Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 68.6% → 99.8%

Time: 14.8s

Alternatives: 22

Speedup: 1.0×

Specification

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

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

Initial Program: 68.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

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

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* (/ y (+ y x)) (/ x (+ y (+ x 1.0)))) (+ y x)))

C

double code(double x, double y) {
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
}

Fortran

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

Java

public static double code(double x, double y) {
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
}

Python

def code(x, y):
	return ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x)

Julia

function code(x, y)
	return Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + Float64(x + 1.0)))) / Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = ((y / (y + x)) * (x / (y + (x + 1.0)))) / (y + x);
end

Wolfram

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

Derivation

1. Initial program 65.0%

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

   1. +-commutative 65.0%

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

   2. +-commutative 65.0%

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

   3. +-commutative 65.0%

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

   4. *-commutative 65.0%

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

   5. distribute-rgt1-in 54.5%

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

   6. fma-define 65.0%

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

   7. +-commutative 65.0%

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

   8. +-commutative 65.0%

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

   9. cube-unmult 65.0%

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

   10. +-commutative 65.0%

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

3. Simplified 65.0%

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

   1. *-commutative 65.0%

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

   2. fma-define 54.5%

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

   3. cube-mult 54.5%

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

   4. distribute-rgt1-in 65.0%

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

   5. *-commutative 65.0%

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

   6. associate-*l* 65.0%

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

   7. times-frac 92.4%

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

   8. associate-+r+ 92.4%

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

6. Applied egg-rr 92.4%

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

   1. frac-times 65.0%

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

   2. *-commutative 65.0%

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

   3. *-un-lft-identity 65.0%

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

   4. frac-times 70.9%

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

   5. associate-*l/ 70.9%

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

   6. *-un-lft-identity 70.9%

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

   7. associate-/l* 92.4%

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

   8. +-commutative 92.4%

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

   9. +-commutative 92.4%

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

8. Applied egg-rr 92.4%

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

   1. associate-*r/ 70.9%

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

   2. +-commutative 70.9%

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

10. Applied egg-rr 70.9%

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

    1. *-commutative 70.9%

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

    2. times-frac 99.8%

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

    3. associate-+r+ 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

    \[\leadsto \frac{\frac{y}{y + x} \cdot \frac{x}{y + \left(x + 1\right)}}{y + x} \]
14. Add Preprocessing

Alternative 2: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\frac{t\_0 \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= y 5e-294)
     (/ (* t_0 (/ x (+ x 1.0))) (+ y x))
     (if (<= y 1.55e+136)
       (* t_0 (/ x (* (+ y x) (+ x (+ y 1.0)))))
       (/ (/ (- x (* x (* 2.0 (/ x y)))) y) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5e-294) {
		tmp = (t_0 * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = t_0 * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (y + x)
    if (y <= 5d-294) then
        tmp = (t_0 * (x / (x + 1.0d0))) / (y + x)
    else if (y <= 1.55d+136) then
        tmp = t_0 * (x / ((y + x) * (x + (y + 1.0d0))))
    else
        tmp = ((x - (x * (2.0d0 * (x / y)))) / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (y <= 5e-294) {
		tmp = (t_0 * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = t_0 * (x / ((y + x) * (x + (y + 1.0))));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	tmp = 0
	if y <= 5e-294:
		tmp = (t_0 * (x / (x + 1.0))) / (y + x)
	elif y <= 1.55e+136:
		tmp = t_0 * (x / ((y + x) * (x + (y + 1.0))))
	else:
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (y <= 5e-294)
		tmp = Float64(Float64(t_0 * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 1.55e+136)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(2.0 * Float64(x / y)))) / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	tmp = 0.0;
	if (y <= 5e-294)
		tmp = (t_0 * (x / (x + 1.0))) / (y + x);
	elseif (y <= 1.55e+136)
		tmp = t_0 * (x / ((y + x) * (x + (y + 1.0))));
	else
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5e-294], N[(N[(t$95$0 * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+136], N[(t$95$0 * N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.0000000000000003e-294

   1. Initial program 61.8%

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

      1. +-commutative 61.8%

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

      2. +-commutative 61.8%

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

      3. +-commutative 61.8%

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

      4. *-commutative 61.8%

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

      5. distribute-rgt1-in 48.7%

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

      6. fma-define 61.8%

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

      7. +-commutative 61.8%

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

      8. +-commutative 61.8%

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

      9. cube-unmult 61.8%

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

      10. +-commutative 61.8%

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

   3. Simplified 61.8%

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

      1. *-commutative 61.8%

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

      2. fma-define 48.7%

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

      3. cube-mult 48.7%

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

      4. distribute-rgt1-in 61.8%

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

      5. *-commutative 61.8%

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

      6. associate-*l* 61.8%

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

      7. times-frac 89.2%

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

      8. associate-+r+ 89.2%

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

   6. Applied egg-rr 89.2%

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

      1. frac-times 61.8%

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

      2. *-commutative 61.8%

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

      3. *-un-lft-identity 61.8%

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

      4. frac-times 66.6%

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

      5. associate-*l/ 66.6%

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

      6. *-un-lft-identity 66.6%

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

      7. associate-/l* 89.2%

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

      8. +-commutative 89.2%

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

      9. +-commutative 89.2%

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

   8. Applied egg-rr 89.2%

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

      1. associate-*r/ 66.6%

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

      2. +-commutative 66.6%

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

   10. Applied egg-rr 66.6%

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

       1. *-commutative 66.6%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 82.9%

       \[\leadsto \frac{\frac{y}{y + x} \cdot \color{blue}{\frac{x}{1 + x}}}{y + x} \]

   if 5.0000000000000003e-294 < y < 1.54999999999999992e136

   1. Initial program 72.1%

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

      1. +-commutative 72.1%

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

      2. +-commutative 72.1%

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

      3. +-commutative 72.1%

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

      4. *-commutative 72.1%

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

      5. distribute-rgt1-in 62.0%

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

      6. fma-define 72.2%

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

      7. +-commutative 72.2%

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

      8. +-commutative 72.2%

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

      9. cube-unmult 72.1%

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

      10. +-commutative 72.1%

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

   3. Simplified 72.1%

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

      1. *-commutative 72.1%

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

      2. fma-define 61.9%

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

      3. cube-mult 62.0%

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

      4. distribute-rgt1-in 72.1%

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

      5. *-commutative 72.1%

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

      6. associate-*l* 72.1%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 1.54999999999999992e136 < y

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      2. +-commutative 58.5%

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

      3. +-commutative 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-rgt1-in 58.5%

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

      6. fma-define 58.5%

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

      7. +-commutative 58.5%

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

      8. +-commutative 58.5%

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

      9. cube-unmult 58.5%

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

      10. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. *-commutative 58.5%

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

      2. fma-define 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 58.5%

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

      5. *-commutative 58.5%

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

      6. associate-*l* 58.5%

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

      7. times-frac 85.3%

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

      8. associate-+r+ 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. frac-times 58.5%

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

      2. *-commutative 58.5%

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

      3. *-un-lft-identity 58.5%

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

      4. frac-times 64.5%

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

      5. associate-*l/ 64.5%

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

      6. *-un-lft-identity 64.5%

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

      7. associate-/l* 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

   8. Applied egg-rr 85.3%

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

   9. Taylor expanded in y around inf 74.6%

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

       1. mul-1-neg 74.6%

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

       2. unsub-neg 74.6%

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

       3. associate-/l* 90.6%

          \[\leadsto \frac{\frac{x - \color{blue}{x \cdot \frac{1 + 2 \cdot x}{y}}}{y}}{y + x} \]

       4. *-commutative 90.6%

          \[\leadsto \frac{\frac{x - x \cdot \frac{1 + \color{blue}{x \cdot 2}}{y}}{y}}{y + x} \]

   11. Simplified 90.6%

       \[\leadsto \frac{\color{blue}{\frac{x - x \cdot \frac{1 + x \cdot 2}{y}}{y}}}{y + x} \]

   12. Taylor expanded in x around inf 90.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-294}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.6e+27)
   (/ (/ 1.0 (+ y x)) (/ x y))
   (if (<= y 4.4e+129)
     (* (/ y (* (+ y x) (+ x (+ y 1.0)))) (/ x (+ y x)))
     (/ (/ (- x (* x (* 2.0 (/ x y)))) y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.4e+129) {
		tmp = (y / ((y + x) * (x + (y + 1.0)))) * (x / (y + x));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + 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 <= (-8.6d+27)) then
        tmp = (1.0d0 / (y + x)) / (x / y)
    else if (y <= 4.4d+129) then
        tmp = (y / ((y + x) * (x + (y + 1.0d0)))) * (x / (y + x))
    else
        tmp = ((x - (x * (2.0d0 * (x / y)))) / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.6e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.4e+129) {
		tmp = (y / ((y + x) * (x + (y + 1.0)))) * (x / (y + x));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.6e+27:
		tmp = (1.0 / (y + x)) / (x / y)
	elif y <= 4.4e+129:
		tmp = (y / ((y + x) * (x + (y + 1.0)))) * (x / (y + x))
	else:
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.6e+27)
		tmp = Float64(Float64(1.0 / Float64(y + x)) / Float64(x / y));
	elseif (y <= 4.4e+129)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0)))) * Float64(x / Float64(y + x)));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(2.0 * Float64(x / y)))) / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.6e+27)
		tmp = (1.0 / (y + x)) / (x / y);
	elseif (y <= 4.4e+129)
		tmp = (y / ((y + x) * (x + (y + 1.0)))) * (x / (y + x));
	else
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.6e+27], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+129], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.60000000000000017e27

   1. Initial program 44.4%

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

      1. +-commutative 44.4%

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

      2. +-commutative 44.4%

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

      3. +-commutative 44.4%

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

      4. *-commutative 44.4%

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

      5. distribute-rgt1-in 22.7%

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

      6. fma-define 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

      9. cube-unmult 44.4%

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

      10. +-commutative 44.4%

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

   3. Simplified 44.4%

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

      1. *-commutative 44.4%

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

      2. fma-define 22.7%

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

      3. cube-mult 22.7%

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

      4. distribute-rgt1-in 44.4%

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

      5. *-commutative 44.4%

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

      6. associate-*l* 44.4%

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

      7. times-frac 71.0%

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

      8. associate-+r+ 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. frac-times 44.4%

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

      2. *-commutative 44.4%

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

      3. *-un-lft-identity 44.4%

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

      4. frac-times 49.9%

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

      5. clear-num 49.9%

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

      6. frac-times 49.9%

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

      7. metadata-eval 49.9%

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

      8. +-commutative 49.9%

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

      9. *-commutative 49.9%

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

      10. times-frac 98.4%

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

      11. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 33.4%

       \[\leadsto \frac{\frac{1}{y + x}}{\color{blue}{\frac{x}{y}}} \]

   if -8.60000000000000017e27 < y < 4.3999999999999999e129

   1. Initial program 71.9%

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

      1. +-commutative 71.9%

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

      2. +-commutative 71.9%

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

      3. +-commutative 71.9%

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

      4. *-commutative 71.9%

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

      5. distribute-rgt1-in 62.7%

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

      6. fma-define 71.9%

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

      7. +-commutative 71.9%

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

      8. +-commutative 71.9%

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

      9. cube-unmult 71.9%

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

      10. +-commutative 71.9%

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

   3. Simplified 71.9%

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

      1. fma-define 62.7%

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

      2. cube-mult 62.7%

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

      3. distribute-rgt1-in 71.9%

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

      4. *-commutative 71.9%

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

      5. associate-*l* 71.9%

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

      6. times-frac 99.8%

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

      7. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   if 4.3999999999999999e129 < y

   1. Initial program 59.8%

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

      1. +-commutative 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. *-commutative 59.8%

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

      5. distribute-rgt1-in 59.8%

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

      6. fma-define 59.8%

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

      7. +-commutative 59.8%

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

      8. +-commutative 59.8%

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

      9. cube-unmult 59.8%

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

      10. +-commutative 59.8%

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

   3. Simplified 59.8%

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

      1. *-commutative 59.8%

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

      2. fma-define 59.8%

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

      3. cube-mult 59.8%

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

      4. distribute-rgt1-in 59.8%

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

      5. *-commutative 59.8%

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

      6. associate-*l* 59.8%

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

      7. times-frac 85.8%

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

      8. associate-+r+ 85.8%

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

   6. Applied egg-rr 85.8%

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

      1. frac-times 59.8%

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

      2. *-commutative 59.8%

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

      3. *-un-lft-identity 59.8%

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

      4. frac-times 65.6%

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

      5. associate-*l/ 65.6%

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

      6. *-un-lft-identity 65.6%

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

      7. associate-/l* 85.8%

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

      8. +-commutative 85.8%

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

      9. +-commutative 85.8%

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

   8. Applied egg-rr 85.8%

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

   9. Taylor expanded in y around inf 75.4%

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

       1. mul-1-neg 75.4%

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

       2. unsub-neg 75.4%

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

       3. associate-/l* 90.9%

          \[\leadsto \frac{\frac{x - \color{blue}{x \cdot \frac{1 + 2 \cdot x}{y}}}{y}}{y + x} \]

       4. *-commutative 90.9%

          \[\leadsto \frac{\frac{x - x \cdot \frac{1 + \color{blue}{x \cdot 2}}{y}}{y}}{y + x} \]

   11. Simplified 90.9%

       \[\leadsto \frac{\color{blue}{\frac{x - x \cdot \frac{1 + x \cdot 2}{y}}{y}}}{y + x} \]

   12. Taylor expanded in x around inf 90.9%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+129}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-17)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (if (<= y 1.7e+94)
     (* x (/ y (* (+ x (+ y 1.0)) (* (+ y x) (+ y x)))))
     (/ (/ (- x (* x (* 2.0 (/ x y)))) y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-17) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.7e+94) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + 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 <= 4.3d-17) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else if (y <= 1.7d+94) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = ((x - (x * (2.0d0 * (x / y)))) / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-17) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.7e+94) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.3e-17:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	elif y <= 1.7e+94:
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))))
	else:
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-17)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 1.7e+94)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(2.0 * Float64(x / y)))) / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e-17)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	elseif (y <= 1.7e+94)
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	else
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.3e-17], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.7e+94], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.30000000000000023e-17

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt1-in 52.2%

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

      6. fma-define 64.6%

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

      7. +-commutative 64.6%

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

      8. +-commutative 64.6%

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

      9. cube-unmult 64.6%

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

      10. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. fma-define 52.2%

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

      3. cube-mult 52.2%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. associate-*l* 64.6%

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

      7. times-frac 92.3%

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

      8. associate-+r+ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. frac-times 64.6%

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

      2. *-commutative 64.6%

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

      3. *-un-lft-identity 64.6%

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

      4. frac-times 68.1%

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

      5. associate-*l/ 68.1%

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

      6. *-un-lft-identity 68.1%

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

      7. associate-/l* 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-*r/ 68.1%

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

      2. +-commutative 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. *-commutative 68.1%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 87.8%

       \[\leadsto \frac{\frac{y}{y + x} \cdot \color{blue}{\frac{x}{1 + x}}}{y + x} \]

   if 4.30000000000000023e-17 < y < 1.7000000000000001e94

   1. Initial program 90.5%

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

      1. associate-/l* 95.1%

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

      2. associate-+l+ 95.1%

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

   3. Simplified 95.1%

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

   if 1.7000000000000001e94 < y

   1. Initial program 54.5%

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

      1. +-commutative 54.5%

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

      2. +-commutative 54.5%

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

      3. +-commutative 54.5%

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

      4. *-commutative 54.5%

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

      5. distribute-rgt1-in 54.5%

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

      6. fma-define 54.5%

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

      7. +-commutative 54.5%

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

      8. +-commutative 54.5%

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

      9. cube-unmult 54.5%

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

      10. +-commutative 54.5%

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

   3. Simplified 54.5%

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

      1. *-commutative 54.5%

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

      2. fma-define 54.5%

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

      3. cube-mult 54.5%

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

      4. distribute-rgt1-in 54.5%

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

      5. *-commutative 54.5%

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

      6. associate-*l* 54.5%

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

      7. times-frac 89.3%

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

      8. associate-+r+ 89.3%

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

   6. Applied egg-rr 89.3%

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

      1. frac-times 54.5%

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

      2. *-commutative 54.5%

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

      3. *-un-lft-identity 54.5%

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

      4. frac-times 71.9%

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

      5. associate-*l/ 71.9%

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

      6. *-un-lft-identity 71.9%

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

      7. associate-/l* 89.3%

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

      8. +-commutative 89.3%

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

      9. +-commutative 89.3%

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

   8. Applied egg-rr 89.3%

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

   9. Taylor expanded in y around inf 70.8%

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

       1. mul-1-neg 70.8%

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

       2. unsub-neg 70.8%

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

       3. associate-/l* 82.4%

          \[\leadsto \frac{\frac{x - \color{blue}{x \cdot \frac{1 + 2 \cdot x}{y}}}{y}}{y + x} \]

       4. *-commutative 82.4%

          \[\leadsto \frac{\frac{x - x \cdot \frac{1 + \color{blue}{x \cdot 2}}{y}}{y}}{y + x} \]

   11. Simplified 82.4%

       \[\leadsto \frac{\color{blue}{\frac{x - x \cdot \frac{1 + x \cdot 2}{y}}{y}}}{y + x} \]

   12. Taylor expanded in x around inf 82.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-17}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8.5e+27)
   (/ (/ 1.0 (+ y x)) (/ x y))
   (if (<= y 1.05e-32)
     (* (/ x (+ y x)) (/ y (* (+ y x) (+ x 1.0))))
     (if (<= y 1.2e+150)
       (/ x (* (+ y x) (+ x (+ y 1.0))))
       (/ (/ x (+ y (+ x 1.0))) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 1.05e-32) {
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	} else if (y <= 1.2e+150) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + 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 <= (-8.5d+27)) then
        tmp = (1.0d0 / (y + x)) / (x / y)
    else if (y <= 1.05d-32) then
        tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0d0)))
    else if (y <= 1.2d+150) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8.5e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 1.05e-32) {
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	} else if (y <= 1.2e+150) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8.5e+27:
		tmp = (1.0 / (y + x)) / (x / y)
	elif y <= 1.05e-32:
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)))
	elif y <= 1.2e+150:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8.5e+27)
		tmp = Float64(Float64(1.0 / Float64(y + x)) / Float64(x / y));
	elseif (y <= 1.05e-32)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * Float64(x + 1.0))));
	elseif (y <= 1.2e+150)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8.5e+27)
		tmp = (1.0 / (y + x)) / (x / y);
	elseif (y <= 1.05e-32)
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	elseif (y <= 1.2e+150)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8.5e+27], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e-32], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+150], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -8.5e27

   1. Initial program 44.4%

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

      1. +-commutative 44.4%

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

      2. +-commutative 44.4%

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

      3. +-commutative 44.4%

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

      4. *-commutative 44.4%

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

      5. distribute-rgt1-in 22.7%

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

      6. fma-define 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

      9. cube-unmult 44.4%

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

      10. +-commutative 44.4%

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

   3. Simplified 44.4%

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

      1. *-commutative 44.4%

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

      2. fma-define 22.7%

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

      3. cube-mult 22.7%

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

      4. distribute-rgt1-in 44.4%

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

      5. *-commutative 44.4%

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

      6. associate-*l* 44.4%

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

      7. times-frac 71.0%

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

      8. associate-+r+ 71.0%

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

   6. Applied egg-rr 71.0%

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

      1. frac-times 44.4%

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

      2. *-commutative 44.4%

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

      3. *-un-lft-identity 44.4%

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

      4. frac-times 49.9%

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

      5. clear-num 49.9%

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

      6. frac-times 49.9%

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

      7. metadata-eval 49.9%

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

      8. +-commutative 49.9%

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

      9. *-commutative 49.9%

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

      10. times-frac 98.4%

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

      11. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 33.4%

       \[\leadsto \frac{\frac{1}{y + x}}{\color{blue}{\frac{x}{y}}} \]

   if -8.5e27 < y < 1.05e-32

   1. Initial program 71.7%

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

   3. Taylor expanded in x around inf 70.3%

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

      1. associate-*l* 70.3%

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

      2. times-frac 98.5%

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

      3. +-commutative 98.5%

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

      4. +-commutative 98.5%

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

   5. Applied egg-rr 98.5%

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

   if 1.05e-32 < y < 1.20000000000000001e150

   1. Initial program 73.2%

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

      1. +-commutative 73.2%

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

      2. +-commutative 73.2%

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

      3. +-commutative 73.2%

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

      4. *-commutative 73.2%

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

      5. distribute-rgt1-in 64.9%

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

      6. fma-define 73.3%

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

      7. +-commutative 73.3%

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

      8. +-commutative 73.3%

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

      9. cube-unmult 73.2%

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

      10. +-commutative 73.2%

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

   3. Simplified 73.2%

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

      1. *-commutative 73.2%

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

      2. fma-define 64.8%

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

      3. cube-mult 64.9%

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

      4. distribute-rgt1-in 73.2%

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

      5. *-commutative 73.2%

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

      6. associate-*l* 73.2%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 77.1%

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

   if 1.20000000000000001e150 < y

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. +-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. distribute-rgt1-in 57.4%

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

      6. fma-define 57.4%

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

      7. +-commutative 57.4%

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

      8. +-commutative 57.4%

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

      9. cube-unmult 57.4%

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

      10. +-commutative 57.4%

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

   3. Simplified 57.4%

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

      1. *-commutative 57.4%

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

      2. fma-define 57.4%

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

      3. cube-mult 57.4%

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

      4. distribute-rgt1-in 57.4%

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

      5. *-commutative 57.4%

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

      6. associate-*l* 57.4%

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

      7. times-frac 83.7%

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

      8. associate-+r+ 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. frac-times 57.4%

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

      2. *-commutative 57.4%

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

      3. *-un-lft-identity 57.4%

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

      4. frac-times 60.7%

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

      5. associate-*l/ 60.7%

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

      6. *-un-lft-identity 60.7%

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

      7. associate-/l* 83.7%

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

      8. +-commutative 83.7%

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

      9. +-commutative 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. associate-*r/ 60.7%

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

      2. +-commutative 60.7%

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

   10. Applied egg-rr 60.7%

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

       1. *-commutative 60.7%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around inf 89.2%

       \[\leadsto \frac{\color{blue}{1} \cdot \frac{x}{\left(x + 1\right) + y}}{y + x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-32)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (if (<= y 1.55e+136)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ (- x (* x (* 2.0 (/ x y)))) y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + 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 <= 1.05d-32) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else if (y <= 1.55d+136) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = ((x - (x * (2.0d0 * (x / y)))) / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-32:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	elif y <= 1.55e+136:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-32)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 1.55e+136)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(2.0 * Float64(x / y)))) / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-32)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	elseif (y <= 1.55e+136)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = ((x - (x * (2.0 * (x / y)))) / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-32], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+136], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(2.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-32

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt1-in 52.2%

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

      6. fma-define 64.6%

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

      7. +-commutative 64.6%

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

      8. +-commutative 64.6%

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

      9. cube-unmult 64.6%

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

      10. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. fma-define 52.2%

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

      3. cube-mult 52.2%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. associate-*l* 64.6%

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

      7. times-frac 92.3%

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

      8. associate-+r+ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. frac-times 64.6%

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

      2. *-commutative 64.6%

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

      3. *-un-lft-identity 64.6%

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

      4. frac-times 68.1%

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

      5. associate-*l/ 68.1%

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

      6. *-un-lft-identity 68.1%

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

      7. associate-/l* 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-*r/ 68.1%

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

      2. +-commutative 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. *-commutative 68.1%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 87.8%

       \[\leadsto \frac{\frac{y}{y + x} \cdot \color{blue}{\frac{x}{1 + x}}}{y + x} \]

   if 1.05e-32 < y < 1.54999999999999992e136

   1. Initial program 73.6%

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

      1. +-commutative 73.6%

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

      2. +-commutative 73.6%

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

      3. +-commutative 73.6%

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

      4. *-commutative 73.6%

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

      5. distribute-rgt1-in 64.6%

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

      6. fma-define 73.7%

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

      7. +-commutative 73.7%

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

      8. +-commutative 73.7%

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

      9. cube-unmult 73.6%

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

      10. +-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. *-commutative 73.6%

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

      2. fma-define 64.4%

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

      3. cube-mult 64.6%

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

      4. distribute-rgt1-in 73.6%

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

      5. *-commutative 73.6%

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

      6. associate-*l* 73.6%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 75.0%

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

   if 1.54999999999999992e136 < y

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      2. +-commutative 58.5%

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

      3. +-commutative 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-rgt1-in 58.5%

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

      6. fma-define 58.5%

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

      7. +-commutative 58.5%

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

      8. +-commutative 58.5%

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

      9. cube-unmult 58.5%

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

      10. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. *-commutative 58.5%

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

      2. fma-define 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 58.5%

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

      5. *-commutative 58.5%

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

      6. associate-*l* 58.5%

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

      7. times-frac 85.3%

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

      8. associate-+r+ 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. frac-times 58.5%

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

      2. *-commutative 58.5%

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

      3. *-un-lft-identity 58.5%

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

      4. frac-times 64.5%

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

      5. associate-*l/ 64.5%

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

      6. *-un-lft-identity 64.5%

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

      7. associate-/l* 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

   8. Applied egg-rr 85.3%

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

   9. Taylor expanded in y around inf 74.6%

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

       1. mul-1-neg 74.6%

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

       2. unsub-neg 74.6%

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

       3. associate-/l* 90.6%

          \[\leadsto \frac{\frac{x - \color{blue}{x \cdot \frac{1 + 2 \cdot x}{y}}}{y}}{y + x} \]

       4. *-commutative 90.6%

          \[\leadsto \frac{\frac{x - x \cdot \frac{1 + \color{blue}{x \cdot 2}}{y}}{y}}{y + x} \]

   11. Simplified 90.6%

       \[\leadsto \frac{\color{blue}{\frac{x - x \cdot \frac{1 + x \cdot 2}{y}}{y}}}{y + x} \]

   12. Taylor expanded in x around inf 90.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \left(2 \cdot \frac{x}{y}\right)}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-32)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (if (<= y 1.26e+150)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y (+ x 1.0))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.26e+150) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + 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 <= 1.05d-32) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else if (y <= 1.26d+150) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-32) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else if (y <= 1.26e+150) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-32:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	elif y <= 1.26e+150:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-32)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	elseif (y <= 1.26e+150)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-32)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	elseif (y <= 1.26e+150)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-32], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.26e+150], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05e-32

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

      3. +-commutative 64.6%

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

      4. *-commutative 64.6%

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

      5. distribute-rgt1-in 52.2%

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

      6. fma-define 64.6%

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

      7. +-commutative 64.6%

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

      8. +-commutative 64.6%

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

      9. cube-unmult 64.6%

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

      10. +-commutative 64.6%

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

   3. Simplified 64.6%

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

      1. *-commutative 64.6%

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

      2. fma-define 52.2%

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

      3. cube-mult 52.2%

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

      4. distribute-rgt1-in 64.6%

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

      5. *-commutative 64.6%

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

      6. associate-*l* 64.6%

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

      7. times-frac 92.3%

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

      8. associate-+r+ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. frac-times 64.6%

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

      2. *-commutative 64.6%

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

      3. *-un-lft-identity 64.6%

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

      4. frac-times 68.1%

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

      5. associate-*l/ 68.1%

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

      6. *-un-lft-identity 68.1%

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

      7. associate-/l* 92.3%

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

      8. +-commutative 92.3%

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

      9. +-commutative 92.3%

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

   8. Applied egg-rr 92.3%

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

      1. associate-*r/ 68.1%

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

      2. +-commutative 68.1%

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

   10. Applied egg-rr 68.1%

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

       1. *-commutative 68.1%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 87.8%

       \[\leadsto \frac{\frac{y}{y + x} \cdot \color{blue}{\frac{x}{1 + x}}}{y + x} \]

   if 1.05e-32 < y < 1.26e150

   1. Initial program 73.2%

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

      1. +-commutative 73.2%

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

      2. +-commutative 73.2%

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

      3. +-commutative 73.2%

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

      4. *-commutative 73.2%

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

      5. distribute-rgt1-in 64.9%

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

      6. fma-define 73.3%

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

      7. +-commutative 73.3%

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

      8. +-commutative 73.3%

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

      9. cube-unmult 73.2%

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

      10. +-commutative 73.2%

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

   3. Simplified 73.2%

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

      1. *-commutative 73.2%

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

      2. fma-define 64.8%

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

      3. cube-mult 64.9%

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

      4. distribute-rgt1-in 73.2%

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

      5. *-commutative 73.2%

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

      6. associate-*l* 73.2%

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

      7. times-frac 99.8%

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

      8. associate-+r+ 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 77.1%

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

   if 1.26e150 < y

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. +-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. distribute-rgt1-in 57.4%

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

      6. fma-define 57.4%

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

      7. +-commutative 57.4%

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

      8. +-commutative 57.4%

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

      9. cube-unmult 57.4%

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

      10. +-commutative 57.4%

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

   3. Simplified 57.4%

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

      1. *-commutative 57.4%

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

      2. fma-define 57.4%

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

      3. cube-mult 57.4%

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

      4. distribute-rgt1-in 57.4%

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

      5. *-commutative 57.4%

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

      6. associate-*l* 57.4%

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

      7. times-frac 83.7%

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

      8. associate-+r+ 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. frac-times 57.4%

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

      2. *-commutative 57.4%

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

      3. *-un-lft-identity 57.4%

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

      4. frac-times 60.7%

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

      5. associate-*l/ 60.7%

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

      6. *-un-lft-identity 60.7%

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

      7. associate-/l* 83.7%

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

      8. +-commutative 83.7%

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

      9. +-commutative 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. associate-*r/ 60.7%

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

      2. +-commutative 60.7%

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

   10. Applied egg-rr 60.7%

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

       1. *-commutative 60.7%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around inf 89.2%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-32}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+150}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.75e-162)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 3.1e+149)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y (+ x 1.0))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-162) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 3.1e+149) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + 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 <= 1.75d-162) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 3.1d+149) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-162) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 3.1e+149) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.75e-162:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 3.1e+149:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.75e-162)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 3.1e+149)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.75e-162)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 3.1e+149)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + (x + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.75e-162], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.1e+149], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.74999999999999995e-162

   1. Initial program 62.8%

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

      1. +-commutative 62.8%

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

      2. +-commutative 62.8%

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

      3. +-commutative 62.8%

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

      4. *-commutative 62.8%

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

      5. distribute-rgt1-in 50.5%

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

      6. fma-define 62.8%

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

      7. +-commutative 62.8%

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

      8. +-commutative 62.8%

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

      9. cube-unmult 62.8%

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

      10. +-commutative 62.8%

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

   3. Simplified 62.8%

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

      1. *-commutative 62.8%

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

      2. fma-define 50.5%

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

      3. cube-mult 50.5%

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

      4. distribute-rgt1-in 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*l* 62.8%

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

      7. times-frac 91.0%

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

      8. associate-+r+ 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. frac-times 62.8%

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

      2. *-commutative 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. frac-times 66.9%

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

      5. associate-*l/ 66.9%

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

      6. *-un-lft-identity 66.9%

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

      7. associate-/l* 90.9%

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

      8. +-commutative 90.9%

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

      9. +-commutative 90.9%

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

   8. Applied egg-rr 90.9%

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

   9. Taylor expanded in y around 0 63.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]
   10. Step-by-step derivation

       1. +-commutative 63.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{y + x} \]

   11. Simplified 63.7%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{y + x} \]

   if 1.74999999999999995e-162 < y < 3.09999999999999987e149

   1. Initial program 73.7%

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

      1. +-commutative 73.7%

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

      2. +-commutative 73.7%

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

      3. +-commutative 73.7%

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

      4. *-commutative 73.7%

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

      5. distribute-rgt1-in 63.1%

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

      6. fma-define 73.8%

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

      7. +-commutative 73.8%

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

      8. +-commutative 73.8%

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

      9. cube-unmult 73.6%

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

      10. +-commutative 73.6%

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

   3. Simplified 73.6%

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

      1. *-commutative 73.6%

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

      2. fma-define 63.0%

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

      3. cube-mult 63.1%

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

      4. distribute-rgt1-in 73.7%

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

      5. *-commutative 73.7%

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

      6. associate-*l* 73.7%

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

      7. times-frac 99.7%

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

      8. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 68.9%

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

   if 3.09999999999999987e149 < y

   1. Initial program 57.4%

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

      1. +-commutative 57.4%

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

      2. +-commutative 57.4%

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

      3. +-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. distribute-rgt1-in 57.4%

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

      6. fma-define 57.4%

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

      7. +-commutative 57.4%

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

      8. +-commutative 57.4%

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

      9. cube-unmult 57.4%

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

      10. +-commutative 57.4%

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

   3. Simplified 57.4%

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

      1. *-commutative 57.4%

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

      2. fma-define 57.4%

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

      3. cube-mult 57.4%

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

      4. distribute-rgt1-in 57.4%

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

      5. *-commutative 57.4%

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

      6. associate-*l* 57.4%

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

      7. times-frac 83.7%

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

      8. associate-+r+ 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. frac-times 57.4%

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

      2. *-commutative 57.4%

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

      3. *-un-lft-identity 57.4%

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

      4. frac-times 60.7%

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

      5. associate-*l/ 60.7%

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

      6. *-un-lft-identity 60.7%

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

      7. associate-/l* 83.7%

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

      8. +-commutative 83.7%

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

      9. +-commutative 83.7%

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

   8. Applied egg-rr 83.7%

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

      1. associate-*r/ 60.7%

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

      2. +-commutative 60.7%

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

   10. Applied egg-rr 60.7%

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

       1. *-commutative 60.7%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around inf 89.2%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e-161)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= y 1.55e+136)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-161) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + 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 <= 5.2d-161) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (y <= 1.55d+136) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.2e-161) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (y <= 1.55e+136) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.2e-161:
		tmp = (y / (x + 1.0)) / (y + x)
	elif y <= 1.55e+136:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e-161)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (y <= 1.55e+136)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.2e-161)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (y <= 1.55e+136)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e-161], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e+136], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.19999999999999991e-161

   1. Initial program 62.8%

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

      1. +-commutative 62.8%

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

      2. +-commutative 62.8%

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

      3. +-commutative 62.8%

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

      4. *-commutative 62.8%

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

      5. distribute-rgt1-in 50.5%

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

      6. fma-define 62.8%

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

      7. +-commutative 62.8%

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

      8. +-commutative 62.8%

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

      9. cube-unmult 62.8%

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

      10. +-commutative 62.8%

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

   3. Simplified 62.8%

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

      1. *-commutative 62.8%

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

      2. fma-define 50.5%

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

      3. cube-mult 50.5%

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

      4. distribute-rgt1-in 62.8%

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

      5. *-commutative 62.8%

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

      6. associate-*l* 62.8%

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

      7. times-frac 91.0%

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

      8. associate-+r+ 91.0%

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

   6. Applied egg-rr 91.0%

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

      1. frac-times 62.8%

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

      2. *-commutative 62.8%

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

      3. *-un-lft-identity 62.8%

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

      4. frac-times 66.9%

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

      5. associate-*l/ 66.9%

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

      6. *-un-lft-identity 66.9%

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

      7. associate-/l* 90.9%

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

      8. +-commutative 90.9%

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

      9. +-commutative 90.9%

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

   8. Applied egg-rr 90.9%

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

   9. Taylor expanded in y around 0 63.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]
   10. Step-by-step derivation

       1. +-commutative 63.7%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{y + x} \]

   11. Simplified 63.7%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{y + x} \]

   if 5.19999999999999991e-161 < y < 1.54999999999999992e136

   1. Initial program 73.9%

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

      1. +-commutative 73.9%

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

      2. +-commutative 73.9%

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

      3. +-commutative 73.9%

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

      4. *-commutative 73.9%

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

      5. distribute-rgt1-in 62.9%

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

      6. fma-define 74.0%

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

      7. +-commutative 74.0%

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

      8. +-commutative 74.0%

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

      9. cube-unmult 73.9%

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

      10. +-commutative 73.9%

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

   3. Simplified 73.9%

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

      1. *-commutative 73.9%

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

      2. fma-define 62.7%

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

      3. cube-mult 62.9%

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

      4. distribute-rgt1-in 73.9%

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

      5. *-commutative 73.9%

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

      6. associate-*l* 73.9%

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

      7. times-frac 99.7%

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

      8. associate-+r+ 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 67.4%

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

   if 1.54999999999999992e136 < y

   1. Initial program 58.5%

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

      1. +-commutative 58.5%

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

      2. +-commutative 58.5%

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

      3. +-commutative 58.5%

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

      4. *-commutative 58.5%

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

      5. distribute-rgt1-in 58.5%

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

      6. fma-define 58.5%

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

      7. +-commutative 58.5%

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

      8. +-commutative 58.5%

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

      9. cube-unmult 58.5%

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

      10. +-commutative 58.5%

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

   3. Simplified 58.5%

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

      1. *-commutative 58.5%

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

      2. fma-define 58.5%

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

      3. cube-mult 58.5%

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

      4. distribute-rgt1-in 58.5%

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

      5. *-commutative 58.5%

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

      6. associate-*l* 58.5%

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

      7. times-frac 85.3%

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

      8. associate-+r+ 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. frac-times 58.5%

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

      2. *-commutative 58.5%

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

      3. *-un-lft-identity 58.5%

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

      4. frac-times 64.5%

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

      5. associate-*l/ 64.5%

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

      6. *-un-lft-identity 64.5%

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

      7. associate-/l* 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

   8. Applied egg-rr 85.3%

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

      1. associate-*r/ 64.5%

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

      2. +-commutative 64.5%

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

   10. Applied egg-rr 64.5%

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

       1. *-commutative 64.5%

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

       2. times-frac 99.8%

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

       3. associate-+r+ 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 89.8%

       \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   14. Step-by-step derivation

       1. +-commutative 89.8%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   15. Simplified 89.8%

       \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+27)
   (/ (/ 1.0 (+ y x)) (/ x y))
   (if (<= y 4.8e-137) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + 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 <= (-7.8d+27)) then
        tmp = (1.0d0 / (y + x)) / (x / y)
    else if (y <= 4.8d-137) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+27:
		tmp = (1.0 / (y + x)) / (x / y)
	elif y <= 4.8e-137:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+27)
		tmp = Float64(Float64(1.0 / Float64(y + x)) / Float64(x / y));
	elseif (y <= 4.8e-137)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+27)
		tmp = (1.0 / (y + x)) / (x / y);
	elseif (y <= 4.8e-137)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+27], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e-137], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999997e27

   1. Initial program 44.4%

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

      1. +-commutative 44.4%

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

      2. +-commutative 44.4%

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

      3. +-commutative 44.4%

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

      4. *-commutative 44.4%

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

      5. distribute-rgt1-in 22.7%

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

      6. fma-define 44.4%

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

      7. +-commutative 44.4%

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

      8. +-commutative 44.4%

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

      9. cube-unmult 44.4%

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

      10. +-commutative 44.4%

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

   3. Simplified 44.4%

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

      1. *-commutative 44.4%

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

      2. fma-define 22.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 22.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 71.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 71.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 44.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 44.4%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 44.4%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 49.9%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. clear-num 49.9%

         \[\leadsto \frac{1}{x + y} \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}}} \]

      6. frac-times 49.9%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}}} \]

      7. metadata-eval 49.9%

         \[\leadsto \frac{\color{blue}{1}}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}} \]

      8. +-commutative 49.9%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right)} \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}} \]

      9. *-commutative 49.9%

         \[\leadsto \frac{1}{\left(y + x\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{\color{blue}{y \cdot x}}} \]

      10. times-frac 98.4%

          \[\leadsto \frac{1}{\left(y + x\right) \cdot \color{blue}{\left(\frac{x + y}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)}} \]

      11. +-commutative 98.4%

          \[\leadsto \frac{1}{\left(y + x\right) \cdot \left(\frac{\color{blue}{y + x}}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)} \]

   8. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{1}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{y + x}}{\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}}} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{y + x}}{\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}}} \]

   11. Taylor expanded in x around inf 33.4%

       \[\leadsto \frac{\frac{1}{y + x}}{\color{blue}{\frac{x}{y}}} \]

   if -7.7999999999999997e27 < y < 4.8000000000000001e-137

   1. Initial program 71.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 77.6%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.8000000000000001e-137 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 62.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 67.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 67.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 67.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 67.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 62.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 62.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 94.7%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 94.7%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 67.8%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 67.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 77.4%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-*l/ 77.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}} \]

      6. *-un-lft-identity 77.4%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      7. associate-/l* 94.6%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      8. +-commutative 94.6%

         \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + \left(y + 1\right)\right)}}{x + y} \]

      9. +-commutative 94.6%

         \[\leadsto \frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{\color{blue}{y + x}} \]

   8. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}}{y + x} \]

      2. +-commutative 77.4%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \color{blue}{\left(1 + y\right)}\right)}}{y + x} \]

   10. Applied egg-rr 77.4%

       \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}}{y + x} \]
   11. Step-by-step derivation

       1. *-commutative 77.4%

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}{y + x} \]

       2. times-frac 99.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{x + \left(1 + y\right)}}}{y + x} \]

       3. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(x + 1\right) + y}}}{y + x} \]

   12. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + 1\right) + y}}}{y + x} \]

   13. Taylor expanded in x around 0 62.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   14. Step-by-step derivation

       1. +-commutative 62.4%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   15. Simplified 62.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+27)
   (/ (/ 1.0 (+ y x)) (/ x y))
   (if (<= y 4.9e-138) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.9e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.8d+27)) then
        tmp = (1.0d0 / (y + x)) / (x / y)
    else if (y <= 4.9d-138) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (1.0 / (y + x)) / (x / y);
	} else if (y <= 4.9e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+27:
		tmp = (1.0 / (y + x)) / (x / y)
	elif y <= 4.9e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+27)
		tmp = Float64(Float64(1.0 / Float64(y + x)) / Float64(x / y));
	elseif (y <= 4.9e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+27)
		tmp = (1.0 / (y + x)) / (x / y);
	elseif (y <= 4.9e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+27], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999997e27

   1. Initial program 44.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 22.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 44.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 22.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 22.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 71.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 71.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 44.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 44.4%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 44.4%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 49.9%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. clear-num 49.9%

         \[\leadsto \frac{1}{x + y} \cdot \color{blue}{\frac{1}{\frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}}} \]

      6. frac-times 49.9%

         \[\leadsto \color{blue}{\frac{1 \cdot 1}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}}} \]

      7. metadata-eval 49.9%

         \[\leadsto \frac{\color{blue}{1}}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}} \]

      8. +-commutative 49.9%

         \[\leadsto \frac{1}{\color{blue}{\left(y + x\right)} \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{x \cdot y}} \]

      9. *-commutative 49.9%

         \[\leadsto \frac{1}{\left(y + x\right) \cdot \frac{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}{\color{blue}{y \cdot x}}} \]

      10. times-frac 98.4%

          \[\leadsto \frac{1}{\left(y + x\right) \cdot \color{blue}{\left(\frac{x + y}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)}} \]

      11. +-commutative 98.4%

          \[\leadsto \frac{1}{\left(y + x\right) \cdot \left(\frac{\color{blue}{y + x}}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)} \]

   8. Applied egg-rr 98.4%

      \[\leadsto \color{blue}{\frac{1}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{1}{y + x}}{\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}}} \]

   10. Simplified 99.8%

       \[\leadsto \color{blue}{\frac{\frac{1}{y + x}}{\frac{y + x}{y} \cdot \frac{x + \left(y + 1\right)}{x}}} \]

   11. Taylor expanded in x around inf 33.4%

       \[\leadsto \frac{\frac{1}{y + x}}{\color{blue}{\frac{x}{y}}} \]

   if -7.7999999999999997e27 < y < 4.90000000000000016e-138

   1. Initial program 71.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 77.6%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.90000000000000016e-138 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 60.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 60.3%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 61.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 61.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 61.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 61.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -7.8e+27)
   (/ (/ y x) (+ y x))
   (if (<= y 2.35e-138) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (y / x) / (y + x);
	} else if (y <= 2.35e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-7.8d+27)) then
        tmp = (y / x) / (y + x)
    else if (y <= 2.35d-138) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -7.8e+27) {
		tmp = (y / x) / (y + x);
	} else if (y <= 2.35e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -7.8e+27:
		tmp = (y / x) / (y + x)
	elif y <= 2.35e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -7.8e+27)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (y <= 2.35e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -7.8e+27)
		tmp = (y / x) / (y + x);
	elseif (y <= 2.35e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -7.8e+27], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.35e-138], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999997e27

   1. Initial program 44.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 22.7%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 44.4%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 44.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 22.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 22.7%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 44.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 71.0%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 71.0%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 44.4%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 44.4%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 44.4%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 49.9%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-*l/ 49.9%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}} \]

      6. *-un-lft-identity 49.9%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      7. associate-/l* 71.0%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      8. +-commutative 71.0%

         \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + \left(y + 1\right)\right)}}{x + y} \]

      9. +-commutative 71.0%

         \[\leadsto \frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{\color{blue}{y + x}} \]

   8. Applied egg-rr 71.0%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 49.9%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}}{y + x} \]

      2. +-commutative 49.9%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \color{blue}{\left(1 + y\right)}\right)}}{y + x} \]

   10. Applied egg-rr 49.9%

       \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}}{y + x} \]
   11. Step-by-step derivation

       1. *-commutative 49.9%

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}{y + x} \]

       2. times-frac 99.7%

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{x + \left(1 + y\right)}}}{y + x} \]

       3. associate-+r+ 99.7%

          \[\leadsto \frac{\frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(x + 1\right) + y}}}{y + x} \]

   12. Simplified 99.7%

       \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + 1\right) + y}}}{y + x} \]

   13. Taylor expanded in x around inf 33.2%

       \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + x} \]

   if -7.7999999999999997e27 < y < 2.3500000000000001e-138

   1. Initial program 71.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 77.6%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 2.3500000000000001e-138 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 60.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 60.3%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 61.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 61.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 61.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 61.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+27}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-137)
   (/ y (* x (+ x 1.0)))
   (if (<= y 5e+133) (/ x (* y (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5e+133) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / 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 <= 4.8d-137) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 5d+133) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5e+133) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-137:
		tmp = y / (x * (x + 1.0))
	elif y <= 5e+133:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-137)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 5e+133)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-137)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 5e+133)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-137], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+133], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.8000000000000001e-137

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 77.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 77.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 60.5%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.8000000000000001e-137 < y < 4.99999999999999961e133

   1. Initial program 72.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 85.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 85.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 46.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if 4.99999999999999961e133 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 85.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 85.3%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 89.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 89.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 89.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 89.7%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]

   10. Taylor expanded in y around inf 89.7%

       \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-137) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + 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 <= 4.8d-137) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-137:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-137)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-137)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-137], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8000000000000001e-137

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 50.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 63.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 63.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 63.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 63.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 63.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 50.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 50.5%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 63.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 91.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 91.2%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 63.5%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 63.5%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 63.5%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 67.5%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-*l/ 67.5%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}} \]

      6. *-un-lft-identity 67.5%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      7. associate-/l* 91.2%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      8. +-commutative 91.2%

         \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + \left(y + 1\right)\right)}}{x + y} \]

      9. +-commutative 91.2%

         \[\leadsto \frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{\color{blue}{y + x}} \]

   8. Applied egg-rr 91.2%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}} \]

   9. Taylor expanded in y around 0 64.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]
   10. Step-by-step derivation

       1. +-commutative 64.4%

          \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{y + x} \]

   11. Simplified 64.4%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{y + x} \]

   if 4.8000000000000001e-137 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 67.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 62.2%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 67.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 67.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 67.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 67.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 67.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 67.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 62.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 62.2%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 67.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 94.7%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 94.7%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 94.7%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   7. Step-by-step derivation

      1. frac-times 67.8%

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)}} \]

      2. *-commutative 67.8%

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      3. *-un-lft-identity 67.8%

         \[\leadsto \frac{\color{blue}{1 \cdot \left(x \cdot y\right)}}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)\right)} \]

      4. frac-times 77.4%

         \[\leadsto \color{blue}{\frac{1}{x + y} \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      5. associate-*l/ 77.4%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}} \]

      6. *-un-lft-identity 77.4%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      7. associate-/l* 94.6%

         \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}}{x + y} \]

      8. +-commutative 94.6%

         \[\leadsto \frac{x \cdot \frac{y}{\color{blue}{\left(y + x\right)} \cdot \left(x + \left(y + 1\right)\right)}}{x + y} \]

      9. +-commutative 94.6%

         \[\leadsto \frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{\color{blue}{y + x}} \]

   8. Applied egg-rr 94.6%

      \[\leadsto \color{blue}{\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}{y + x}} \]
   9. Step-by-step derivation

      1. associate-*r/ 77.4%

         \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}}}{y + x} \]

      2. +-commutative 77.4%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \color{blue}{\left(1 + y\right)}\right)}}{y + x} \]

   10. Applied egg-rr 77.4%

       \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}}{y + x} \]
   11. Step-by-step derivation

       1. *-commutative 77.4%

          \[\leadsto \frac{\frac{\color{blue}{y \cdot x}}{\left(y + x\right) \cdot \left(x + \left(1 + y\right)\right)}}{y + x} \]

       2. times-frac 99.8%

          \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{x + \left(1 + y\right)}}}{y + x} \]

       3. associate-+r+ 99.8%

          \[\leadsto \frac{\frac{y}{y + x} \cdot \frac{x}{\color{blue}{\left(x + 1\right) + y}}}{y + x} \]

   12. Simplified 99.8%

       \[\leadsto \frac{\color{blue}{\frac{y}{y + x} \cdot \frac{x}{\left(x + 1\right) + y}}}{y + x} \]

   13. Taylor expanded in x around 0 62.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   14. Step-by-step derivation

       1. +-commutative 62.4%

          \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   15. Simplified 62.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-137) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.8d-137) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-137:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-137)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-137)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-137], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.8000000000000001e-137

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 77.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 77.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 60.5%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.8000000000000001e-137 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 60.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 60.3%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 61.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 61.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 61.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 61.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 61.8%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 61.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.25 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.25e-137) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.25e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (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 (y <= 4.25d-137) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.25e-137) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.25e-137:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.25e-137)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.25e-137)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.25e-137], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.2500000000000001e-137

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 77.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 77.9%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 77.9%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   6. Step-by-step derivation

      1. +-commutative 60.5%

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

   7. Simplified 60.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + 1\right)}} \]

   if 4.2500000000000001e-137 < y

   1. Initial program 67.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 83.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 83.0%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 83.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 60.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 61.8%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 61.8%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 61.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 49.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e+133) (/ x (* y (+ y 1.0))) (/ (/ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e+133) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / 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 <= 5d+133) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e+133) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e+133:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e+133)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e+133)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e+133], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999961e133

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 37.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   if 4.99999999999999961e133 < y

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 79.3%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 79.3%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 79.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 85.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 85.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 85.3%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 89.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 89.7%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 89.7%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 89.7%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 89.7%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]

   10. Taylor expanded in y around inf 89.7%

       \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 37.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ (/ x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / 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 <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 34.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 25.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 64.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 78.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 78.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 71.8%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 71.8%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 74.0%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]
   8. Step-by-step derivation

      1. associate-*l/ 74.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + 1}}{y}} \]

      2. *-un-lft-identity 74.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y} \]

      3. +-commutative 74.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y} \]

   9. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{1 + y}}{y}} \]

   10. Taylor expanded in y around inf 72.0%

       \[\leadsto \frac{\frac{x}{\color{blue}{y}}}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 36.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * 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 <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 80.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 80.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 80.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 34.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 25.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 64.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 78.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 78.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 78.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 71.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 69.8%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ y x)))

C

double code(double x, double y) {
	return 1.0 / (y / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y / x)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (y / x);
}

Python

def code(x, y):
	return 1.0 / (y / x)

Julia

function code(x, y)
	return Float64(1.0 / Float64(y / x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (y / x);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 79.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 79.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 79.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.6%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 25.3%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Step-by-step derivation

   1. clear-num 25.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

   2. inv-pow 25.8%

      \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]

8. Applied egg-rr 25.8%

   \[\leadsto \color{blue}{{\left(\frac{y}{x}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 25.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]

10. Simplified 25.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{y}{x}}} \]
11. Add Preprocessing

Alternative 21: 25.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 65.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/l* 79.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. associate-+l+ 79.7%

      \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

3. Simplified 79.7%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.6%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 25.3%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
7. Add Preprocessing

Alternative 22: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 54.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 65.0%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 65.0%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 65.0%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 65.0%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 65.0%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 65.0%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 54.5%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 54.5%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 65.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 65.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 65.0%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 92.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 92.4%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 92.4%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

7. Taylor expanded in y around 0 55.6%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{1 + x}} \]
8. Step-by-step derivation

   1. +-commutative 55.6%

      \[\leadsto \frac{y}{x + y} \cdot \frac{1}{\color{blue}{x + 1}} \]

9. Simplified 55.6%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x + 1}} \]

10. Taylor expanded in x around 0 3.5%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024146 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))