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

Percentage Accurate: 99.4% → 99.4%

Time: 12.0s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

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. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. expm1-log1p-u 47.6%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)\right)} \]

   2. expm1-undefine 47.6%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)} - 1\right)} \]

6. Applied egg-rr 47.6%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)} - 1\right)} \]
7. Step-by-step derivation

   1. sub-neg 47.6%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)} + \left(-1\right)\right)} \]

   2. metadata-eval 47.6%

      \[\leadsto \sqrt{x} \cdot \left(e^{\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)} + \color{blue}{-1}\right) \]

   3. +-commutative 47.6%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\left(-1 + e^{\mathsf{log1p}\left(\mathsf{fma}\left(3, y, -3 + \frac{0.3333333333333333}{x}\right)\right)}\right)} \]

   4. log1p-undefine 47.6%

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

   5. rem-exp-log 99.5%

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

   6. fma-define 99.4%

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

   7. +-commutative 99.4%

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

   8. associate-+l+ 99.4%

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

   9. associate-+r+ 99.4%

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

   10. metadata-eval 99.4%

       \[\leadsto \sqrt{x} \cdot \left(-1 + \left(\color{blue}{-2} + \left(\frac{0.3333333333333333}{x} + 3 \cdot y\right)\right)\right) \]

   11. +-commutative 99.4%

       \[\leadsto \sqrt{x} \cdot \left(-1 + \left(-2 + \color{blue}{\left(3 \cdot y + \frac{0.3333333333333333}{x}\right)}\right)\right) \]

   12. fma-define 99.5%

       \[\leadsto \sqrt{x} \cdot \left(-1 + \left(-2 + \color{blue}{\mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)}\right)\right) \]

8. Simplified 99.5%

   \[\leadsto \sqrt{x} \cdot \color{blue}{\left(-1 + \left(-2 + \mathsf{fma}\left(3, y, \frac{0.3333333333333333}{x}\right)\right)\right)} \]

9. Taylor expanded in y around 0 99.4%

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

    1. sub-neg 99.4%

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

    2. associate-*r/ 99.4%

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

    3. metadata-eval 99.4%

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

    4. metadata-eval 99.4%

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

    5. +-commutative 99.4%

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

    6. fma-undefine 99.5%

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

    7. +-commutative 99.5%

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

11. Simplified 99.5%

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

Alternative 3: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+43} \lor \neg \left(y \leq 8 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3e+43) (not (<= y 8e+45)))
   (* y (sqrt (* x 9.0)))
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3e+43) || !(y <= 8e+45)) {
		tmp = y * sqrt((x * 9.0));
	} else {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-3d+43)) .or. (.not. (y <= 8d+45))) then
        tmp = y * sqrt((x * 9.0d0))
    else
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -3e+43) || !(y <= 8e+45)) {
		tmp = y * Math.sqrt((x * 9.0));
	} else {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -3e+43) or not (y <= 8e+45):
		tmp = y * math.sqrt((x * 9.0))
	else:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -3e+43) || !(y <= 8e+45))
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	else
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -3e+43) || ~((y <= 8e+45)))
		tmp = y * sqrt((x * 9.0));
	else
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -3e+43], N[Not[LessEqual[y, 8e+45]], $MachinePrecision]], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000017e43 or 7.9999999999999994e45 < y

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. associate--l+ 99.4%

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

      3. *-commutative 99.4%

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

      4. associate-/r* 99.4%

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

      5. metadata-eval 99.4%

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

      6. sub-neg 99.4%

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

      7. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

      1. *-commutative 99.4%

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

      2. metadata-eval 99.4%

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

      3. sqrt-prod 99.5%

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

      4. pow1/2 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. unpow1/2 99.5%

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

   8. Simplified 99.5%

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

   9. Taylor expanded in y around inf 72.5%

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

   if -3.00000000000000017e43 < y < 7.9999999999999994e45

   1. Initial program 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate--l+ 99.4%

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

      4. distribute-lft-in 99.4%

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

      5. fma-define 99.4%

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

      6. sub-neg 99.4%

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

      7. +-commutative 99.4%

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

      8. distribute-lft-in 99.4%

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

      9. metadata-eval 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. associate-/r* 99.3%

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

      13. associate-*r/ 99.4%

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

      14. metadata-eval 99.4%

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

      15. metadata-eval 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 95.6%

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

      1. sub-neg 95.6%

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

      2. metadata-eval 95.6%

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

      3. associate-*r/ 95.7%

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

      4. metadata-eval 95.7%

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

      5. +-commutative 95.7%

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

      6. metadata-eval 95.7%

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

      7. distribute-neg-frac 95.7%

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

      8. unsub-neg 95.7%

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

   7. Simplified 95.7%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+43} \lor \neg \left(y \leq 8 \cdot 10^{+45}\right):\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+233}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 2.9e+233) (* (sqrt x) -3.0) (* (sqrt x) (* 3.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 2.9e+233) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 2.9d+233) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = sqrt(x) * (3.0d0 * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 2.9e+233) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = Math.sqrt(x) * (3.0 * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 2.9e+233:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = math.sqrt(x) * (3.0 * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 2.9e+233)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(sqrt(x) * Float64(3.0 * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 2.9e+233)
		tmp = sqrt(x) * -3.0;
	else
		tmp = sqrt(x) * (3.0 * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 2.9e+233], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * N[(3.0 * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.2%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 74.1%

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

      1. metadata-eval 74.1%

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

      2. sqrt-prod 74.2%

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

      3. div-inv 74.3%

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

      4. pow1/2 74.3%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 74.3%

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

   9. Simplified 74.3%

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

   if 0.110000000000000001 < x < 2.90000000000000012e233

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 98.9%

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

   6. Taylor expanded in y around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   if 2.90000000000000012e233 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.9%

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

      5. fma-define 99.9%

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

      6. sub-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-lft-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. associate-/r* 99.9%

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

      13. associate-*r/ 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l* 60.4%

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

      3. *-commutative 60.4%

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

   7. Simplified 60.4%

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

Alternative 5: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+232}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11)
   (sqrt (/ 0.1111111111111111 x))
   (if (<= x 1.3e+232) (* (sqrt x) -3.0) (* 3.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = sqrt((0.1111111111111111 / x));
	} else if (x <= 1.3e+232) {
		tmp = sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (sqrt(x) * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else if (x <= 1.3d+232) then
        tmp = sqrt(x) * (-3.0d0)
    else
        tmp = 3.0d0 * (sqrt(x) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else if (x <= 1.3e+232) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = math.sqrt((0.1111111111111111 / x))
	elif x <= 1.3e+232:
		tmp = math.sqrt(x) * -3.0
	else:
		tmp = 3.0 * (math.sqrt(x) * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	elseif (x <= 1.3e+232)
		tmp = Float64(sqrt(x) * -3.0);
	else
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sqrt((0.1111111111111111 / x));
	elseif (x <= 1.3e+232)
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], If[LessEqual[x, 1.3e+232], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.2%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 74.1%

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

      1. metadata-eval 74.1%

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

      2. sqrt-prod 74.2%

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

      3. div-inv 74.3%

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

      4. pow1/2 74.3%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 74.3%

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

   9. Simplified 74.3%

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

   if 0.110000000000000001 < x < 1.29999999999999987e232

   1. Initial program 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-*l* 99.5%

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

      3. associate--l+ 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. fma-define 99.5%

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

      6. sub-neg 99.5%

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

      7. +-commutative 99.5%

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

      8. distribute-lft-in 99.5%

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

      9. metadata-eval 99.5%

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

      10. metadata-eval 99.5%

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

      11. *-commutative 99.5%

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

      12. associate-/r* 99.5%

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

      13. associate-*r/ 99.5%

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

      14. metadata-eval 99.5%

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

      15. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around inf 98.9%

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

   6. Taylor expanded in y around 0 60.3%

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

      1. *-commutative 60.3%

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

   8. Simplified 60.3%

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

   if 1.29999999999999987e232 < x

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l* 99.8%

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

      3. associate--l+ 99.8%

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

      4. distribute-lft-in 99.9%

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

      5. fma-define 99.9%

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

      6. sub-neg 99.9%

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

      7. +-commutative 99.9%

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

      8. distribute-lft-in 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

      11. *-commutative 99.9%

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

      12. associate-/r* 99.9%

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

      13. associate-*r/ 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 60.4%

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

Alternative 6: 86.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(3 \cdot \left(0.1111111111111111 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.2e-7)
   (* (sqrt (/ 1.0 x)) (* 3.0 (- 0.1111111111111111 x)))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.2e-7) {
		tmp = sqrt((1.0 / x)) * (3.0 * (0.1111111111111111 - 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 <= 6.2d-7) then
        tmp = sqrt((1.0d0 / x)) * (3.0d0 * (0.1111111111111111d0 - 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 <= 6.2e-7) {
		tmp = Math.sqrt((1.0 / x)) * (3.0 * (0.1111111111111111 - x));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.2e-7:
		tmp = math.sqrt((1.0 / x)) * (3.0 * (0.1111111111111111 - x))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.2e-7)
		tmp = Float64(sqrt(Float64(1.0 / x)) * Float64(3.0 * Float64(0.1111111111111111 - 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 <= 6.2e-7)
		tmp = sqrt((1.0 / x)) * (3.0 * (0.1111111111111111 - x));
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.2e-7], N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(3.0 * N[(0.1111111111111111 - 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 < 6.1999999999999999e-7

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/r* 99.2%

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

      5. metadata-eval 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 99.2%

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

   6. Taylor expanded in y around 0 75.9%

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

      1. *-commutative 75.9%

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

      2. associate-*l* 75.9%

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

      3. neg-mul-1 75.9%

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

      4. unsub-neg 75.9%

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

   8. Simplified 75.9%

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

   if 6.1999999999999999e-7 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 98.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.2 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{1}{x}} \cdot \left(3 \cdot \left(0.1111111111111111 - x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5e-6)
   (* (+ (/ 0.1111111111111111 x) -1.0) (sqrt (* x 9.0)))
   (* (sqrt x) (- (* 3.0 y) 3.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-6) {
		tmp = ((0.1111111111111111 / x) + -1.0) * sqrt((x * 9.0));
	} 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 <= 5d-6) then
        tmp = ((0.1111111111111111d0 / x) + (-1.0d0)) * sqrt((x * 9.0d0))
    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 <= 5e-6) {
		tmp = ((0.1111111111111111 / x) + -1.0) * Math.sqrt((x * 9.0));
	} else {
		tmp = Math.sqrt(x) * ((3.0 * y) - 3.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e-6:
		tmp = ((0.1111111111111111 / x) + -1.0) * math.sqrt((x * 9.0))
	else:
		tmp = math.sqrt(x) * ((3.0 * y) - 3.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-6)
		tmp = Float64(Float64(Float64(0.1111111111111111 / x) + -1.0) * sqrt(Float64(x * 9.0)));
	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 <= 5e-6)
		tmp = ((0.1111111111111111 / x) + -1.0) * sqrt((x * 9.0));
	else
		tmp = sqrt(x) * ((3.0 * y) - 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-6], N[(N[(N[(0.1111111111111111 / x), $MachinePrecision] + -1.0), $MachinePrecision] * N[Sqrt[N[(x * 9.0), $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 < 5.00000000000000041e-6

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate--l+ 99.3%

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

      3. *-commutative 99.3%

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

      4. associate-/r* 99.2%

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

      5. metadata-eval 99.2%

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

      6. sub-neg 99.2%

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

      7. metadata-eval 99.2%

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

   3. Simplified 99.2%

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

      1. *-commutative 99.2%

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

      2. metadata-eval 99.2%

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

      3. sqrt-prod 99.3%

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

      4. pow1/2 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. unpow1/2 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in y around 0 75.8%

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

       1. sub-neg 75.8%

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

       2. associate-*r/ 75.9%

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

       3. metadata-eval 75.9%

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

       4. metadata-eval 75.9%

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

       5. +-commutative 75.9%

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

   11. Simplified 75.9%

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

   if 5.00000000000000041e-6 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 98.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\left(\frac{0.1111111111111111}{x} + -1\right) \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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. Add Preprocessing

3. Final simplification 99.4%

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

Alternative 9: 85.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.8 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.8e-6)
   (* (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 <= 3.8e-6) {
		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 <= 3.8d-6) 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 <= 3.8e-6) {
		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 <= 3.8e-6:
		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 <= 3.8e-6)
		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 <= 3.8e-6)
		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, 3.8e-6], 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 < 3.8e-6

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.2%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in y around 0 75.8%

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

      1. sub-neg 75.8%

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

      2. metadata-eval 75.8%

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

      3. associate-*r/ 75.9%

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

      4. metadata-eval 75.9%

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

      5. +-commutative 75.9%

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

      6. metadata-eval 75.9%

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

      7. distribute-neg-frac 75.9%

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

      8. unsub-neg 75.9%

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

   7. Simplified 75.9%

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

   if 3.8e-6 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 98.7%

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

Alternative 10: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 11: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. associate--l+ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. sqrt-prod 99.2%

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

   4. pow1/2 99.2%

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

6. Applied egg-rr 99.2%

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

   1. unpow1/2 99.2%

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

8. Simplified 99.2%

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

9. Taylor expanded in y around 0 99.3%

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

    1. distribute-lft-out 99.3%

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

    2. distribute-lft-out 99.3%

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

    3. sub-neg 99.3%

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

    4. associate-*r/ 99.3%

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

    5. metadata-eval 99.3%

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

    6. metadata-eval 99.3%

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

    7. +-commutative 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 12: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.11:\\ \;\;\;\;\sqrt{\frac{0.1111111111111111}{x}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.11) (sqrt (/ 0.1111111111111111 x)) (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = sqrt((0.1111111111111111 / x));
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 0.11d0) then
        tmp = sqrt((0.1111111111111111d0 / x))
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.11) {
		tmp = Math.sqrt((0.1111111111111111 / x));
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.11:
		tmp = math.sqrt((0.1111111111111111 / x))
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.11)
		tmp = sqrt(Float64(0.1111111111111111 / x));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.11)
		tmp = sqrt((0.1111111111111111 / x));
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.11], N[Sqrt[N[(0.1111111111111111 / x), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.110000000000000001

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

      3. associate--l+ 99.3%

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

      4. distribute-lft-in 99.2%

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

      5. fma-define 99.3%

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

      6. sub-neg 99.3%

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

      7. +-commutative 99.3%

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

      8. distribute-lft-in 99.3%

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

      9. metadata-eval 99.3%

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

      10. metadata-eval 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-/r* 99.2%

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

      13. associate-*r/ 99.3%

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

      14. metadata-eval 99.3%

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

      15. metadata-eval 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in x around 0 74.1%

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

      1. metadata-eval 74.1%

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

      2. sqrt-prod 74.2%

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

      3. div-inv 74.3%

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

      4. pow1/2 74.3%

         \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

   7. Applied egg-rr 74.3%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
   8. Step-by-step derivation

      1. unpow1/2 74.3%

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

   9. Simplified 74.3%

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

   if 0.110000000000000001 < x

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. associate-*l* 99.6%

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

      3. associate--l+ 99.6%

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

      4. distribute-lft-in 99.6%

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

      5. fma-define 99.6%

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

      6. sub-neg 99.6%

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

      7. +-commutative 99.6%

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

      8. distribute-lft-in 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

      11. *-commutative 99.6%

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

      12. associate-/r* 99.6%

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

      13. associate-*r/ 99.6%

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

      14. metadata-eval 99.6%

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

      15. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around inf 99.2%

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

   6. Taylor expanded in y around 0 55.2%

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

      1. *-commutative 55.2%

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

   8. Simplified 55.2%

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

Alternative 13: 36.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around 0 37.0%

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

   1. metadata-eval 37.0%

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

   2. sqrt-prod 37.1%

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

   3. div-inv 37.1%

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

   4. pow1/2 37.1%

      \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]

7. Applied egg-rr 37.1%

   \[\leadsto \color{blue}{{\left(\frac{0.1111111111111111}{x}\right)}^{0.5}} \]
8. Step-by-step derivation

   1. unpow1/2 37.1%

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

9. Simplified 37.1%

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

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. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. associate--l+ 99.4%

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

   4. distribute-lft-in 99.4%

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

   5. fma-define 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-lft-in 99.4%

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

   9. metadata-eval 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

   12. associate-/r* 99.4%

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

   13. associate-*r/ 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in x around inf 62.0%

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

6. Taylor expanded in y around 0 29.1%

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

   1. *-commutative 29.1%

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

8. Simplified 29.1%

   \[\leadsto \color{blue}{\sqrt{x} \cdot -3} \]
9. 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%

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

10. Applied egg-rr 3.1%

    \[\leadsto \color{blue}{\sqrt{x \cdot 9}} \]
11. Add Preprocessing

Developer Target 1: 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 2024165 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (* 3 (+ (* y (sqrt x)) (* (- (/ 1 (* x 9)) 1) (sqrt x)))))

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