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

Percentage Accurate: 68.9% → 99.8%

Time: 10.5s

Alternatives: 21

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% 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

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (* (/ y (+ (+ x y) 1.0)) (/ x (+ x y))) (+ x y)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((y / ((x + y) + 1.0d0)) * (x / (x + y))) / (x + y)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((y / ((x + y) + 1.0)) * (x / (x + y))) / (x + y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(y / Float64(Float64(x + y) + 1.0)) * Float64(x / Float64(x + y))) / Float64(x + y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((y / ((x + y) + 1.0)) * (x / (x + y))) / (x + y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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. *-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. Final simplification 99.9%

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

Alternative 2: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ t_1 := \frac{y}{t\_0}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{t\_1}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{t\_0 \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{\frac{y}{x} \cdot \left(x + y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)) (t_1 (/ y t_0)))
   (if (<= x -1.9e+154)
     (/ t_1 (fma 2.0 y x))
     (if (<= x 2.7e-139)
       (* (/ x (* t_0 (+ x y))) (/ y (+ x y)))
       (/ t_1 (* (/ y x) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double t_1 = y / t_0;
	double tmp;
	if (x <= -1.9e+154) {
		tmp = t_1 / fma(2.0, y, x);
	} else if (x <= 2.7e-139) {
		tmp = (x / (t_0 * (x + y))) * (y / (x + y));
	} else {
		tmp = t_1 / ((y / x) * (x + y));
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	t_1 = Float64(y / t_0)
	tmp = 0.0
	if (x <= -1.9e+154)
		tmp = Float64(t_1 / fma(2.0, y, x));
	elseif (x <= 2.7e-139)
		tmp = Float64(Float64(x / Float64(t_0 * Float64(x + y))) * Float64(y / Float64(x + y)));
	else
		tmp = Float64(t_1 / Float64(Float64(y / x) * Float64(x + y)));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(y / t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.9e+154], N[(t$95$1 / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.7e-139], N[(N[(x / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 / N[(N[(y / x), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.8999999999999999e154

   1. Initial program 54.4%

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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 81.6

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

   9. Applied rewrites 81.6%

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

   if -1.8999999999999999e154 < x < 2.6999999999999998e-139

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.4

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.4

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 96.4

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if 2.6999999999999998e-139 < x

   1. Initial program 65.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in y around inf

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

      1. lower-/.f64 39.5

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

   9. Applied rewrites 39.5%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-139}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\frac{y}{x} \cdot \left(x + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)} \cdot t\_0\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot t\_0}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ x y) 1.0)))
   (if (<= y -5.8e-81)
     (/ (/ y t_1) (fma 2.0 y x))
     (if (<= y 2.8e-23)
       (* (/ y (* (+ x 1.0) (+ x y))) t_0)
       (if (<= y 7.1e+75)
         (/ (* x y) (* (* (+ x y) (+ x y)) t_1))
         (/ (* 1.0 t_0) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-81) {
		tmp = (y / t_1) / fma(2.0, y, x);
	} else if (y <= 2.8e-23) {
		tmp = (y / ((x + 1.0) * (x + y))) * t_0;
	} else if (y <= 7.1e+75) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_1);
	} else {
		tmp = (1.0 * t_0) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -5.8e-81)
		tmp = Float64(Float64(y / t_1) / fma(2.0, y, x));
	elseif (y <= 2.8e-23)
		tmp = Float64(Float64(y / Float64(Float64(x + 1.0) * Float64(x + y))) * t_0);
	elseif (y <= 7.1e+75)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * t_1));
	else
		tmp = Float64(Float64(1.0 * t_0) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-81], N[(N[(y / t$95$1), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.8e-23], N[(N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y, 7.1e+75], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.79999999999999978e-81

   1. Initial program 55.1%

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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.0

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

   9. Applied rewrites 39.0%

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

   if -5.79999999999999978e-81 < y < 2.7999999999999997e-23

   1. Initial program 72.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 100.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 100.0

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

   7. Applied rewrites 100.0%

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

   if 2.7999999999999997e-23 < y < 7.09999999999999982e75

   1. Initial program 95.2%

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

   if 7.09999999999999982e75 < y

   1. Initial program 54.8%

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

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

      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. *-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 inf

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

   7. Applied rewrites 80.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{y}{\left(x + 1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{elif}\;y \leq 7.1 \cdot 10^{+75}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+70}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= x -8.5e+139)
     (/ (/ y t_0) (fma 2.0 y x))
     (if (<= x -1.06e+70)
       (* 1.0 (/ y (* t_0 (+ x y))))
       (if (<= x -2.7e-143)
         (/ (* x y) (* (* (+ x y) (+ x y)) t_0))
         (/ (/ x (+ 1.0 y)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -8.5e+139) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else if (x <= -1.06e+70) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else if (x <= -2.7e-143) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (x <= -8.5e+139)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	elseif (x <= -1.06e+70)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(x + y))));
	elseif (x <= -2.7e-143)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * t_0));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -8.5e+139], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.06e+70], N[(1.0 * N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.7e-143], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.5e139

   1. Initial program 49.7%

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

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

      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. *-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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if -8.5e139 < x < -1.06e70

   1. Initial program 50.8%

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

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 76.6

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 76.6%

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

   if -1.06e70 < x < -2.70000000000000009e-143

   1. Initial program 90.4%

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

   if -2.70000000000000009e-143 < x

   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

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

      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. *-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 59.0

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

   7. Applied rewrites 59.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -1.06 \cdot 10^{+70}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -2.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{t\_0 \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y -5.8e-81)
     (/ (/ y t_0) (fma 2.0 y x))
     (if (<= y 1.6e+154)
       (* (/ x (* t_0 (+ x y))) (/ y (+ x y)))
       (/ (* 1.0 (/ x (+ x y))) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-81) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else if (y <= 1.6e+154) {
		tmp = (x / (t_0 * (x + y))) * (y / (x + y));
	} else {
		tmp = (1.0 * (x / (x + y))) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -5.8e-81)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	elseif (y <= 1.6e+154)
		tmp = Float64(Float64(x / Float64(t_0 * Float64(x + y))) * Float64(y / Float64(x + y)));
	else
		tmp = Float64(Float64(1.0 * Float64(x / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-81], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+154], N[(N[(x / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999978e-81

   1. Initial program 55.1%

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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.0

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

   9. Applied rewrites 39.0%

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

   if -5.79999999999999978e-81 < y < 1.6e154

   1. Initial program 75.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 97.9

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 97.9

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 97.9

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

   4. Applied rewrites 97.9%

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

   if 1.6e154 < y

   1. Initial program 48.5%

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

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

      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. *-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 inf

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

   7. Applied rewrites 85.7%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ t_1 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{t\_1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{t\_1 \cdot \left(x + y\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot t\_0}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))) (t_1 (+ (+ x y) 1.0)))
   (if (<= y -5.8e-81)
     (/ (/ y t_1) (fma 2.0 y x))
     (if (<= y 1.6e+154)
       (* (/ y (* t_1 (+ x y))) t_0)
       (/ (* 1.0 t_0) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double t_1 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-81) {
		tmp = (y / t_1) / fma(2.0, y, x);
	} else if (y <= 1.6e+154) {
		tmp = (y / (t_1 * (x + y))) * t_0;
	} else {
		tmp = (1.0 * t_0) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	t_1 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (y <= -5.8e-81)
		tmp = Float64(Float64(y / t_1) / fma(2.0, y, x));
	elseif (y <= 1.6e+154)
		tmp = Float64(Float64(y / Float64(t_1 * Float64(x + y))) * t_0);
	else
		tmp = Float64(Float64(1.0 * t_0) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-81], N[(N[(y / t$95$1), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+154], N[(N[(y / N[(t$95$1 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(1.0 * t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.79999999999999978e-81

   1. Initial program 55.1%

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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 97.9%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 39.0

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

   9. Applied rewrites 39.0%

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

   if -5.79999999999999978e-81 < y < 1.6e154

   1. Initial program 75.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 97.8

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 97.8%

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

   if 1.6e154 < y

   1. Initial program 48.5%

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

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

      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. *-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 inf

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

   7. Applied rewrites 85.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2 + \frac{y}{x}, y, x\right)} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ y (+ (+ x y) 1.0)) (fma (+ 2.0 (/ y x)) y x)))

C

assert(x < y);
double code(double x, double y) {
	return (y / ((x + y) + 1.0)) / fma((2.0 + (y / x)), y, x);
}

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y / Float64(Float64(x + y) + 1.0)) / fma(Float64(2.0 + Float64(y / x)), y, x))
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(2.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 65.6%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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. *-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

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. frac-times N/A

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

   8. lift-/.f64 N/A

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

   9. clear-num N/A

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

   10. div-inv N/A

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

   11. clear-num N/A

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

   12. lift-/.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-+.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lower-+.f64 N/A

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

   20. lower-*.f64 N/A

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 99.1

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

9. Applied rewrites 99.1%

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

10. Final simplification 99.1%

    \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2 + \frac{y}{x}, y, x\right)} \]
11. Add Preprocessing

Alternative 8: 79.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+136)
   (/ (/ y x) (+ x y))
   (if (<= x -5.2e-88)
     (* (/ y (* (fma x x x) x)) x)
     (if (<= x 1.32e-42) (/ x (fma y y y)) (/ (/ x y) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+136) {
		tmp = (y / x) / (x + y);
	} else if (x <= -5.2e-88) {
		tmp = (y / (fma(x, x, x) * x)) * x;
	} else if (x <= 1.32e-42) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+136)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -5.2e-88)
		tmp = Float64(Float64(y / Float64(fma(x, x, x) * x)) * x);
	elseif (x <= 1.32e-42)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+136], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(N[(y / N[(N[(x * x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[x, 1.32e-42], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.99999999999999958e136

   1. Initial program 49.7%

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

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

      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. *-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

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

      1. lower-/.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -5.99999999999999958e136 < x < -5.20000000000000027e-88

   1. Initial program 79.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 57.5

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

   5. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 64.7

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

   7. Applied rewrites 64.7%

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

   if -5.20000000000000027e-88 < x < 1.32000000000000006e-42

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.32000000000000006e-42 < x

   1. Initial program 59.3%

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

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

      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. *-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 inf

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

      1. lower-/.f64 29.8

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

   7. Applied rewrites 29.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e+14)
   (/ (/ y x) (+ x y))
   (if (<= x -5.2e-88)
     (/ y (fma x x x))
     (if (<= x 1.32e-42) (/ x (fma y y y)) (/ (/ x y) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.05e+14) {
		tmp = (y / x) / (x + y);
	} else if (x <= -5.2e-88) {
		tmp = y / fma(x, x, x);
	} else if (x <= 1.32e-42) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.05e+14)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -5.2e-88)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 1.32e-42)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.05e+14], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-42], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.05e14

   1. Initial program 54.2%

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

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

      1. lower-/.f64 74.7

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

   7. Applied rewrites 74.7%

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

   if -1.05e14 < x < -5.20000000000000027e-88

   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. 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 57.8

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

   5. Applied rewrites 57.8%

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

   if -5.20000000000000027e-88 < x < 1.32000000000000006e-42

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.32000000000000006e-42 < x

   1. Initial program 59.3%

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

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

      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. *-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 inf

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

      1. lower-/.f64 29.8

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

   7. Applied rewrites 29.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+67)
   (/ (/ y x) x)
   (if (<= x -5.2e-88)
     (/ y (fma x x x))
     (if (<= x 1.32e-42) (/ x (fma y y y)) (/ (/ x y) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+67) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-88) {
		tmp = y / fma(x, x, x);
	} else if (x <= 1.32e-42) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+67)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-88)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 1.32e-42)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+67], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-42], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.99999999999999976e67

   1. Initial program 50.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y}{x}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \frac{\frac{y}{x}}{x} \]

   if -4.99999999999999976e67 < x < -5.20000000000000027e-88

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -5.20000000000000027e-88 < x < 1.32000000000000006e-42

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.32000000000000006e-42 < x

   1. Initial program 59.3%

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

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

      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. *-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 inf

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

      1. lower-/.f64 29.8

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

   7. Applied rewrites 29.8%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y + x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 87.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;1 \cdot \frac{y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= x -8.5e+139)
     (/ (/ y t_0) (fma 2.0 y x))
     (if (<= x -3.8e-116)
       (* 1.0 (/ y (* t_0 (+ x y))))
       (/ (/ x (+ 1.0 y)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (x <= -8.5e+139) {
		tmp = (y / t_0) / fma(2.0, y, x);
	} else if (x <= -3.8e-116) {
		tmp = 1.0 * (y / (t_0 * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(Float64(x + y) + 1.0)
	tmp = 0.0
	if (x <= -8.5e+139)
		tmp = Float64(Float64(y / t_0) / fma(2.0, y, x));
	elseif (x <= -3.8e-116)
		tmp = Float64(1.0 * Float64(y / Float64(t_0 * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -8.5e+139], N[(N[(y / t$95$0), $MachinePrecision] / N[(2.0 * y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-116], N[(1.0 * N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5e139

   1. Initial program 49.7%

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

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

      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. *-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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. frac-times N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. div-inv N/A

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

      11. clear-num N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lower-+.f64 N/A

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

      20. lower-*.f64 N/A

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

   6. Applied rewrites 99.2%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if -8.5e139 < x < -3.8000000000000001e-116

   1. Initial program 79.8%

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

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 94.2

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 73.0%

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

   if -3.8000000000000001e-116 < 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

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

      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. *-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 59.6

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

   7. Applied rewrites 59.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{y}{\left(x + y\right) + 1}}{\mathsf{fma}\left(2, y, x\right)}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -7.8e+147)
   (/ (/ y x) (+ x y))
   (if (<= x -3.8e-116)
     (* 1.0 (/ y (* (+ (+ x y) 1.0) (+ x y))))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -7.8e+147) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.8e-116) {
		tmp = 1.0 * (y / (((x + y) + 1.0) * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.8d+147)) then
        tmp = (y / x) / (x + y)
    else if (x <= (-3.8d-116)) then
        tmp = 1.0d0 * (y / (((x + y) + 1.0d0) * (x + y)))
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -7.8e+147) {
		tmp = (y / x) / (x + y);
	} else if (x <= -3.8e-116) {
		tmp = 1.0 * (y / (((x + y) + 1.0) * (x + y)));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -7.8e+147:
		tmp = (y / x) / (x + y)
	elif x <= -3.8e-116:
		tmp = 1.0 * (y / (((x + y) + 1.0) * (x + y)))
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -7.8e+147)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -3.8e-116)
		tmp = Float64(1.0 * Float64(y / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y))));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.8e+147)
		tmp = (y / x) / (x + y);
	elseif (x <= -3.8e-116)
		tmp = 1.0 * (y / (((x + y) + 1.0) * (x + y)));
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -7.8e+147], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.8e-116], N[(1.0 * N[(y / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.80000000000000033e147

   1. Initial program 51.9%

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

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

      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. *-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

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

      1. lower-/.f64 79.5

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

   7. Applied rewrites 79.5%

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

   if -7.80000000000000033e147 < x < -3.8000000000000001e-116

   1. Initial program 75.5%

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

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 91.9

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 91.9%

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

   5. Taylor expanded in y around 0

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

   7. Applied rewrites 70.8%

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

   if -3.8000000000000001e-116 < 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

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

      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. *-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 59.6

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

   7. Applied rewrites 59.6%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -3.8 \cdot 10^{-116}:\\ \;\;\;\;1 \cdot \frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+67)
   (/ (/ y x) x)
   (if (<= x -5.2e-88)
     (/ y (fma x x x))
     (if (<= x 1.32e-42) (/ x (fma y y y)) (/ (/ x y) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+67) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-88) {
		tmp = y / fma(x, x, x);
	} else if (x <= 1.32e-42) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+67)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-88)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 1.32e-42)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+67], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.32e-42], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.99999999999999976e67

   1. Initial program 50.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y}{x}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \frac{\frac{y}{x}}{x} \]

   if -4.99999999999999976e67 < x < -5.20000000000000027e-88

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -5.20000000000000027e-88 < x < 1.32000000000000006e-42

   1. Initial program 73.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 85.9

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

   5. Applied rewrites 85.9%

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

   if 1.32000000000000006e-42 < x

   1. Initial program 59.3%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 26.3

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

   5. Applied rewrites 26.3%

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

   7. Applied rewrites 29.0%

      \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 79.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -6e+136)
   (/ (/ y x) (+ x y))
   (if (<= x -5.2e-88)
     (* (/ y (* (fma x x x) x)) x)
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -6e+136) {
		tmp = (y / x) / (x + y);
	} else if (x <= -5.2e-88) {
		tmp = (y / (fma(x, x, x) * x)) * x;
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -6e+136)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (x <= -5.2e-88)
		tmp = Float64(Float64(y / Float64(fma(x, x, x) * x)) * x);
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -6e+136], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-88], N[(N[(y / N[(N[(x * x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999958e136

   1. Initial program 49.7%

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

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

      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. *-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

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

      1. lower-/.f64 77.3

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

   7. Applied rewrites 77.3%

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

   if -5.99999999999999958e136 < x < -5.20000000000000027e-88

   1. Initial program 79.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. *-rgt-identity N/A

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

      9. lower-fma.f64 57.5

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

   5. Applied rewrites 57.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 64.7

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

   7. Applied rewrites 64.7%

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

   if -5.20000000000000027e-88 < x

   1. Initial program 67.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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. *-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 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right) \cdot x} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -42000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-180}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -42000000.0)
     (/ y (* x x))
     (if (<= x -3.4e-180) t_0 (if (<= x 4.8e-137) (/ x y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -42000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.4e-180) {
		tmp = t_0;
	} else if (x <= 4.8e-137) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (y * y)
    if (x <= (-42000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-3.4d-180)) then
        tmp = t_0
    else if (x <= 4.8d-137) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -42000000.0) {
		tmp = y / (x * x);
	} else if (x <= -3.4e-180) {
		tmp = t_0;
	} else if (x <= 4.8e-137) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -42000000.0:
		tmp = y / (x * x)
	elif x <= -3.4e-180:
		tmp = t_0
	elif x <= 4.8e-137:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -42000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.4e-180)
		tmp = t_0;
	elseif (x <= 4.8e-137)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -42000000.0)
		tmp = y / (x * x);
	elseif (x <= -3.4e-180)
		tmp = t_0;
	elseif (x <= 4.8e-137)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -42000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.4e-180], t$95$0, If[LessEqual[x, 4.8e-137], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2e7

   1. Initial program 55.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -4.2e7 < x < -3.39999999999999981e-180 or 4.8000000000000001e-137 < x

   1. Initial program 70.8%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 38.7

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

   5. Applied rewrites 38.7%

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

   if -3.39999999999999981e-180 < x < 4.8000000000000001e-137

   1. Initial program 63.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 88.4

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

   5. Applied rewrites 88.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 77.4%

      \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 82.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-94) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x (+ 1.0 y)) (+ x y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-94) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-94)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / (1.0d0 + y)) / (x + y)
    end if
    code = tmp
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.7e-94:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / (1.0 + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-94)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / Float64(1.0 + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-94)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / (1.0 + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.7e-94], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(1.0 + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-94

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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. +-commutative N/A

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

      3. lower-+.f64 70.2

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

   7. Applied rewrites 70.2%

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

   if -1.6999999999999999e-94 < x

   1. Initial program 67.0%

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

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

      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. *-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 60.0

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

   7. Applied rewrites 60.0%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5e+67)
   (/ (/ y x) x)
   (if (<= x -5.2e-88) (/ y (fma x x x)) (/ x (fma y y y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5e+67) {
		tmp = (y / x) / x;
	} else if (x <= -5.2e-88) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5e+67)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -5.2e-88)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5e+67], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -5.2e-88], 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 < -4.99999999999999976e67

   1. Initial program 50.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. lower-fma.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\frac{y}{x}}{x} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.4%

      \[\leadsto \frac{\frac{y}{x}}{x} \]

   if -4.99999999999999976e67 < x < -5.20000000000000027e-88

   1. Initial program 90.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 62.9

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

   5. Applied rewrites 62.9%

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

   if -5.20000000000000027e-88 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in 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 58.7

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

   5. Applied rewrites 58.7%

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

Alternative 18: 47.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -3.8e-12) t_0 (if (<= y 1.0) (/ x y) t_0))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -3.8e-12) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 = x / (y * y)
    if (y <= (-3.8d-12)) then
        tmp = t_0
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -3.8e-12) {
		tmp = t_0;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -3.8e-12:
		tmp = t_0
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -3.8e-12)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -3.8e-12)
		tmp = t_0;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.8e-12], t$95$0, If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.79999999999999996e-12 or 1 < y

   1. Initial program 56.5%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. 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 60.8

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

   5. Applied rewrites 60.8%

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

   if -3.79999999999999996e-12 < y < 1

   1. Initial program 76.3%

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

   3. 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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 31.1%

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

Alternative 19: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-88}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-88) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-88) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-88)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.2e-88], 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 < -5.20000000000000027e-88

   1. Initial program 61.6%

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

   3. Taylor expanded in 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 68.6

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

   5. Applied rewrites 68.6%

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

   if -5.20000000000000027e-88 < x

   1. Initial program 67.2%

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

   3. Taylor expanded in 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 58.7

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

   5. Applied rewrites 58.7%

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

Alternative 20: 76.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -42000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -42000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -42000000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -42000000.0)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -42000000.0], 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 < -4.2e7

   1. Initial program 55.8%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

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

      3. lower-*.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if -4.2e7 < x

   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 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 58.2

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

   5. Applied rewrites 58.2%

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

Alternative 21: 26.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

assert(x < y);
double code(double x, double y) {
	return x / y;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

assert x < y;
public static double code(double x, double y) {
	return x / y;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in x around 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 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 26.4%

   \[\leadsto \frac{x}{\color{blue}{y}} \]
9. 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 2024277 
(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))))