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

Percentage Accurate: 99.4% → 99.4%

Time: 9.2s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   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. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in y around 0 99.4%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

   3. *-commutative 99.5%

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

   4. sub-neg 99.5%

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

   5. associate-*r/ 99.5%

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

   6. metadata-eval 99.5%

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

   7. metadata-eval 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   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. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 61.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \sqrt{x \cdot 9}\\ \mathbf{if}\;y \leq -80000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (sqrt (* x 9.0)))))
   (if (<= y -80000000.0)
     t_0
     (if (<= y 8.8e-149)
       (* (sqrt x) -3.0)
       (if (<= y 2.25e+52) (* (sqrt x) (/ 0.3333333333333333 x)) t_0)))))

C

double code(double x, double y) {
	double t_0 = y * sqrt((x * 9.0));
	double tmp;
	if (y <= -80000000.0) {
		tmp = t_0;
	} else if (y <= 8.8e-149) {
		tmp = sqrt(x) * -3.0;
	} else if (y <= 2.25e+52) {
		tmp = sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * sqrt((x * 9.0d0))
    if (y <= (-80000000.0d0)) then
        tmp = t_0
    else if (y <= 8.8d-149) then
        tmp = sqrt(x) * (-3.0d0)
    else if (y <= 2.25d+52) then
        tmp = sqrt(x) * (0.3333333333333333d0 / x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * Math.sqrt((x * 9.0));
	double tmp;
	if (y <= -80000000.0) {
		tmp = t_0;
	} else if (y <= 8.8e-149) {
		tmp = Math.sqrt(x) * -3.0;
	} else if (y <= 2.25e+52) {
		tmp = Math.sqrt(x) * (0.3333333333333333 / x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * math.sqrt((x * 9.0))
	tmp = 0
	if y <= -80000000.0:
		tmp = t_0
	elif y <= 8.8e-149:
		tmp = math.sqrt(x) * -3.0
	elif y <= 2.25e+52:
		tmp = math.sqrt(x) * (0.3333333333333333 / x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * sqrt(Float64(x * 9.0)))
	tmp = 0.0
	if (y <= -80000000.0)
		tmp = t_0;
	elseif (y <= 8.8e-149)
		tmp = Float64(sqrt(x) * -3.0);
	elseif (y <= 2.25e+52)
		tmp = Float64(sqrt(x) * Float64(0.3333333333333333 / x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * sqrt((x * 9.0));
	tmp = 0.0;
	if (y <= -80000000.0)
		tmp = t_0;
	elseif (y <= 8.8e-149)
		tmp = sqrt(x) * -3.0;
	elseif (y <= 2.25e+52)
		tmp = sqrt(x) * (0.3333333333333333 / x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -80000000.0], t$95$0, If[LessEqual[y, 8.8e-149], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision], If[LessEqual[y, 2.25e+52], N[(N[Sqrt[x], $MachinePrecision] * N[(0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8e7 or 2.25e52 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 59.6%

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

      2. pow2 59.6%

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

   5. Applied egg-rr 59.6%

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

      1. unpow2 59.6%

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

      2. add-sqr-sqrt 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. add-sqr-sqrt 99.2%

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

      5. sqrt-unprod 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. swap-sqr 99.4%

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

      9. add-sqr-sqrt 99.6%

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

      10. metadata-eval 99.6%

          \[\leadsto \sqrt{x \cdot \color{blue}{9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   7. Applied egg-rr 98.8%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around inf 75.8%

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

   if -8e7 < y < 8.7999999999999993e-149

   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. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 59.9%

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

   5. Taylor expanded in y around 0 59.5%

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

      1. *-commutative 59.5%

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

   7. Simplified 59.5%

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

   if 8.7999999999999993e-149 < y < 2.25e52

   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. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.2%

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

      7. cancel-sign-sub 99.2%

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

      8. *-commutative 99.2%

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

      9. associate-*r* 99.3%

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

      10. *-commutative 99.3%

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

      11. distribute-rgt-out-- 99.2%

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

      12. distribute-lft-neg-in 99.2%

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

      13. cancel-sign-sub 99.2%

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

      14. +-commutative 99.2%

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

      15. *-commutative 99.2%

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

      16. distribute-rgt-in 99.2%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 65.7%

      \[\leadsto \sqrt{x} \cdot \color{blue}{\frac{0.3333333333333333}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80000000:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-149}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{x} \cdot \frac{0.3333333333333333}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \end{array} \]

Alternative 5: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+24} \lor \neg \left(y \leq 3.35 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -3.9e+24) (not (<= y 3.35e+52)))
   (* y (sqrt (* x 9.0)))
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -3.9e+24) || !(y <= 3.35e+52)) {
		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 <= (-3.9d+24)) .or. (.not. (y <= 3.35d+52))) 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 <= -3.9e+24) || !(y <= 3.35e+52)) {
		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 <= -3.9e+24) or not (y <= 3.35e+52):
		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 <= -3.9e+24) || !(y <= 3.35e+52))
		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 <= -3.9e+24) || ~((y <= 3.35e+52)))
		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, -3.9e+24], N[Not[LessEqual[y, 3.35e+52]], $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 < -3.8999999999999998e24 or 3.34999999999999998e52 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.5%

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

      11. distribute-neg-frac 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.5%

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

   3. Simplified 99.5%

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

      1. add-sqr-sqrt 60.5%

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

      2. pow2 60.5%

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

   5. Applied egg-rr 60.5%

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

      1. unpow2 60.5%

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

      2. add-sqr-sqrt 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. add-sqr-sqrt 99.2%

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

      5. sqrt-unprod 99.5%

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

      6. *-commutative 99.5%

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

      7. *-commutative 99.5%

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

      8. swap-sqr 99.4%

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

      9. add-sqr-sqrt 99.6%

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

      10. metadata-eval 99.6%

          \[\leadsto \sqrt{x \cdot \color{blue}{9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   7. Applied egg-rr 98.8%

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

      1. distribute-lft-out 99.6%

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

      2. *-commutative 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-+l+ 99.6%

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

      5. +-commutative 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in y around inf 77.2%

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

   if -3.8999999999999998e24 < y < 3.34999999999999998e52

   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. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.3%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 91.6%

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

      1. *-commutative 91.6%

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

      2. sub-neg 91.6%

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

      3. associate-*r/ 91.6%

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

      4. metadata-eval 91.6%

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

      5. metadata-eval 91.6%

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

      6. +-commutative 91.6%

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

      7. metadata-eval 91.6%

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

      8. distribute-neg-frac 91.6%

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

      9. unsub-neg 91.6%

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

   6. Simplified 91.6%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{+24} \lor \neg \left(y \leq 3.35 \cdot 10^{+52}\right):\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \end{array} \]

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[Sqrt[x], $MachinePrecision] * N[(N[(N[(0.3333333333333333 / x), $MachinePrecision] + -3.0), $MachinePrecision] + N[(3.0 * y), $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. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.5%

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

   1. fma-udef 99.4%

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

   2. +-commutative 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 7: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80000000 \lor \neg \left(y \leq 9.5 \cdot 10^{-18}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -80000000.0) (not (<= y 9.5e-18)))
   (* 3.0 (* (sqrt x) y))
   (* (sqrt x) -3.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -80000000.0) || !(y <= 9.5e-18)) {
		tmp = 3.0 * (sqrt(x) * y);
	} else {
		tmp = sqrt(x) * -3.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= (-80000000.0d0)) .or. (.not. (y <= 9.5d-18))) then
        tmp = 3.0d0 * (sqrt(x) * y)
    else
        tmp = sqrt(x) * (-3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -80000000.0) || !(y <= 9.5e-18)) {
		tmp = 3.0 * (Math.sqrt(x) * y);
	} else {
		tmp = Math.sqrt(x) * -3.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -80000000.0) or not (y <= 9.5e-18):
		tmp = 3.0 * (math.sqrt(x) * y)
	else:
		tmp = math.sqrt(x) * -3.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -80000000.0) || !(y <= 9.5e-18))
		tmp = Float64(3.0 * Float64(sqrt(x) * y));
	else
		tmp = Float64(sqrt(x) * -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -80000000.0) || ~((y <= 9.5e-18)))
		tmp = 3.0 * (sqrt(x) * y);
	else
		tmp = sqrt(x) * -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -80000000.0], N[Not[LessEqual[y, 9.5e-18]], $MachinePrecision]], N[(3.0 * N[(N[Sqrt[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[x], $MachinePrecision] * -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8e7 or 9.5000000000000003e-18 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 70.2%

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

   if -8e7 < y < 9.5000000000000003e-18

   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. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 54.6%

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

   5. Taylor expanded in y around 0 54.2%

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

      1. *-commutative 54.2%

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

   7. Simplified 54.2%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80000000 \lor \neg \left(y \leq 9.5 \cdot 10^{-18}\right):\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \end{array} \]

Alternative 8: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -80000000:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -80000000.0)
   (* y (sqrt (* x 9.0)))
   (if (<= y 9.5e-18) (* (sqrt x) -3.0) (* 3.0 (* (sqrt x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -80000000.0) {
		tmp = y * sqrt((x * 9.0));
	} else if (y <= 9.5e-18) {
		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 (y <= (-80000000.0d0)) then
        tmp = y * sqrt((x * 9.0d0))
    else if (y <= 9.5d-18) 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 (y <= -80000000.0) {
		tmp = y * Math.sqrt((x * 9.0));
	} else if (y <= 9.5e-18) {
		tmp = Math.sqrt(x) * -3.0;
	} else {
		tmp = 3.0 * (Math.sqrt(x) * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -80000000.0:
		tmp = y * math.sqrt((x * 9.0))
	elif y <= 9.5e-18:
		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 (y <= -80000000.0)
		tmp = Float64(y * sqrt(Float64(x * 9.0)));
	elseif (y <= 9.5e-18)
		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 (y <= -80000000.0)
		tmp = y * sqrt((x * 9.0));
	elseif (y <= 9.5e-18)
		tmp = sqrt(x) * -3.0;
	else
		tmp = 3.0 * (sqrt(x) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -80000000.0], N[(y * N[Sqrt[N[(x * 9.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e-18], 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 y < -8e7

   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. distribute-rgt-in 99.5%

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

      4. remove-double-neg 99.5%

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

      5. distribute-lft-neg-in 99.5%

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

      6. distribute-rgt-neg-in 99.5%

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

      7. mul-1-neg 99.5%

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

      8. metadata-eval 99.5%

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

      9. *-commutative 99.5%

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

      10. associate-/r* 99.4%

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

      11. distribute-neg-frac 99.4%

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

      12. *-commutative 99.4%

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

      13. associate-/r/ 99.5%

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

      14. associate-/l/ 99.5%

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

      15. associate-/r/ 99.4%

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

   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. Step-by-step derivation

      1. add-sqr-sqrt 23.2%

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

      2. pow2 23.2%

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

   5. Applied egg-rr 23.2%

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

      1. unpow2 23.2%

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

      2. add-sqr-sqrt 99.4%

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

      3. distribute-lft-in 99.4%

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

      4. add-sqr-sqrt 99.2%

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

      5. sqrt-unprod 99.4%

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

      6. *-commutative 99.4%

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

      7. *-commutative 99.4%

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

      8. swap-sqr 99.3%

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

      9. add-sqr-sqrt 99.5%

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

      10. metadata-eval 99.5%

          \[\leadsto \sqrt{x \cdot \color{blue}{9}} \cdot y + \left(3 \cdot \sqrt{x}\right) \cdot \left(\frac{0.1111111111111111}{x} + -1\right) \]

   7. Applied egg-rr 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. *-commutative 99.5%

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

      3. +-commutative 99.5%

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

      4. associate-+l+ 99.5%

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

      5. +-commutative 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in y around inf 74.1%

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

   if -8e7 < y < 9.5000000000000003e-18

   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. distribute-lft-in 99.4%

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

      4. +-commutative 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.4%

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

      12. distribute-lft-neg-in 99.4%

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

      13. cancel-sign-sub 99.4%

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

      14. +-commutative 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-in 99.4%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 54.6%

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

   5. Taylor expanded in y around 0 54.2%

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

      1. *-commutative 54.2%

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

   7. Simplified 54.2%

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

   if 9.5000000000000003e-18 < y

   1. Initial program 99.5%

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

      1. +-commutative 99.5%

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

      2. associate--l+ 99.5%

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

      3. distribute-lft-in 99.5%

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

      4. +-commutative 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.4%

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

      7. cancel-sign-sub 99.4%

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

      8. *-commutative 99.4%

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

      9. associate-*r* 99.4%

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

      10. *-commutative 99.4%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around inf 67.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -80000000:\\ \;\;\;\;y \cdot \sqrt{x \cdot 9}\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-18}:\\ \;\;\;\;\sqrt{x} \cdot -3\\ \mathbf{else}:\\ \;\;\;\;3 \cdot \left(\sqrt{x} \cdot y\right)\\ \end{array} \]

Alternative 9: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.7e-8)
   (* (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 <= 5.7e-8) {
		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 <= 5.7d-8) 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 <= 5.7e-8) {
		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 <= 5.7e-8:
		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 <= 5.7e-8)
		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 <= 5.7e-8)
		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, 5.7e-8], 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 < 5.70000000000000009e-8

   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. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-out-- 99.2%

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

      12. distribute-lft-neg-in 99.2%

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

      13. cancel-sign-sub 99.2%

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

      14. +-commutative 99.2%

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

      15. *-commutative 99.2%

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

      16. distribute-rgt-in 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. sub-neg 73.2%

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

      3. associate-*r/ 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

      6. +-commutative 73.3%

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

      7. metadata-eval 73.3%

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

      8. distribute-neg-frac 73.3%

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

      9. unsub-neg 73.3%

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

   6. Simplified 73.3%

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

   if 5.70000000000000009e-8 < 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%

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

      2. associate--l+ 99.6%

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

      3. distribute-lft-in 99.6%

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

      4. +-commutative 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-*r* 99.6%

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

      7. cancel-sign-sub 99.6%

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

      8. *-commutative 99.6%

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

      9. associate-*r* 99.5%

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

      10. *-commutative 99.5%

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

      11. distribute-rgt-out-- 99.5%

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

      12. distribute-lft-neg-in 99.5%

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

      13. cancel-sign-sub 99.5%

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

      14. +-commutative 99.5%

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

      15. *-commutative 99.5%

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

      16. distribute-rgt-in 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 98.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.7 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(3 \cdot y - 3\right)\\ \end{array} \]

Alternative 10: 86.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-5}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e-5)
   (* (sqrt x) (- -3.0 (/ -0.3333333333333333 x)))
   (* (* 3.0 (sqrt x)) (+ y -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e-5) {
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1d-5) then
        tmp = sqrt(x) * ((-3.0d0) - ((-0.3333333333333333d0) / x))
    else
        tmp = (3.0d0 * sqrt(x)) * (y + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e-5) {
		tmp = Math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	} else {
		tmp = (3.0 * Math.sqrt(x)) * (y + -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e-5:
		tmp = math.sqrt(x) * (-3.0 - (-0.3333333333333333 / x))
	else:
		tmp = (3.0 * math.sqrt(x)) * (y + -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e-5)
		tmp = Float64(sqrt(x) * Float64(-3.0 - Float64(-0.3333333333333333 / x)));
	else
		tmp = Float64(Float64(3.0 * sqrt(x)) * Float64(y + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e-5)
		tmp = sqrt(x) * (-3.0 - (-0.3333333333333333 / x));
	else
		tmp = (3.0 * sqrt(x)) * (y + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e-5], N[(N[Sqrt[x], $MachinePrecision] * N[(-3.0 - N[(-0.3333333333333333 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(3.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[(y + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.00000000000000008e-5

   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. distribute-lft-in 99.3%

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

      4. +-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. associate-*r* 99.3%

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

      7. cancel-sign-sub 99.3%

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

      8. *-commutative 99.3%

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

      9. associate-*r* 99.2%

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

      10. *-commutative 99.2%

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

      11. distribute-rgt-out-- 99.2%

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

      12. distribute-lft-neg-in 99.2%

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

      13. cancel-sign-sub 99.2%

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

      14. +-commutative 99.2%

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

      15. *-commutative 99.2%

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

      16. distribute-rgt-in 99.2%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in y around 0 73.2%

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

      1. *-commutative 73.2%

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

      2. sub-neg 73.2%

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

      3. associate-*r/ 73.3%

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

      4. metadata-eval 73.3%

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

      5. metadata-eval 73.3%

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

      6. +-commutative 73.3%

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

      7. metadata-eval 73.3%

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

      8. distribute-neg-frac 73.3%

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

      9. unsub-neg 73.3%

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

   6. Simplified 73.3%

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

   if 1.00000000000000008e-5 < 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%

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

      2. associate--l+ 99.6%

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

      3. distribute-rgt-in 99.6%

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

      4. remove-double-neg 99.6%

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

      5. distribute-lft-neg-in 99.6%

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

      6. distribute-rgt-neg-in 99.6%

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

      7. mul-1-neg 99.6%

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

      8. metadata-eval 99.6%

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

      9. *-commutative 99.6%

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

      10. associate-/r* 99.6%

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

      11. distribute-neg-frac 99.6%

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

      12. *-commutative 99.6%

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

      13. associate-/r/ 99.6%

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

      14. associate-/l/ 99.6%

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

      15. associate-/r/ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 98.6%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{-5}:\\ \;\;\;\;\sqrt{x} \cdot \left(-3 - \frac{-0.3333333333333333}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(3 \cdot \sqrt{x}\right) \cdot \left(y + -1\right)\\ \end{array} \]

Alternative 11: 26.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. +-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. distribute-lft-in 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*r* 99.4%

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

   7. cancel-sign-sub 99.4%

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

   8. *-commutative 99.4%

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

   9. associate-*r* 99.4%

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

   10. *-commutative 99.4%

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

   11. distribute-rgt-out-- 99.4%

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

   12. distribute-lft-neg-in 99.4%

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

   13. cancel-sign-sub 99.4%

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

   14. +-commutative 99.4%

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

   15. *-commutative 99.4%

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

   16. distribute-rgt-in 99.4%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around inf 64.1%

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

5. Taylor expanded in y around 0 26.0%

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

   1. *-commutative 26.0%

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

7. Simplified 26.0%

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

8. Final simplification 26.0%

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

Developer target: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023230 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

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

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