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

Percentage Accurate: 69.5% → 99.8%

Time: 9.9s

Alternatives: 19

Speedup: 0.8×

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: 69.5% 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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/r* N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f64 N/A

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

4. Applied rewrites 99.9%

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

Alternative 2: 70.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{t\_0} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-307}:\\ \;\;\;\;\frac{y}{\frac{y + x}{x} \cdot \left(\left(1 + y\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -5.5e+72)
     (/ (* (/ y t_0) 1.0) (+ y x))
     (if (<= x -1.5e-123)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (if (<= x -1.75e-307)
         (/ y (* (/ (+ y x) x) (* (+ 1.0 y) (+ y x))))
         (* (/ (/ x (+ y x)) t_0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -1.5e-123) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else if (x <= -1.75e-307) {
		tmp = y / (((y + x) / x) * ((1.0 + y) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y + x)
    if (x <= (-5.5d+72)) then
        tmp = ((y / t_0) * 1.0d0) / (y + x)
    else if (x <= (-1.5d-123)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else if (x <= (-1.75d-307)) then
        tmp = y / (((y + x) / x) * ((1.0d0 + y) * (y + x)))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -1.5e-123) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else if (x <= -1.75e-307) {
		tmp = y / (((y + x) / x) * ((1.0 + y) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -5.5e+72:
		tmp = ((y / t_0) * 1.0) / (y + x)
	elif x <= -1.5e-123:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	elif x <= -1.75e-307:
		tmp = y / (((y + x) / x) * ((1.0 + y) * (y + x)))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -5.5e+72)
		tmp = Float64(Float64(Float64(y / t_0) * 1.0) / Float64(y + x));
	elseif (x <= -1.5e-123)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	elseif (x <= -1.75e-307)
		tmp = Float64(y / Float64(Float64(Float64(y + x) / x) * Float64(Float64(1.0 + y) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -5.5e+72)
		tmp = ((y / t_0) * 1.0) / (y + x);
	elseif (x <= -1.5e-123)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	elseif (x <= -1.75e-307)
		tmp = y / (((y + x) / x) * ((1.0 + y) * (y + x)));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+72], N[(N[(N[(y / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-123], 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], If[LessEqual[x, -1.75e-307], N[(y / N[(N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision] * N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.5e72

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 79.7%

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

   if -5.5e72 < x < -1.49999999999999992e-123

   1. Initial program 88.2%

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

   if -1.49999999999999992e-123 < x < -1.7500000000000001e-307

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 99.9%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/l* N/A

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

       4. lift-/.f64 N/A

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

       5. clear-num N/A

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

       6. frac-times N/A

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

       7. *-lft-identity N/A

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

       8. lower-/.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-/.f64 97.6

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

   11. Applied rewrites 97.6%

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

   if -1.7500000000000001e-307 < x

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.8%

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

Alternative 3: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{t\_0} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -8.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(t\_0 \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{\left(1 + y\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -5.5e+72)
     (/ (* (/ y t_0) 1.0) (+ y x))
     (if (<= x -8.8e-6)
       (/ (* x y) (* (+ y x) (* t_0 (+ y x))))
       (/ (* (/ x (+ y x)) y) (* (+ 1.0 y) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -8.8e-6) {
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) * y) / ((1.0 + 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 = 1.0d0 + (y + x)
    if (x <= (-5.5d+72)) then
        tmp = ((y / t_0) * 1.0d0) / (y + x)
    else if (x <= (-8.8d-6)) then
        tmp = (x * y) / ((y + x) * (t_0 * (y + x)))
    else
        tmp = ((x / (y + x)) * y) / ((1.0d0 + y) * (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -8.8e-6) {
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) * y) / ((1.0 + y) * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -5.5e+72:
		tmp = ((y / t_0) * 1.0) / (y + x)
	elif x <= -8.8e-6:
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)))
	else:
		tmp = ((x / (y + x)) * y) / ((1.0 + y) * (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -5.5e+72)
		tmp = Float64(Float64(Float64(y / t_0) * 1.0) / Float64(y + x));
	elseif (x <= -8.8e-6)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(Float64(1.0 + y) * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -5.5e+72)
		tmp = ((y / t_0) * 1.0) / (y + x);
	elseif (x <= -8.8e-6)
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	else
		tmp = ((x / (y + x)) * y) / ((1.0 + y) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+72], N[(N[(N[(y / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.8e-6], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(1.0 + y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e72

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 79.7%

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

   if -5.5e72 < x < -8.8000000000000004e-6

   1. Initial program 85.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 85.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if -8.8000000000000004e-6 < x

   1. Initial program 69.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 83.0

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

   7. Applied rewrites 83.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. *-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. lower-/.f64 N/A

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

   9. Applied rewrites 84.0%

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

Alternative 4: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ t_1 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_0}{t\_1} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0 \cdot y}{t\_1 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))) (t_1 (+ 1.0 (+ y x))))
   (if (<= x -1.05e+154)
     (* (/ t_0 t_1) (/ y x))
     (/ (* t_0 y) (* t_1 (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = 1.0 + (y + x);
	double tmp;
	if (x <= -1.05e+154) {
		tmp = (t_0 / t_1) * (y / x);
	} else {
		tmp = (t_0 * y) / (t_1 * (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) :: t_1
    real(8) :: tmp
    t_0 = x / (y + x)
    t_1 = 1.0d0 + (y + x)
    if (x <= (-1.05d+154)) then
        tmp = (t_0 / t_1) * (y / x)
    else
        tmp = (t_0 * y) / (t_1 * (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double t_1 = 1.0 + (y + x);
	double tmp;
	if (x <= -1.05e+154) {
		tmp = (t_0 / t_1) * (y / x);
	} else {
		tmp = (t_0 * y) / (t_1 * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	t_1 = 1.0 + (y + x)
	tmp = 0
	if x <= -1.05e+154:
		tmp = (t_0 / t_1) * (y / x)
	else:
		tmp = (t_0 * y) / (t_1 * (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	t_1 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -1.05e+154)
		tmp = Float64(Float64(t_0 / t_1) * Float64(y / x));
	else
		tmp = Float64(Float64(t_0 * y) / Float64(t_1 * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	t_1 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -1.05e+154)
		tmp = (t_0 / t_1) * (y / x);
	else
		tmp = (t_0 * y) / (t_1 * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+154], N[(N[(t$95$0 / t$95$1), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * y), $MachinePrecision] / N[(t$95$1 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04999999999999997e154

   1. Initial program 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around inf

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

       1. lower-/.f64 92.8

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

   11. Applied rewrites 92.8%

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

   if -1.04999999999999997e154 < x

   1. Initial program 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.6

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 94.6

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 69.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{t\_0} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -5.5e+72)
     (/ (* (/ y t_0) 1.0) (+ y x))
     (if (<= x -9.5e-133)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (* (/ (/ x (+ y x)) t_0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -9.5e-133) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y + x)
    if (x <= (-5.5d+72)) then
        tmp = ((y / t_0) * 1.0d0) / (y + x)
    else if (x <= (-9.5d-133)) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -5.5e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -9.5e-133) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -5.5e+72:
		tmp = ((y / t_0) * 1.0) / (y + x)
	elif x <= -9.5e-133:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -5.5e+72)
		tmp = Float64(Float64(Float64(y / t_0) * 1.0) / Float64(y + x));
	elseif (x <= -9.5e-133)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -5.5e+72)
		tmp = ((y / t_0) * 1.0) / (y + x);
	elseif (x <= -9.5e-133)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.5e+72], N[(N[(N[(y / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-133], 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], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5e72

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 79.7%

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

   if -5.5e72 < x < -9.4999999999999992e-133

   1. Initial program 86.9%

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

   if -9.4999999999999992e-133 < x

   1. Initial program 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 60.4%

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

Alternative 6: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0} \cdot 1}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot y}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -1.35e+154)
     (/ (* (/ y t_0) 1.0) (+ y x))
     (/ (* (/ x (+ y x)) y) (* t_0 (+ y x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else {
		tmp = ((x / (y + x)) * y) / (t_0 * (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 = 1.0d0 + (y + x)
    if (x <= (-1.35d+154)) then
        tmp = ((y / t_0) * 1.0d0) / (y + x)
    else
        tmp = ((x / (y + x)) * y) / (t_0 * (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -1.35e+154) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else {
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -1.35e+154:
		tmp = ((y / t_0) * 1.0) / (y + x)
	else:
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -1.35e+154)
		tmp = Float64(Float64(Float64(y / t_0) * 1.0) / Float64(y + x));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * y) / Float64(t_0 * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -1.35e+154)
		tmp = ((y / t_0) * 1.0) / (y + x);
	else
		tmp = ((x / (y + x)) * y) / (t_0 * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+154], N[(N[(N[(y / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.35000000000000003e154

   1. Initial program 57.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 86.8%

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

   if -1.35000000000000003e154 < x

   1. Initial program 70.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 94.6

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 94.6

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 94.6

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 94.6

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

   4. Applied rewrites 94.6%

      \[\leadsto \color{blue}{\frac{\frac{x}{y + x} \cdot y}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 91.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\frac{1 + \left(y + x\right)}{y} \cdot \left(y + x\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e+84)
   (/ (/ y x) (+ y x))
   (/ x (* (+ y x) (* (/ (+ 1.0 (+ y x)) y) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+84) {
		tmp = (y / x) / (y + x);
	} else {
		tmp = x / ((y + x) * (((1.0 + (y + x)) / 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 (x <= (-1.7d+84)) then
        tmp = (y / x) / (y + x)
    else
        tmp = x / ((y + x) * (((1.0d0 + (y + x)) / y) * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e+84) {
		tmp = (y / x) / (y + x);
	} else {
		tmp = x / ((y + x) * (((1.0 + (y + x)) / y) * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e+84:
		tmp = (y / x) / (y + x)
	else:
		tmp = x / ((y + x) * (((1.0 + (y + x)) / y) * (y + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e+84)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(Float64(1.0 + Float64(y + x)) / y) * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e+84)
		tmp = (y / x) / (y + x);
	else
		tmp = x / ((y + x) * (((1.0 + (y + x)) / y) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e+84], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e84

   1. Initial program 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 82.2

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

   9. Applied rewrites 82.2%

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

   if -1.6999999999999999e84 < x

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lower-*.f64 91.5

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 91.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 91.5

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 91.5

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

   8. Applied rewrites 91.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\frac{1 + \left(y + x\right)}{y} \cdot \left(y + x\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 9: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;x \leq -4 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{y}{t\_0} \cdot 1}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= x -4e+72)
     (/ (* (/ y t_0) 1.0) (+ y x))
     (if (<= x -2.3e-68)
       (/ (* x y) (* (+ y x) (* (+ 1.0 x) (+ y x))))
       (* (/ (/ x (+ y x)) t_0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -4e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (y + x)
    if (x <= (-4d+72)) then
        tmp = ((y / t_0) * 1.0d0) / (y + x)
    else if (x <= (-2.3d-68)) then
        tmp = (x * y) / ((y + x) * ((1.0d0 + x) * (y + x)))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (x <= -4e+72) {
		tmp = ((y / t_0) * 1.0) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if x <= -4e+72:
		tmp = ((y / t_0) * 1.0) / (y + x)
	elif x <= -2.3e-68:
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (x <= -4e+72)
		tmp = Float64(Float64(Float64(y / t_0) * 1.0) / Float64(y + x));
	elseif (x <= -2.3e-68)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (x <= -4e+72)
		tmp = ((y / t_0) * 1.0) / (y + x);
	elseif (x <= -2.3e-68)
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4e+72], N[(N[(N[(y / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-68], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.99999999999999978e72

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 79.7%

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

   if -3.99999999999999978e72 < x < -2.29999999999999997e-68

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 88.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2.29999999999999997e-68 < x

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.5%

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

Alternative 10: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + \left(y + x\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+71)
   (/ (/ y x) (+ y x))
   (if (<= x -2.3e-68)
     (/ (* x y) (* (+ y x) (* (+ 1.0 x) (+ y x))))
     (* (/ (/ x (+ y x)) (+ 1.0 (+ y x))) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+71) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.8d+71)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-2.3d-68)) then
        tmp = (x * y) / ((y + x) * ((1.0d0 + x) * (y + x)))
    else
        tmp = ((x / (y + x)) / (1.0d0 + (y + x))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+71) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e+71:
		tmp = (y / x) / (y + x)
	elif x <= -2.3e-68:
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)))
	else:
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+71)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -2.3e-68)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(1.0 + Float64(y + x))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e+71)
		tmp = (y / x) / (y + x);
	elseif (x <= -2.3e-68)
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	else
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e+71], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-68], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7999999999999997e71

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 79.5

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

   9. Applied rewrites 79.5%

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

   if -6.7999999999999997e71 < x < -2.29999999999999997e-68

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 88.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2.29999999999999997e-68 < x

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/r/ N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. lower-/.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lift-+.f64 99.9

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 61.5%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + \left(y + x\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.8e+71)
   (/ (/ y x) (+ y x))
   (if (<= x -2.3e-68)
     (/ (* x y) (* (+ y x) (* (+ 1.0 x) (+ y x))))
     (/ (/ x (+ 1.0 y)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+71) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = (x / (1.0 + 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 (x <= (-6.8d+71)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-2.3d-68)) then
        tmp = (x * y) / ((y + x) * ((1.0d0 + x) * (y + x)))
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.8e+71) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.3e-68) {
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.8e+71:
		tmp = (y / x) / (y + x)
	elif x <= -2.3e-68:
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)))
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.8e+71)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -2.3e-68)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(Float64(1.0 + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.8e+71)
		tmp = (y / x) / (y + x);
	elseif (x <= -2.3e-68)
		tmp = (x * y) / ((y + x) * ((1.0 + x) * (y + x)));
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.8e+71], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.3e-68], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(1.0 + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7999999999999997e71

   1. Initial program 58.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.9

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 79.5

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

   9. Applied rewrites 79.5%

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

   if -6.7999999999999997e71 < x < -2.29999999999999997e-68

   1. Initial program 87.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 88.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 88.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 88.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 88.0

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

   4. Applied rewrites 88.0%

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

   5. Taylor expanded in y around 0

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

      1. lower-+.f64 73.1

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

   7. Applied rewrites 73.1%

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

   if -2.29999999999999997e-68 < x

   1. Initial program 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 60.8

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

   7. Applied rewrites 60.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{+71}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(\left(1 + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e+15)
   (/ (/ y x) (+ y x))
   (if (<= x -7.5e-39) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+15) {
		tmp = (y / x) / (y + x);
	} else if (x <= -7.5e-39) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e+15)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -7.5e-39)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e+15], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-39], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-+.f64 N/A

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

      10. lower-/.f64 99.8

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 72.2

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

   9. Applied rewrites 72.2%

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

   if -4.4e15 < x < -7.49999999999999971e-39

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 55.6

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

   5. Applied rewrites 55.6%

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

   if -7.49999999999999971e-39 < x

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]

      2. lower-+.f64 61.7

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y + x} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-39) (/ (/ y (+ 1.0 x)) (+ y x)) (/ (/ x (+ 1.0 y)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-39) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + 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 (x <= (-7.5d-39)) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else
        tmp = (x / (1.0d0 + y)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-39) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else {
		tmp = (x / (1.0 + y)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.5e-39:
		tmp = (y / (1.0 + x)) / (y + x)
	else:
		tmp = (x / (1.0 + y)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-39)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.5e-39)
		tmp = (y / (1.0 + x)) / (y + x);
	else
		tmp = (x / (1.0 + y)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-39], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999971e-39

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]

      2. lower-+.f64 70.0

         \[\leadsto \frac{\frac{y}{\color{blue}{1 + x}}}{y + x} \]

   7. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]

   if -7.49999999999999971e-39 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]

      2. lower-+.f64 61.7

         \[\leadsto \frac{\frac{x}{\color{blue}{1 + y}}}{y + x} \]

   7. Applied rewrites 61.7%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 63.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.4e+15)
   (/ (/ y x) (+ y x))
   (if (<= x -7.5e-39) (/ y (fma x x x)) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.4e+15) {
		tmp = (y / x) / (y + x);
	} else if (x <= -7.5e-39) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.4e+15)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -7.5e-39)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.4e+15], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7.5e-39], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4e15

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}}{y + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x} \cdot \frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{y}{1 + \left(y + x\right)}}}{y + x} \]

      4. clear-num N/A

         \[\leadsto \frac{\frac{x}{y + x} \cdot \color{blue}{\frac{1}{\frac{1 + \left(y + x\right)}{y}}}}{y + x} \]

      5. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + x}}{\frac{1 + \left(y + x\right)}{y}}}}{y + x} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\frac{x}{y + x}}{\frac{1 + \left(y + x\right)}{y}}}}{y + x} \]

      7. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{x}{\color{blue}{y + x}}}{\frac{1 + \left(y + x\right)}{y}}}{y + x} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{\frac{x}{\color{blue}{x + y}}}{\frac{1 + \left(y + x\right)}{y}}}{y + x} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\frac{x}{\color{blue}{x + y}}}{\frac{1 + \left(y + x\right)}{y}}}{y + x} \]

      10. lower-/.f64 99.8

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\color{blue}{\frac{1 + \left(y + x\right)}{y}}}}{y + x} \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{1 + \left(y + x\right)}}{y}}}{y + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right) + 1}}{y}}}{y + x} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(y + x\right)} + 1}{y}}}{y + x} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y}}}{y + x} \]

      15. lift-+.f64 N/A

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right)} + 1}{y}}}{y + x} \]

      16. lift-+.f64 99.8

          \[\leadsto \frac{\frac{\frac{x}{x + y}}{\frac{\color{blue}{\left(x + y\right) + 1}}{y}}}{y + x} \]

   6. Applied rewrites 99.8%

      \[\leadsto \frac{\color{blue}{\frac{\frac{x}{x + y}}{\frac{\left(x + y\right) + 1}{y}}}}{y + x} \]

   7. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + x} \]
   8. Step-by-step derivation

      1. lower-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + x} \]

   9. Applied rewrites 72.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{y + x} \]

   if -4.4e15 < x < -7.49999999999999971e-39

   1. Initial program 90.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 55.6

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 55.6%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -7.49999999999999971e-39 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 61.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9e+50)
   (/ (/ y x) x)
   (if (<= x -7.5e-39) (/ y (fma x x x)) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9e+50) {
		tmp = (y / x) / x;
	} else if (x <= -7.5e-39) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9e+50)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -7.5e-39)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9e+50], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -7.5e-39], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.00000000000000027e50

   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. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 77.0

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -9.00000000000000027e50 < x < -7.49999999999999971e-39

   1. Initial program 81.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 47.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -7.49999999999999971e-39 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 61.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-39) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-39) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.5e-39)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.5e-39], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999971e-39

   1. Initial program 68.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 65.4

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 65.4%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -7.49999999999999971e-39 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 61.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 61.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2950:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2950.0) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2950.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2950.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2950.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2950

   1. Initial program 65.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 71.6

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 66.9%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if -2950 < x

   1. Initial program 69.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 60.8

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 60.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e+22) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+22) {
		tmp = y / (x * x);
	} 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 <= 6.2d+22) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.2e+22:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e+22)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e+22)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e+22], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000004e22

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      2. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      4. lower-/.f64 43.1

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
   6. Step-by-step derivation

   7. Applied rewrites 42.5%

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   if 6.2000000000000004e22 < y

   1. Initial program 62.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{1 + \left(y + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 75.0

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   7. Applied rewrites 75.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{y}{x \cdot x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ y (* x x)))

C

double code(double x, double y) {
	return y / (x * x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / (x * x)
end function

Java

public static double code(double x, double y) {
	return y / (x * x);
}

Python

def code(x, y):
	return y / (x * x)

Julia

function code(x, y)
	return Float64(y / Float64(x * x))
end

MATLAB

function tmp = code(x, y)
	tmp = y / (x * x);
end

Wolfram

code[x_, y_] := N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   2. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   4. lower-/.f64 37.7

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

5. Applied rewrites 37.7%

   \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]
6. Step-by-step derivation

7. Applied rewrites 35.8%

   \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× 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 2024318 
(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))))