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

Percentage Accurate: 99.4% → 99.4%

Time: 10.2s

Alternatives: 16

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. distribute-lft-out-- 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-*l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out-- 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. associate--l+ 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. sub-neg 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

   19. fma-def 99.4%

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

3. Simplified 99.4%

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

   1. clear-num 99.3%

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

   2. inv-pow 99.3%

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

   3. div-inv 99.4%

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

   4. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. sub-neg 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. +-commutative 99.4%

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

   6. metadata-eval 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. fma-def 99.3%

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

   10. *-commutative 99.3%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

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) * Float64(fma(3.0, y, -3.0) + Float64(0.3333333333333333 / x)))
end

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. distribute-lft-out-- 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-*l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out-- 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. associate--l+ 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. sub-neg 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

   19. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 62.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+73} \lor \neg \left(x \leq 4.1 \cdot 10^{+95}\right) \land \left(x \leq 1.75 \cdot 10^{+224} \lor \neg \left(x \leq 10^{+258}\right)\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.3e-13)
   (* 3.0 (sqrt (/ 0.012345679012345678 x)))
   (if (or (<= x 2.6e+73)
           (and (not (<= x 4.1e+95))
                (or (<= x 1.75e+224) (not (<= x 1e+258)))))
     (* 3.0 (* (sqrt x) y))
     (* (sqrt x) -3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.3e-13) {
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	} else if ((x <= 2.6e+73) || (!(x <= 4.1e+95) && ((x <= 1.75e+224) || !(x <= 1e+258)))) {
		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 <= 2.3d-13) then
        tmp = 3.0d0 * sqrt((0.012345679012345678d0 / x))
    else if ((x <= 2.6d+73) .or. (.not. (x <= 4.1d+95)) .and. (x <= 1.75d+224) .or. (.not. (x <= 1d+258))) 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 <= 2.3e-13) {
		tmp = 3.0 * Math.sqrt((0.012345679012345678 / x));
	} else if ((x <= 2.6e+73) || (!(x <= 4.1e+95) && ((x <= 1.75e+224) || !(x <= 1e+258)))) {
		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 <= 2.3e-13:
		tmp = 3.0 * math.sqrt((0.012345679012345678 / x))
	elif (x <= 2.6e+73) or (not (x <= 4.1e+95) and ((x <= 1.75e+224) or not (x <= 1e+258))):
		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 <= 2.3e-13)
		tmp = Float64(3.0 * sqrt(Float64(0.012345679012345678 / x)));
	elseif ((x <= 2.6e+73) || (!(x <= 4.1e+95) && ((x <= 1.75e+224) || !(x <= 1e+258))))
		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 <= 2.3e-13)
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	elseif ((x <= 2.6e+73) || (~((x <= 4.1e+95)) && ((x <= 1.75e+224) || ~((x <= 1e+258)))))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.3e-13], N[(3.0 * N[Sqrt[N[(0.012345679012345678 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 2.6e+73], And[N[Not[LessEqual[x, 4.1e+95]], $MachinePrecision], Or[LessEqual[x, 1.75e+224], N[Not[LessEqual[x, 1e+258]], $MachinePrecision]]]], 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 < 2.29999999999999979e-13

   1. Initial program 99.3%

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

      1. associate-*l* 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. add-sqr-sqrt 86.9%

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

      2. sqrt-unprod 81.8%

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

      3. swap-sqr 33.1%

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

      4. add-sqr-sqrt 33.2%

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

      5. pow2 33.2%

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

      6. associate-+r+ 33.2%

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

      7. clear-num 33.1%

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

      8. div-inv 33.2%

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

      9. metadata-eval 33.2%

         \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(\frac{1}{x \cdot \color{blue}{9}} + y\right) + -1\right)}^{2}} \]

      10. +-commutative 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)}^{2}} \]

      11. metadata-eval 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)}^{2}} \]

      12. sub-neg 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}}^{2}} \]

   5. Applied egg-rr 33.2%

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

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]

   if 2.29999999999999979e-13 < x < 2.6000000000000001e73 or 4.09999999999999986e95 < x < 1.75e224 or 1.00000000000000006e258 < x

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 61.9%

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

   if 2.6000000000000001e73 < x < 4.09999999999999986e95 or 1.75e224 < x < 1.00000000000000006e258

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. associate--l+ 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

      18. metadata-eval 99.4%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in x around inf 76.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{+73} \lor \neg \left(x \leq 4.1 \cdot 10^{+95}\right) \land \left(x \leq 1.75 \cdot 10^{+224} \lor \neg \left(x \leq 10^{+258}\right)\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 5: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+95} \lor \neg \left(x \leq 1.65 \cdot 10^{+224}\right) \land x \leq 2.8 \cdot 10^{+258}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.1e-13)
   (* 3.0 (sqrt (/ 0.012345679012345678 x)))
   (if (<= x 1.1e+74)
     (* y (sqrt (* x 9.0)))
     (if (or (<= x 3.3e+95) (and (not (<= x 1.65e+224)) (<= x 2.8e+258)))
       (* (sqrt x) -3.0)
       (* 3.0 (* (sqrt x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.1e-13) {
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	} else if (x <= 1.1e+74) {
		tmp = y * sqrt((x * 9.0));
	} else if ((x <= 3.3e+95) || (!(x <= 1.65e+224) && (x <= 2.8e+258))) {
		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 <= 2.1d-13) then
        tmp = 3.0d0 * sqrt((0.012345679012345678d0 / x))
    else if (x <= 1.1d+74) then
        tmp = y * sqrt((x * 9.0d0))
    else if ((x <= 3.3d+95) .or. (.not. (x <= 1.65d+224)) .and. (x <= 2.8d+258)) 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 <= 2.1e-13) {
		tmp = 3.0 * Math.sqrt((0.012345679012345678 / x));
	} else if (x <= 1.1e+74) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if ((x <= 3.3e+95) || (!(x <= 1.65e+224) && (x <= 2.8e+258))) {
		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 <= 2.1e-13:
		tmp = 3.0 * math.sqrt((0.012345679012345678 / x))
	elif x <= 1.1e+74:
		tmp = y * math.sqrt((x * 9.0))
	elif (x <= 3.3e+95) or (not (x <= 1.65e+224) and (x <= 2.8e+258)):
		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 <= 2.1e-13)
		tmp = Float64(3.0 * sqrt(Float64(0.012345679012345678 / x)));
	elseif (x <= 1.1e+74)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif ((x <= 3.3e+95) || (!(x <= 1.65e+224) && (x <= 2.8e+258)))
		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 <= 2.1e-13)
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	elseif (x <= 1.1e+74)
		tmp = y * sqrt((x * 9.0));
	elseif ((x <= 3.3e+95) || (~((x <= 1.65e+224)) && (x <= 2.8e+258)))
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.1e-13], N[(3.0 * N[Sqrt[N[(0.012345679012345678 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.1e+74], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, 3.3e+95], And[N[Not[LessEqual[x, 1.65e+224]], $MachinePrecision], LessEqual[x, 2.8e+258]]], 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 < 2.09999999999999989e-13

   1. Initial program 99.3%

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

      1. associate-*l* 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. add-sqr-sqrt 86.9%

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

      2. sqrt-unprod 81.8%

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

      3. swap-sqr 33.1%

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

      4. add-sqr-sqrt 33.2%

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

      5. pow2 33.2%

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

      6. associate-+r+ 33.2%

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

      7. clear-num 33.1%

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

      8. div-inv 33.2%

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

      9. metadata-eval 33.2%

         \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(\frac{1}{x \cdot \color{blue}{9}} + y\right) + -1\right)}^{2}} \]

      10. +-commutative 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)}^{2}} \]

      11. metadata-eval 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)}^{2}} \]

      12. sub-neg 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}}^{2}} \]

   5. Applied egg-rr 33.2%

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

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]

   if 2.09999999999999989e-13 < x < 1.1000000000000001e74

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.3%

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

      6. distribute-rgt-out-- 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. associate--l+ 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

      18. metadata-eval 99.4%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.3%

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

      2. inv-pow 99.3%

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

      3. div-inv 99.4%

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

      4. metadata-eval 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

   7. Simplified 99.4%

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

   8. Taylor expanded in y around inf 51.7%

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

      1. associate-*r* 51.8%

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

   10. Simplified 51.8%

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

       1. expm1-log1p-u 50.1%

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

       2. expm1-udef 48.5%

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

       3. *-commutative 48.5%

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

       4. metadata-eval 48.5%

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

       5. sqrt-prod 48.5%

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

   12. Applied egg-rr 48.5%

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

       1. expm1-def 50.0%

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

       2. expm1-log1p 51.7%

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

   14. Simplified 51.7%

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

   if 1.1000000000000001e74 < x < 3.2999999999999998e95 or 1.64999999999999998e224 < x < 2.79999999999999982e258

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. associate--l+ 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

      18. metadata-eval 99.4%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in x around inf 76.3%

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

   if 3.2999999999999998e95 < x < 1.64999999999999998e224 or 2.79999999999999982e258 < x

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 66.3%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+74}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+95} \lor \neg \left(x \leq 1.65 \cdot 10^{+224}\right) \land x \leq 2.8 \cdot 10^{+258}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 6: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{x} \cdot -3\\ t_1 := \sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{if}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+224}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+258}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (sqrt x) -3.0)) (t_1 (* (sqrt x) (* 3.0 y))))
   (if (<= x 9e-14)
     (* 3.0 (sqrt (/ 0.012345679012345678 x)))
     (if (<= x 8.5e+72)
       t_1
       (if (<= x 3.3e+95)
         t_0
         (if (<= x 2.5e+224)
           t_1
           (if (<= x 1.85e+258) t_0 (* 3.0 (* (sqrt x) y)))))))))

C

double code(double x, double y) {
	double t_0 = sqrt(x) * -3.0;
	double t_1 = sqrt(x) * (3.0 * y);
	double tmp;
	if (x <= 9e-14) {
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	} else if (x <= 8.5e+72) {
		tmp = t_1;
	} else if (x <= 3.3e+95) {
		tmp = t_0;
	} else if (x <= 2.5e+224) {
		tmp = t_1;
	} else if (x <= 1.85e+258) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt(x) * (-3.0d0)
    t_1 = sqrt(x) * (3.0d0 * y)
    if (x <= 9d-14) then
        tmp = 3.0d0 * sqrt((0.012345679012345678d0 / x))
    else if (x <= 8.5d+72) then
        tmp = t_1
    else if (x <= 3.3d+95) then
        tmp = t_0
    else if (x <= 2.5d+224) then
        tmp = t_1
    else if (x <= 1.85d+258) then
        tmp = t_0
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sqrt(x) * -3.0;
	double t_1 = Math.sqrt(x) * (3.0 * y);
	double tmp;
	if (x <= 9e-14) {
		tmp = 3.0 * Math.sqrt((0.012345679012345678 / x));
	} else if (x <= 8.5e+72) {
		tmp = t_1;
	} else if (x <= 3.3e+95) {
		tmp = t_0;
	} else if (x <= 2.5e+224) {
		tmp = t_1;
	} else if (x <= 1.85e+258) {
		tmp = t_0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sqrt(x) * -3.0
	t_1 = math.sqrt(x) * (3.0 * y)
	tmp = 0
	if x <= 9e-14:
		tmp = 3.0 * math.sqrt((0.012345679012345678 / x))
	elif x <= 8.5e+72:
		tmp = t_1
	elif x <= 3.3e+95:
		tmp = t_0
	elif x <= 2.5e+224:
		tmp = t_1
	elif x <= 1.85e+258:
		tmp = t_0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sqrt(x) * -3.0)
	t_1 = Float64(sqrt(x) * Float64(3.0 * y))
	tmp = 0.0
	if (x <= 9e-14)
		tmp = Float64(3.0 * sqrt(Float64(0.012345679012345678 / x)));
	elseif (x <= 8.5e+72)
		tmp = t_1;
	elseif (x <= 3.3e+95)
		tmp = t_0;
	elseif (x <= 2.5e+224)
		tmp = t_1;
	elseif (x <= 1.85e+258)
		tmp = t_0;
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sqrt(x) * -3.0;
	t_1 = sqrt(x) * (3.0 * y);
	tmp = 0.0;
	if (x <= 9e-14)
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	elseif (x <= 8.5e+72)
		tmp = t_1;
	elseif (x <= 3.3e+95)
		tmp = t_0;
	elseif (x <= 2.5e+224)
		tmp = t_1;
	elseif (x <= 1.85e+258)
		tmp = t_0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 9e-14], N[(3.0 * N[Sqrt[N[(0.012345679012345678 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e+72], t$95$1, If[LessEqual[x, 3.3e+95], t$95$0, If[LessEqual[x, 2.5e+224], t$95$1, If[LessEqual[x, 1.85e+258], t$95$0, N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < 8.9999999999999995e-14

   1. Initial program 99.3%

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

      1. associate-*l* 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. add-sqr-sqrt 86.9%

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

      2. sqrt-unprod 81.8%

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

      3. swap-sqr 33.1%

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

      4. add-sqr-sqrt 33.2%

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

      5. pow2 33.2%

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

      6. associate-+r+ 33.2%

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

      7. clear-num 33.1%

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

      8. div-inv 33.2%

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

      9. metadata-eval 33.2%

         \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(\frac{1}{x \cdot \color{blue}{9}} + y\right) + -1\right)}^{2}} \]

      10. +-commutative 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)}^{2}} \]

      11. metadata-eval 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)}^{2}} \]

      12. sub-neg 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}}^{2}} \]

   5. Applied egg-rr 33.2%

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

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]

   if 8.9999999999999995e-14 < x < 8.5000000000000004e72 or 3.2999999999999998e95 < x < 2.49999999999999982e224

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*r* 99.4%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-*r* 56.4%

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

   6. Simplified 56.4%

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

   if 8.5000000000000004e72 < x < 3.2999999999999998e95 or 2.49999999999999982e224 < x < 1.8499999999999999e258

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.3%

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

      4. *-commutative 99.3%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. associate--l+ 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-/r* 99.4%

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

      12. associate-*r/ 99.4%

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

      13. metadata-eval 99.4%

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

      14. metadata-eval 99.4%

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

      15. sub-neg 99.4%

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

      16. metadata-eval 99.4%

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

      17. metadata-eval 99.4%

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

      18. metadata-eval 99.4%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 76.3%

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

      1. sub-neg 76.3%

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

      2. associate-*r/ 76.3%

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

      3. metadata-eval 76.3%

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

      4. metadata-eval 76.3%

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

   6. Simplified 76.3%

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

   7. Taylor expanded in x around inf 76.3%

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

   if 1.8499999999999999e258 < 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. distribute-lft-out-- 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 80.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-14}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+72}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{+95}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{+224}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{+258}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 7: 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 99.4%

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

2. Final simplification 99.4%

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

Alternative 8: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 380000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{x \cdot 3} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 380000000.0)
   (* (sqrt x) (+ (/ 1.0 (* x 3.0)) -3.0))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 380000000.0) {
		tmp = sqrt(x) * ((1.0 / (x * 3.0)) + -3.0);
	} 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 <= 380000000.0d0) then
        tmp = sqrt(x) * ((1.0d0 / (x * 3.0d0)) + (-3.0d0))
    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 <= 380000000.0) {
		tmp = Math.sqrt(x) * ((1.0 / (x * 3.0)) + -3.0);
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 380000000.0:
		tmp = math.sqrt(x) * ((1.0 / (x * 3.0)) + -3.0)
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 380000000.0)
		tmp = Float64(sqrt(x) * Float64(Float64(1.0 / Float64(x * 3.0)) + -3.0));
	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 <= 380000000.0)
		tmp = sqrt(x) * ((1.0 / (x * 3.0)) + -3.0);
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 380000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(N[(1.0 / N[(x * 3.0), $MachinePrecision]), $MachinePrecision] + -3.0), $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 < 3.8e8

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. distribute-rgt-out-- 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.3%

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

      9. associate--l+ 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-/r* 99.2%

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

      12. associate-*r/ 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. sub-neg 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

      19. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 76.4%

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

      1. sub-neg 76.4%

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

      2. associate-*r/ 76.5%

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

      3. metadata-eval 76.5%

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

      4. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

      3. div-inv 99.3%

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

      4. metadata-eval 99.3%

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

   8. Applied egg-rr 76.5%

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

      1. unpow-1 99.3%

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

   10. Simplified 76.5%

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

   if 3.8e8 < x

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 380000000:\\ \;\;\;\;\sqrt{x} \cdot \left(\frac{1}{x \cdot 3} + -3\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 9: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l* 99.4%

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

   2. sub-neg 99.4%

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

   3. +-commutative 99.4%

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

   4. associate-+l+ 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-/r* 99.4%

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

   7. metadata-eval 99.4%

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

   8. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 10: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.15e-13)
   (* 3.0 (sqrt (/ 0.012345679012345678 x)))
   (* 3.0 (* (sqrt x) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.15e-13) {
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.15d-13) then
        tmp = 3.0d0 * sqrt((0.012345679012345678d0 / x))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.15e-13) {
		tmp = 3.0 * Math.sqrt((0.012345679012345678 / x));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.15e-13:
		tmp = 3.0 * math.sqrt((0.012345679012345678 / x))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.15e-13)
		tmp = Float64(3.0 * sqrt(Float64(0.012345679012345678 / x)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.15e-13)
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.15e-13], N[(3.0 * N[Sqrt[N[(0.012345679012345678 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1499999999999999e-13

   1. Initial program 99.3%

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

      1. associate-*l* 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. add-sqr-sqrt 86.9%

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

      2. sqrt-unprod 81.8%

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

      3. swap-sqr 33.1%

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

      4. add-sqr-sqrt 33.2%

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

      5. pow2 33.2%

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

      6. associate-+r+ 33.2%

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

      7. clear-num 33.1%

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

      8. div-inv 33.2%

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

      9. metadata-eval 33.2%

         \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(\frac{1}{x \cdot \color{blue}{9}} + y\right) + -1\right)}^{2}} \]

      10. +-commutative 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)}^{2}} \]

      11. metadata-eval 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)}^{2}} \]

      12. sub-neg 33.2%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}}^{2}} \]

   5. Applied egg-rr 33.2%

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

   6. Taylor expanded in x around 0 77.4%

      \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]

   if 2.1499999999999999e-13 < x

   1. Initial program 99.4%

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

      1. associate-*l* 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 95.5%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 11: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 320000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 320000000.0)
   (* (sqrt x) (+ -3.0 (/ 0.3333333333333333 x)))
   (* 3.0 (* (sqrt x) (+ y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 320000000.0) {
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 320000000.0d0) then
        tmp = sqrt(x) * ((-3.0d0) + (0.3333333333333333d0 / x))
    else
        tmp = 3.0d0 * (sqrt(x) * (y + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 320000000.0) {
		tmp = Math.sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	} else {
		tmp = 3.0 * (Math.sqrt(x) * (y + -1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 320000000.0:
		tmp = math.sqrt(x) * (-3.0 + (0.3333333333333333 / x))
	else:
		tmp = 3.0 * (math.sqrt(x) * (y + -1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 320000000.0)
		tmp = Float64(sqrt(x) * Float64(-3.0 + Float64(0.3333333333333333 / x)));
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * Float64(y + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 320000000.0)
		tmp = sqrt(x) * (-3.0 + (0.3333333333333333 / x));
	else
		tmp = 3.0 * (sqrt(x) * (y + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 320000000.0], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 + N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.2e8

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. distribute-rgt-out-- 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.3%

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

      9. associate--l+ 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-/r* 99.2%

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

      12. associate-*r/ 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. sub-neg 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

      19. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 76.4%

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

      1. sub-neg 76.4%

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

      2. associate-*r/ 76.5%

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

      3. metadata-eval 76.5%

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

      4. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

   if 3.2e8 < x

   1. Initial program 99.4%

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

      1. associate-*l* 99.5%

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

      2. sub-neg 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+l+ 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-/r* 99.5%

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

      7. metadata-eval 99.5%

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

      8. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 320000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot \left(y + -1\right)\right)\\ \end{array} \]

Alternative 12: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 340000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 340000000.0)
   (* (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 <= 340000000.0) {
		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 <= 340000000.0d0) 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 <= 340000000.0) {
		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 <= 340000000.0:
		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 <= 340000000.0)
		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 <= 340000000.0)
		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, 340000000.0], 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 < 3.4e8

   1. Initial program 99.3%

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

      1. distribute-lft-out-- 99.3%

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

      2. *-rgt-identity 99.3%

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

      3. associate-*l* 99.2%

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

      4. *-commutative 99.2%

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

      5. associate-*r* 99.2%

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

      6. distribute-rgt-out-- 99.2%

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

      7. +-commutative 99.2%

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

      8. distribute-lft-in 99.3%

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

      9. associate--l+ 99.3%

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

      10. *-commutative 99.3%

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

      11. associate-/r* 99.2%

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

      12. associate-*r/ 99.3%

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

      13. metadata-eval 99.3%

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

      14. metadata-eval 99.3%

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

      15. sub-neg 99.3%

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

      16. metadata-eval 99.3%

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

      17. metadata-eval 99.3%

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

      18. metadata-eval 99.3%

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

      19. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 76.4%

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

      1. sub-neg 76.4%

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

      2. associate-*r/ 76.5%

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

      3. metadata-eval 76.5%

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

      4. metadata-eval 76.5%

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

   6. Simplified 76.5%

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

   if 3.4e8 < x

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 99.3%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 340000000:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 + \frac{0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 13: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.112) (* 3.0 (sqrt (/ 0.012345679012345678 x))) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.112d0) then
        tmp = 3.0d0 * sqrt((0.012345679012345678d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.112) {
		tmp = 3.0 * Math.sqrt((0.012345679012345678 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.112:
		tmp = 3.0 * math.sqrt((0.012345679012345678 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.112)
		tmp = Float64(3.0 * sqrt(Float64(0.012345679012345678 / x)));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.112)
		tmp = 3.0 * sqrt((0.012345679012345678 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.112], N[(3.0 * N[Sqrt[N[(0.012345679012345678 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.112000000000000002

   1. Initial program 99.4%

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

      1. associate-*l* 99.2%

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

      2. sub-neg 99.2%

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

      3. +-commutative 99.2%

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

      4. associate-+l+ 99.2%

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

      5. *-commutative 99.2%

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

      6. associate-/r* 99.2%

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

      7. metadata-eval 99.2%

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

      8. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. add-sqr-sqrt 86.5%

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

      2. sqrt-unprod 81.6%

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

      3. swap-sqr 34.4%

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

      4. add-sqr-sqrt 34.5%

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

      5. pow2 34.5%

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

      6. associate-+r+ 34.5%

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

      7. clear-num 34.4%

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

      8. div-inv 34.5%

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

      9. metadata-eval 34.5%

         \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(\frac{1}{x \cdot \color{blue}{9}} + y\right) + -1\right)}^{2}} \]

      10. +-commutative 34.5%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\color{blue}{\left(y + \frac{1}{x \cdot 9}\right)} + -1\right)}^{2}} \]

      11. metadata-eval 34.5%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\left(\left(y + \frac{1}{x \cdot 9}\right) + \color{blue}{\left(-1\right)}\right)}^{2}} \]

      12. sub-neg 34.5%

          \[\leadsto 3 \cdot \sqrt{x \cdot {\color{blue}{\left(\left(y + \frac{1}{x \cdot 9}\right) - 1\right)}}^{2}} \]

   5. Applied egg-rr 34.5%

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

   6. Taylor expanded in x around 0 75.6%

      \[\leadsto 3 \cdot \sqrt{\color{blue}{\frac{0.012345679012345678}{x}}} \]

   if 0.112000000000000002 < x

   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. distribute-lft-out-- 99.4%

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

      2. *-rgt-identity 99.4%

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

      3. associate-*l* 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*r* 99.5%

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

      6. distribute-rgt-out-- 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. associate--l+ 99.5%

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

      10. *-commutative 99.5%

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

      11. associate-/r* 99.5%

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

      12. associate-*r/ 99.5%

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

      13. metadata-eval 99.5%

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

      14. metadata-eval 99.5%

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

      15. sub-neg 99.5%

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

      16. metadata-eval 99.5%

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

      17. metadata-eval 99.5%

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

      18. metadata-eval 99.5%

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

      19. fma-def 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 45.2%

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

      1. sub-neg 45.2%

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

      2. associate-*r/ 45.2%

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

      3. metadata-eval 45.2%

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

      4. metadata-eval 45.2%

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

   6. Simplified 45.2%

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

   7. Taylor expanded in x around inf 41.9%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.112:\\ \;\;\;\;3 \cdot \sqrt{\frac{0.012345679012345678}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 14: 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 99.4%

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

   1. distribute-lft-out-- 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-*l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out-- 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. associate--l+ 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. sub-neg 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

   19. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 60.3%

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

   1. sub-neg 60.3%

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

   2. associate-*r/ 60.4%

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

   3. metadata-eval 60.4%

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

   4. metadata-eval 60.4%

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

6. Simplified 60.4%

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

7. Taylor expanded in x around inf 22.0%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.1%

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

   3. swap-sqr 3.1%

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

   4. add-sqr-sqrt 3.1%

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

   5. metadata-eval 3.1%

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

9. Applied egg-rr 3.1%

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

10. Final simplification 3.1%

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

Alternative 15: 3.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot 3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (sqrt x) 3.0))

C

double code(double x, double y) {
	return sqrt(x) * 3.0;
}

Fortran

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

Java

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

Python

def code(x, y):
	return math.sqrt(x) * 3.0

Julia

function code(x, y)
	return Float64(sqrt(x) * 3.0)
end

MATLAB

function tmp = code(x, y)
	tmp = sqrt(x) * 3.0;
end

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * 3.0), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. distribute-lft-out-- 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-*l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out-- 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. associate--l+ 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. sub-neg 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

   19. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 60.3%

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

   1. sub-neg 60.3%

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

   2. associate-*r/ 60.4%

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

   3. metadata-eval 60.4%

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

   4. metadata-eval 60.4%

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

6. Simplified 60.4%

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

7. Taylor expanded in x around inf 22.0%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 3.1%

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

   3. *-commutative 3.1%

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

   4. *-commutative 3.1%

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

   5. swap-sqr 3.1%

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

   6. metadata-eval 3.1%

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

   7. metadata-eval 3.1%

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

   8. swap-sqr 3.1%

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

   9. sqrt-unprod 3.2%

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

   10. add-sqr-sqrt 3.2%

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

   11. expm1-log1p-u 3.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3 \cdot \sqrt{x}\right)\right)} \]

   12. expm1-udef 2.4%

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

   13. *-commutative 2.4%

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

9. Applied egg-rr 2.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{x} \cdot 3\right)} - 1} \]
10. Step-by-step derivation

    1. expm1-def 3.2%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{x} \cdot 3\right)\right)} \]

    2. expm1-log1p 3.2%

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

    3. *-commutative 3.2%

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

11. Simplified 3.2%

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

12. Final simplification 3.2%

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

Alternative 16: 25.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{x} \cdot -3 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. distribute-lft-out-- 99.4%

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

   2. *-rgt-identity 99.4%

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

   3. associate-*l* 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. distribute-rgt-out-- 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. associate--l+ 99.4%

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

   10. *-commutative 99.4%

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

   11. associate-/r* 99.3%

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

   12. associate-*r/ 99.4%

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

   13. metadata-eval 99.4%

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

   14. metadata-eval 99.4%

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

   15. sub-neg 99.4%

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

   16. metadata-eval 99.4%

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

   17. metadata-eval 99.4%

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

   18. metadata-eval 99.4%

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

   19. fma-def 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in y around 0 60.3%

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

   1. sub-neg 60.3%

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

   2. associate-*r/ 60.4%

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

   3. metadata-eval 60.4%

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

   4. metadata-eval 60.4%

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

6. Simplified 60.4%

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

7. Taylor expanded in x around inf 22.0%

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

8. Final simplification 22.0%

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

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 2023320 
(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)))