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

Percentage Accurate: 69.4% → 99.8%

Time: 10.2s

Alternatives: 22

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: 69.4% 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

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((x / (x + y)) * (y / ((x + y) + 1.0))) / (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 / (x + y)) * (y / ((x + y) + 1.0d0))) / (x + y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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. Final simplification 99.8%

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

Alternative 2: 96.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - \frac{\mathsf{fma}\left(2, x, 1\right)}{y} \cdot x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-62)
   (/ (/ (fma (fma 3.0 y 1.0) (/ (- y) x) y) x) x)
   (if (<= y 1.15e+106)
     (/ (* (/ x (+ x y)) y) (* (+ (+ x y) 1.0) (+ x y)))
     (/ (/ (- x (* (/ (fma 2.0 x 1.0) y) x)) y) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (fma(fma(3.0, y, 1.0), (-y / x), y) / x) / x;
	} else if (y <= 1.15e+106) {
		tmp = ((x / (x + y)) * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = ((x - ((fma(2.0, x, 1.0) / y) * x)) / y) / (x + y);
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-62)
		tmp = Float64(Float64(fma(fma(3.0, y, 1.0), Float64(Float64(-y) / x), y) / x) / x);
	elseif (y <= 1.15e+106)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x - Float64(Float64(fma(2.0, x, 1.0) / y) * 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[y, -2.1e-62], N[(N[(N[(N[(3.0 * y + 1.0), $MachinePrecision] * N[((-y) / x), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.15e+106], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(N[(N[(2.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

   3. 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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 40.4%

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

   if -2.0999999999999999e-62 < y < 1.1500000000000001e106

   1. Initial program 76.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.1500000000000001e106 < y

   1. Initial program 47.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 88.0

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

   10. Applied rewrites 88.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - \frac{\mathsf{fma}\left(2, x, 1\right)}{y} \cdot x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{t\_0 \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(2, x, 1\right)}{y}}{y} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -2.1e-62)
     (/ (/ (fma (fma 3.0 y 1.0) (/ (- y) x) y) x) x)
     (if (<= y 6.3e+131)
       (/ (* t_0 y) (* (+ (+ x y) 1.0) (+ x y)))
       (* (/ (- 1.0 (/ (fma 2.0 x 1.0) y)) y) t_0)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (fma(fma(3.0, y, 1.0), (-y / x), y) / x) / x;
	} else if (y <= 6.3e+131) {
		tmp = (t_0 * y) / (((x + y) + 1.0) * (x + y));
	} else {
		tmp = ((1.0 - (fma(2.0, x, 1.0) / y)) / y) * t_0;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -2.1e-62)
		tmp = Float64(Float64(fma(fma(3.0, y, 1.0), Float64(Float64(-y) / x), y) / x) / x);
	elseif (y <= 6.3e+131)
		tmp = Float64(Float64(t_0 * y) / Float64(Float64(Float64(x + y) + 1.0) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(fma(2.0, x, 1.0) / y)) / y) * t_0);
	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]}, If[LessEqual[y, -2.1e-62], N[(N[(N[(N[(3.0 * y + 1.0), $MachinePrecision] * N[((-y) / x), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 6.3e+131], N[(N[(t$95$0 * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[(2.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

   3. 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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 40.4%

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

   if -2.0999999999999999e-62 < y < 6.29999999999999996e131

   1. Initial program 75.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 6.29999999999999996e131 < y

   1. Initial program 43.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 75.6

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

      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 75.6%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{1 + 2 \cdot x}{y}}{y}} \cdot \frac{x}{y + x} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 86.4

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

   7. Applied rewrites 86.4%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 6.3 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \frac{\mathsf{fma}\left(2, x, 1\right)}{y}}{y} \cdot \frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{y}{t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y -2.1e-62)
     (/ (/ (fma (fma 3.0 y 1.0) (/ (- y) x) y) x) x)
     (if (<= y 1.15e+106)
       (/ (* (/ x (+ x y)) y) (* t_0 (+ x y)))
       (/ (* (/ x y) (/ y t_0)) (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (fma(fma(3.0, y, 1.0), (-y / x), y) / x) / x;
	} else if (y <= 1.15e+106) {
		tmp = ((x / (x + y)) * y) / (t_0 * (x + y));
	} else {
		tmp = ((x / y) * (y / t_0)) / (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 <= -2.1e-62)
		tmp = Float64(Float64(fma(fma(3.0, y, 1.0), Float64(Float64(-y) / x), y) / x) / x);
	elseif (y <= 1.15e+106)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) * y) / Float64(t_0 * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x / y) * Float64(y / 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[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -2.1e-62], N[(N[(N[(N[(3.0 * y + 1.0), $MachinePrecision] * N[((-y) / x), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.15e+106], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

   3. 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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 40.4%

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

   if -2.0999999999999999e-62 < y < 1.1500000000000001e106

   1. Initial program 76.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.1500000000000001e106 < y

   1. Initial program 47.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-62)
   (/ (/ (fma (fma 3.0 y 1.0) (/ (- y) x) y) x) x)
   (if (<= y 1.15e+106)
     (/ (* (/ x (+ x y)) 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 (y <= -2.1e-62) {
		tmp = (fma(fma(3.0, y, 1.0), (-y / x), y) / x) / x;
	} else if (y <= 1.15e+106) {
		tmp = ((x / (x + y)) * 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 (y <= -2.1e-62)
		tmp = Float64(Float64(fma(fma(3.0, y, 1.0), Float64(Float64(-y) / x), y) / x) / x);
	elseif (y <= 1.15e+106)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) * 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

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, -2.1e-62], N[(N[(N[(N[(3.0 * y + 1.0), $MachinePrecision] * N[((-y) / x), $MachinePrecision] + y), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 1.15e+106], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x + y), $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 y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

   3. 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 33.4

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

   5. Applied rewrites 33.4%

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

   7. Applied rewrites 40.4%

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

   if -2.0999999999999999e-62 < y < 1.1500000000000001e106

   1. Initial program 76.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.1500000000000001e106 < y

   1. Initial program 47.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(3, y, 1\right), \frac{-y}{x}, y\right)}{x}}{x}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{t\_0 \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

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 (<= y -5.8e-63)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 1.15e+106)
       (/ (* (/ x (+ x y)) 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 (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.15e+106) {
		tmp = ((x / (x + y)) * y) / (t_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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= (-5.8d-63)) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 1.15d+106) then
        tmp = ((x / (x + y)) * y) / (t_0 * (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 t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.15e+106) {
		tmp = ((x / (x + y)) * y) / (t_0 * (x + y));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= -5.8e-63:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 1.15e+106:
		tmp = ((x / (x + y)) * 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 (y <= -5.8e-63)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 1.15e+106)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) * y) / Float64(t_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)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -5.8e-63)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 1.15e+106)
		tmp = ((x / (x + y)) * y) / (t_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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-63], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+106], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(t$95$0 * N[(x + y), $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 y < -5.7999999999999995e-63

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 42.9%

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

   if -5.7999999999999995e-63 < y < 1.1500000000000001e106

   1. Initial program 75.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.2

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.2

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.2

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.1500000000000001e106 < y

   1. Initial program 47.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{t\_0 \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y -5.8e-63)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 5.3e+132)
       (* (/ x (* t_0 (+ x y))) (/ y (+ 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 (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 5.3e+132) {
		tmp = (x / (t_0 * (x + y))) * (y / (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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= (-5.8d-63)) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 5.3d+132) then
        tmp = (x / (t_0 * (x + y))) * (y / (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 t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 5.3e+132) {
		tmp = (x / (t_0 * (x + y))) * (y / (x + y));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= -5.8e-63:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 5.3e+132:
		tmp = (x / (t_0 * (x + y))) * (y / (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 (y <= -5.8e-63)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 5.3e+132)
		tmp = Float64(Float64(x / Float64(t_0 * Float64(x + y))) * Float64(y / 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)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -5.8e-63)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 5.3e+132)
		tmp = (x / (t_0 * (x + y))) * (y / (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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-63], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.3e+132], N[(N[(x / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $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 y < -5.7999999999999995e-63

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 42.9%

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

   if -5.7999999999999995e-63 < y < 5.3e132

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. *-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 99.2

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

      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 99.2

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

      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 99.2

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

      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 99.2

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

   4. Applied rewrites 99.2%

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

   if 5.3e132 < y

   1. Initial program 44.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 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 85.8

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

   7. Applied rewrites 85.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 5.3 \cdot 10^{+132}:\\ \;\;\;\;\frac{x}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ x y) 1.0)))
   (if (<= y -5.8e-63)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 1.15e+106)
       (* (/ y (* t_0 (+ x y))) (/ x (+ 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 (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.15e+106) {
		tmp = (y / (t_0 * (x + y))) * (x / (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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= (-5.8d-63)) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 1.15d+106) then
        tmp = (y / (t_0 * (x + y))) * (x / (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 t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= -5.8e-63) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.15e+106) {
		tmp = (y / (t_0 * (x + y))) * (x / (x + y));
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= -5.8e-63:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 1.15e+106:
		tmp = (y / (t_0 * (x + y))) * (x / (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 (y <= -5.8e-63)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 1.15e+106)
		tmp = Float64(Float64(y / Float64(t_0 * Float64(x + y))) * Float64(x / 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)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= -5.8e-63)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 1.15e+106)
		tmp = (y / (t_0 * (x + y))) * (x / (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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, -5.8e-63], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+106], N[(N[(y / N[(t$95$0 * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $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 y < -5.7999999999999995e-63

   1. Initial program 67.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 42.9%

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

   if -5.7999999999999995e-63 < y < 1.1500000000000001e106

   1. Initial program 75.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.2

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

      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 99.2%

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

   if 1.1500000000000001e106 < y

   1. Initial program 47.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 87.9

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

   7. Applied rewrites 87.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+106}:\\ \;\;\;\;\frac{y}{\left(\left(x + y\right) + 1\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 93.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(1 + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e-11)
   (/ (* 1.0 (/ y (+ (+ x y) 1.0))) (+ x y))
   (if (<= x 3e-87)
     (/ (* (/ x (+ x y)) y) (* (+ 1.0 y) (+ x y)))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-11) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (x <= 3e-87) {
		tmp = ((x / (x + y)) * y) / ((1.0 + y) * (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 <= (-1.65d-11)) then
        tmp = (1.0d0 * (y / ((x + y) + 1.0d0))) / (x + y)
    else if (x <= 3d-87) then
        tmp = ((x / (x + y)) * y) / ((1.0d0 + y) * (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 <= -1.65e-11) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (x <= 3e-87) {
		tmp = ((x / (x + y)) * y) / ((1.0 + y) * (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 <= -1.65e-11:
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y)
	elif x <= 3e-87:
		tmp = ((x / (x + y)) * y) / ((1.0 + y) * (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 <= -1.65e-11)
		tmp = Float64(Float64(1.0 * Float64(y / Float64(Float64(x + y) + 1.0))) / Float64(x + y));
	elseif (x <= 3e-87)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) * y) / Float64(Float64(1.0 + y) * 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 <= -1.65e-11)
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	elseif (x <= 3e-87)
		tmp = ((x / (x + y)) * y) / ((1.0 + y) * (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, -1.65e-11], N[(N[(1.0 * N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3e-87], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] / N[(N[(1.0 + y), $MachinePrecision] * N[(x + y), $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 < -1.6500000000000001e-11

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 78.4%

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

   if -1.6500000000000001e-11 < x < 3.00000000000000016e-87

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. times-frac N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 99.9

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 99.9

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 99.9

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 3.00000000000000016e-87 < x

   1. Initial program 67.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 31.8

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

   7. Applied rewrites 31.8%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{x}{x + y} \cdot y}{\left(1 + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq 10^{-203}:\\ \;\;\;\;\frac{y}{\left(1 + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.65e-11)
   (/ (* 1.0 (/ y (+ (+ x y) 1.0))) (+ x y))
   (if (<= x 1e-203)
     (* (/ y (* (+ 1.0 y) (+ x y))) (/ x (+ x y)))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-11) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (x <= 1e-203) {
		tmp = (y / ((1.0 + y) * (x + y))) * (x / (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 <= (-1.65d-11)) then
        tmp = (1.0d0 * (y / ((x + y) + 1.0d0))) / (x + y)
    else if (x <= 1d-203) then
        tmp = (y / ((1.0d0 + y) * (x + y))) * (x / (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 <= -1.65e-11) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (x <= 1e-203) {
		tmp = (y / ((1.0 + y) * (x + y))) * (x / (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 <= -1.65e-11:
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y)
	elif x <= 1e-203:
		tmp = (y / ((1.0 + y) * (x + y))) * (x / (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 <= -1.65e-11)
		tmp = Float64(Float64(1.0 * Float64(y / Float64(Float64(x + y) + 1.0))) / Float64(x + y));
	elseif (x <= 1e-203)
		tmp = Float64(Float64(y / Float64(Float64(1.0 + y) * Float64(x + y))) * Float64(x / 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 <= -1.65e-11)
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	elseif (x <= 1e-203)
		tmp = (y / ((1.0 + y) * (x + y))) * (x / (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, -1.65e-11], N[(N[(1.0 * N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-203], N[(N[(y / N[(N[(1.0 + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + y), $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 < -1.6500000000000001e-11

   1. Initial program 63.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 78.4%

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

   if -1.6500000000000001e-11 < x < 1e-203

   1. Initial program 69.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. 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 99.8

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

      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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-+.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if 1e-203 < x

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.9%

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

   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 41.2

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

   7. Applied rewrites 41.2%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-11}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;x \leq 10^{-203}:\\ \;\;\;\;\frac{y}{\left(1 + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 88.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \left(x + y\right) + 1\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{t\_0}}{x + y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+91}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

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 (<= y 1.65e-144)
     (/ (* 1.0 (/ y t_0)) (+ x y))
     (if (<= y 1.45e+91)
       (/ (* 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 (y <= 1.65e-144) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.45e+91) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} 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) :: t_0
    real(8) :: tmp
    t_0 = (x + y) + 1.0d0
    if (y <= 1.65d-144) then
        tmp = (1.0d0 * (y / t_0)) / (x + y)
    else if (y <= 1.45d+91) then
        tmp = (x * y) / (((x + y) * (x + y)) * t_0)
    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 t_0 = (x + y) + 1.0;
	double tmp;
	if (y <= 1.65e-144) {
		tmp = (1.0 * (y / t_0)) / (x + y);
	} else if (y <= 1.45e+91) {
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	} else {
		tmp = (x / (1.0 + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = (x + y) + 1.0
	tmp = 0
	if y <= 1.65e-144:
		tmp = (1.0 * (y / t_0)) / (x + y)
	elif y <= 1.45e+91:
		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 (y <= 1.65e-144)
		tmp = Float64(Float64(1.0 * Float64(y / t_0)) / Float64(x + y));
	elseif (y <= 1.45e+91)
		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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = (x + y) + 1.0;
	tmp = 0.0;
	if (y <= 1.65e-144)
		tmp = (1.0 * (y / t_0)) / (x + y);
	elseif (y <= 1.45e+91)
		tmp = (x * y) / (((x + y) * (x + y)) * t_0);
	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_] := Block[{t$95$0 = N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.65e-144], N[(N[(1.0 * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e+91], 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 3 regimes

2. if y < 1.64999999999999998e-144

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   7. Applied rewrites 63.3%

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

   if 1.64999999999999998e-144 < y < 1.45000000000000007e91

   1. Initial program 85.7%

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

   if 1.45000000000000007e91 < y

   1. Initial program 50.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 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 87.3

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

   7. Applied rewrites 87.3%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{+91}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 86.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-144)
   (/ (* 1.0 (/ y (+ (+ x y) 1.0))) (+ x y))
   (if (<= y 1.2)
     (/ (* x y) (* (+ x 1.0) (* (+ x y) (+ x y))))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-144) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (y <= 1.2) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 (y <= 1.65d-144) then
        tmp = (1.0d0 * (y / ((x + y) + 1.0d0))) / (x + y)
    else if (y <= 1.2d0) then
        tmp = (x * y) / ((x + 1.0d0) * ((x + y) * (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 (y <= 1.65e-144) {
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	} else if (y <= 1.2) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 y <= 1.65e-144:
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y)
	elif y <= 1.2:
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 (y <= 1.65e-144)
		tmp = Float64(Float64(1.0 * Float64(y / Float64(Float64(x + y) + 1.0))) / Float64(x + y));
	elseif (y <= 1.2)
		tmp = Float64(Float64(x * y) / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * 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 (y <= 1.65e-144)
		tmp = (1.0 * (y / ((x + y) + 1.0))) / (x + y);
	elseif (y <= 1.2)
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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[y, 1.65e-144], N[(N[(1.0 * N[(y / N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2], N[(N[(x * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $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 y < 1.64999999999999998e-144

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

   7. Applied rewrites 63.3%

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

   if 1.64999999999999998e-144 < y < 1.19999999999999996

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if 1.19999999999999996 < y

   1. Initial program 59.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 78.4

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

   7. Applied rewrites 78.4%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-144)
   (/ (/ y (+ x 1.0)) (+ x y))
   (if (<= y 1.2)
     (/ (* x y) (* (+ x 1.0) (* (+ x y) (+ x y))))
     (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-144) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 1.2) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 (y <= 1.65d-144) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else if (y <= 1.2d0) then
        tmp = (x * y) / ((x + 1.0d0) * ((x + y) * (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 (y <= 1.65e-144) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 1.2) {
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 y <= 1.65e-144:
		tmp = (y / (x + 1.0)) / (x + y)
	elif y <= 1.2:
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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 (y <= 1.65e-144)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	elseif (y <= 1.2)
		tmp = Float64(Float64(x * y) / Float64(Float64(x + 1.0) * Float64(Float64(x + y) * 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 (y <= 1.65e-144)
		tmp = (y / (x + 1.0)) / (x + y);
	elseif (y <= 1.2)
		tmp = (x * y) / ((x + 1.0) * ((x + y) * (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[y, 1.65e-144], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2], N[(N[(x * y), $MachinePrecision] / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x + y), $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 y < 1.64999999999999998e-144

   1. Initial program 68.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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. lower-+.f64 62.8

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

   7. Applied rewrites 62.8%

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

   if 1.64999999999999998e-144 < y < 1.19999999999999996

   1. Initial program 86.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-+.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if 1.19999999999999996 < y

   1. Initial program 59.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. 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 78.4

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

   7. Applied rewrites 78.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 1.2:\\ \;\;\;\;\frac{x \cdot y}{\left(x + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 81.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-62)
   (/ (/ y x) (+ x y))
   (if (<= y 2.1e-95) (/ y (fma x x x)) (/ (/ x (+ 1.0 y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (y / x) / (x + y);
	} else if (y <= 2.1e-95) {
		tmp = y / fma(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 (y <= -2.1e-62)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (y <= 2.1e-95)
		tmp = Float64(y / fma(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[y, -2.1e-62], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-95], N[(y / N[(x * x + x), $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 y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 40.6

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

   7. Applied rewrites 40.6%

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

   if -2.0999999999999999e-62 < y < 2.1e-95

   1. Initial program 74.0%

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

   3. 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 82.8

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

   5. Applied rewrites 82.8%

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

   if 2.1e-95 < y

   1. Initial program 63.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 70.3

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

   7. Applied rewrites 70.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-62)
   (/ (/ y x) (+ x y))
   (if (<= y 2.7e+18) (/ y (fma x x x)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (y / x) / (x + y);
	} else if (y <= 2.7e+18) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-62)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif (y <= 2.7e+18)
		tmp = Float64(y / fma(x, x, x));
	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[y, -2.1e-62], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+18], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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 99.7%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 40.6

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

   7. Applied rewrites 40.6%

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

   if -2.0999999999999999e-62 < y < 2.7e18

   1. Initial program 75.0%

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

   3. 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 79.2

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

   5. Applied rewrites 79.2%

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

   if 2.7e18 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around 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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 83.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 79.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y -2.1e-62)
   (/ (/ y x) x)
   (if (<= y 2.7e+18) (/ y (fma x x x)) (/ (/ x y) y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= -2.1e-62) {
		tmp = (y / x) / x;
	} else if (y <= 2.7e+18) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= -2.1e-62)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 2.7e+18)
		tmp = Float64(y / fma(x, x, x));
	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[y, -2.1e-62], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 2.7e+18], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0999999999999999e-62

   1. Initial program 67.1%

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

   3. 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 36.0

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

   5. Applied rewrites 36.0%

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

   7. Applied rewrites 39.9%

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

   if -2.0999999999999999e-62 < y < 2.7e18

   1. Initial program 75.0%

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

   3. 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 79.2

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

   5. Applied rewrites 79.2%

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

   if 2.7e18 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around 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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 83.4%

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

Alternative 17: 81.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-95) (/ (/ 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 (y <= 2.1e-95) {
		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 (y <= 2.1d-95) 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 (y <= 2.1e-95) {
		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 y <= 2.1e-95:
		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 (y <= 2.1e-95)
		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 (y <= 2.1e-95)
		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[y, 2.1e-95], 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 y < 2.1e-95

   1. Initial program 71.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. lower-+.f64 65.0

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

   7. Applied rewrites 65.0%

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

   if 2.1e-95 < y

   1. Initial program 63.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 70.3

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

   7. Applied rewrites 70.3%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1 + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 78.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+18) (/ y (fma x x x)) (/ (/ x y) y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+18) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+18)
		tmp = Float64(y / fma(x, x, x));
	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[y, 2.7e+18], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e18

   1. Initial program 72.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 63.4

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

   5. Applied rewrites 63.4%

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

   if 2.7e18 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around 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 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 83.4%

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

Alternative 19: 78.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-95) (/ y (fma x x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-95) {
		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 (y <= 2.1e-95)
		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[y, 2.1e-95], 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 y < 2.1e-95

   1. Initial program 71.1%

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

   3. Taylor expanded in 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 63.0

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

   5. Applied rewrites 63.0%

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

   if 2.1e-95 < y

   1. Initial program 63.0%

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

   3. 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 64.1

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

   5. Applied rewrites 64.1%

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

Alternative 20: 76.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -350000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -350000000.0) (/ y (* x x)) (/ x (fma y y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -350000000.0) {
		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 <= -350000000.0)
		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, -350000000.0], 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 < -3.5e8

   1. Initial program 62.6%

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

   3. Taylor expanded in x around inf

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

      1. 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 75.5

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

   5. Applied rewrites 75.5%

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

   if -3.5e8 < x

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 55.8

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

   5. Applied rewrites 55.8%

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

Alternative 21: 63.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e+18) (/ y (* x x)) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+18) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * 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 (y <= 2.7d+18) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e+18) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 2.7e+18:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 2.7e+18)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e+18)
		tmp = y / (x * x);
	else
		tmp = x / (y * 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[y, 2.7e+18], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.7e18

   1. Initial program 72.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. 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 48.1

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

   5. Applied rewrites 48.1%

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

   if 2.7e18 < y

   1. Initial program 56.7%

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

   3. Taylor expanded in y around 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 75.3

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

   5. Applied rewrites 75.3%

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

Alternative 22: 36.6% accurate, 2.3× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return x / (y * 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 * y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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 34.2

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

5. Applied rewrites 34.2%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
6. 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 2024296 
(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))))