqrt 75.5%

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

      2. sqrt-unprod 75.7%

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

      3. *-commutative 75.7%

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

      4. *-commutative 75.7%

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

      5. swap-sqr 75.7%

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

      6. swap-sqr 33.2%

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

      7. add-sqr-sqrt 33.1%

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

      8. pow2 33.1%

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

      9. sub-neg 33.1%

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

      10. metadata-eval 33.1%

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

      11. +-commutative 33.1%

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

      12. un-div-inv 33.1%

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

      13. metadata-eval 33.1%

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

   7. Applied egg-rr 33.1%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. sub-neg 75.2%

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

      2. associate-*r/ 75.2%

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

      3. metadata-eval 75.2%

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

      4. metadata-eval 75.2%

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

   10. Simplified 75.2%

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

   if 0.0560000000000000012 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.4%

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

   6. Taylor expanded in y around 0 45.9%

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

      1. *-commutative 45.9%

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

   8. Simplified 45.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.056:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x} + -2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 11200:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 11200.0) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 11200.0) {
		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 <= 11200.0d0) 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 <= 11200.0) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 11200.0:
		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 <= 11200.0)
		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 <= 11200.0)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 11200.0], 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 < 11200

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.3%

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

      3. +-commutative 99.3%

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

      4. associate--l+ 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-/r* 99.3%

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

      7. metadata-eval 99.3%

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

      8. sub-neg 99.3%

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

      9. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 74.6%

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

      1. add-sqr-sqrt 74.4%

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

      2. sqrt-unprod 74.6%

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

      3. *-commutative 74.6%

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

      4. *-commutative 74.6%

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

      5. swap-sqr 74.6%

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

      6. swap-sqr 32.7%

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

      7. add-sqr-sqrt 32.7%

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

      8. pow2 32.7%

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

      9. sub-neg 32.7%

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

      10. metadata-eval 32.7%

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

      11. +-commutative 32.7%

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

      12. un-div-inv 32.7%

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

      13. metadata-eval 32.7%

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

   7. Applied egg-rr 32.7%

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

   8. Taylor expanded in x around 0 72.9%

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

   if 11200 < x

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-def 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 99.4%

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

   6. Taylor expanded in y around 0 46.6%

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

      1. *-commutative 46.6%

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

   8. Simplified 46.6%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 11200:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 3.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.6%

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

   1. add-sqr-sqrt 37.2%

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

   2. sqrt-unprod 38.3%

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

   3. *-commutative 38.3%

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

   4. *-commutative 38.3%

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

   5. swap-sqr 38.3%

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

   6. swap-sqr 17.4%

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

   7. add-sqr-sqrt 17.3%

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

   8. pow2 17.3%

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

   9. sub-neg 17.3%

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

   10. metadata-eval 17.3%

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

   11. +-commutative 17.3%

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

   12. un-div-inv 17.4%

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

   13. metadata-eval 17.4%

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

7. Applied egg-rr 17.4%

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

8. Taylor expanded in x around inf 3.3%

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

9. Final simplification 3.3%

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

Alternative 14: 37.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.6%

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

   1. add-sqr-sqrt 37.2%

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

   2. sqrt-unprod 38.3%

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

   3. *-commutative 38.3%

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

   4. *-commutative 38.3%

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

   5. swap-sqr 38.3%

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

   6. swap-sqr 17.4%

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

   7. add-sqr-sqrt 17.3%

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

   8. pow2 17.3%

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

   9. sub-neg 17.3%

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

   10. metadata-eval 17.3%

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

   11. +-commutative 17.3%

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

   12. un-div-inv 17.4%

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

   13. metadata-eval 17.4%

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

7. Applied egg-rr 17.4%

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

8. Taylor expanded in x around 0 37.4%

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

9. Final simplification 37.4%

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

Alternative 15: 0.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*l* 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. sub-neg 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in y around 0 60.6%

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

   1. add-sqr-sqrt 37.2%

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

   2. sqrt-unprod 38.3%

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

   3. *-commutative 38.3%

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

   4. *-commutative 38.3%

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

   5. swap-sqr 38.3%

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

   6. swap-sqr 17.4%

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

   7. add-sqr-sqrt 17.3%

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

   8. pow2 17.3%

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

   9. sub-neg 17.3%

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

   10. metadata-eval 17.3%

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

   11. +-commutative 17.3%

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

   12. un-div-inv 17.4%

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

   13. metadata-eval 17.4%

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

7. Applied egg-rr 17.4%

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

8. Taylor expanded in x around 0 37.0%

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

   1. sub-neg 37.0%

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

   2. associate-*r/ 37.0%

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

   3. metadata-eval 37.0%

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

   4. metadata-eval 37.0%

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

10. Simplified 37.0%

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

11. Taylor expanded in x around inf 0.0%

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

12. Final simplification 0.0%

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

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024020 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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