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

Percentage Accurate: 69.2% → 99.8%

Time: 14.7s

Alternatives: 22

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 86.9

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

   9. lift-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate-+l+ N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. lower-+.f64 N/A

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

   15. lower-+.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.9

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e+81)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (if (<= x -1.55e-162)
     (* x (/ y (* (+ 1.0 (+ y x)) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.65e+81) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else if (x <= -1.55e-162) {
		tmp = x * (y / ((1.0 + (y + x)) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.65e+81)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	elseif (x <= -1.55e-162)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + Float64(y + x)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.65e+81], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(x * N[(y / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65000000000000014e81

   1. Initial program 60.5%

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

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 85.1

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 85.1

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

   4. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 87.5

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

   11. Applied rewrites 87.5%

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

   if -2.65000000000000014e81 < x < -1.5499999999999999e-162

   1. Initial program 80.0%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 93.2%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 68.2%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.0

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

   7. Applied rewrites 60.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + \left(y + x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -2.65e+81)
     (/ (/ y t_0) (fma y 2.0 x))
     (if (<= x -1.55e-162)
       (* x (/ y (* t_0 (* (+ y x) (+ y x)))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2.65e+81) {
		tmp = (y / t_0) / fma(y, 2.0, x);
	} else if (x <= -1.55e-162) {
		tmp = x * (y / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -2.65e+81)
		tmp = Float64(Float64(y / t_0) / fma(y, 2.0, x));
	elseif (x <= -1.55e-162)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.65e+81], N[(N[(y / t$95$0), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.55e-162], N[(x * N[(y / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65000000000000014e81

   1. Initial program 60.5%

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

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 85.1

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 85.1

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

   4. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 87.5

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

   11. Applied rewrites 87.5%

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

   if -2.65000000000000014e81 < x < -1.5499999999999999e-162

   1. Initial program 80.0%

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

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.2

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. associate-+l+ N/A

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

      12. lower-+.f64 N/A

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

      13. lower-+.f64 93.2

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

   4. Applied rewrites 93.2%

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

   if -1.5499999999999999e-162 < x

   1. Initial program 68.2%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.0

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

   7. Applied rewrites 60.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;x \cdot \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.4e+154)
     (/ (/ y t_0) (fma y 2.0 x))
     (/ (* x (/ y (+ y x))) (* t_0 (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / t_0) / fma(y, 2.0, x);
	} else {
		tmp = (x * (y / (y + x))) / (t_0 * (y + x));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / t_0) / fma(y, 2.0, x));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(t_0 * Float64(y + x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e154

   1. Initial program 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.2

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 91.0

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

   11. Applied rewrites 91.0%

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

   if -1.4e154 < x

   1. Initial program 70.2%

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

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 95.6

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. associate-+l+ N/A

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

      16. +-commutative N/A

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

      17. associate-+l+ N/A

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

      18. lower-+.f64 N/A

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

      19. lower-+.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.4e+154)
     (/ (/ y t_0) (fma y 2.0 x))
     (* (/ x (+ y x)) (/ y (* t_0 (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.4e+154) {
		tmp = (y / t_0) / fma(y, 2.0, x);
	} else {
		tmp = (x / (y + x)) * (y / (t_0 * (y + x)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.4e+154)
		tmp = Float64(Float64(y / t_0) / fma(y, 2.0, x));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(t_0 * Float64(y + x))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e154

   1. Initial program 57.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.2

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 91.0

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

   11. Applied rewrites 91.0%

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

   if -1.4e154 < x

   1. Initial program 70.2%

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

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 95.6

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

   4. Applied rewrites 95.6%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 N/A

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

   19. lower-/.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-+.f64 N/A

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

   9. lower-/.f64 99.8

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f64 N/A

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

   13. associate-+r+ N/A

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

   14. lift-+.f64 N/A

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

   15. lower-+.f64 99.8

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

   16. lift-+.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 99.8

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

   19. lift-+.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 8: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.8

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. associate-+l+ N/A

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

   15. +-commutative N/A

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

   16. associate-+l+ N/A

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

   17. lower-+.f64 N/A

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

   18. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 99.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ y (+ y (+ x 1.0))) (fma y (+ 2.0 (/ y x)) x)))

C

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

Julia

function code(x, y)
	return Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, Float64(2.0 + Float64(y / x)), x))
end

Wolfram

code[x_, y_] := N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * N[(2.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.1%

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

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 86.9

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

   9. lift-+.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. associate-+l+ N/A

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

   12. +-commutative N/A

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

   13. associate-+l+ N/A

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

   14. lower-+.f64 N/A

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

   15. lower-+.f64 86.8

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

4. Applied rewrites 86.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-/.f64 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f64 N/A

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

   9. lower-*.f64 99.9

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 99.9

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.9

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

6. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. frac-times N/A

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

   12. lift-+.f64 N/A

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

   13. lift-+.f64 N/A

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

   14. clear-num N/A

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

   15. un-div-inv N/A

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

   16. lower-/.f64 N/A

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

8. Applied rewrites 98.7%

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

9. Taylor expanded in y around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-+.f64 N/A

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

    4. lower-/.f64 98.7

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

11. Applied rewrites 98.7%

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

Alternative 10: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) \cdot \left(y + x\right)\\ \mathbf{if}\;y \leq 3 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 550000000:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (+ y x) (+ y x))))
   (if (<= y 3e-166)
     (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
     (if (<= y 550000000.0) (* x (/ y (* t_0 (+ x 1.0)))) (/ (* x 1.0) t_0)))))

C

double code(double x, double y) {
	double t_0 = (y + x) * (y + x);
	double tmp;
	if (y <= 3e-166) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else if (y <= 550000000.0) {
		tmp = x * (y / (t_0 * (x + 1.0)));
	} else {
		tmp = (x * 1.0) / t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) * Float64(y + x))
	tmp = 0.0
	if (y <= 3e-166)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	elseif (y <= 550000000.0)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x * 1.0) / t_0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3e-166], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 550000000.0], N[(x * N[(y / N[(t$95$0 * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 1.0), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0000000000000003e-166

   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. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 82.9

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 82.9

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

   4. Applied rewrites 82.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 98.5%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 58.8

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

   11. Applied rewrites 58.8%

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

   if 3.0000000000000003e-166 < y < 5.5e8

   1. Initial program 86.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/r* N/A

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

      7. lift-*.f64 N/A

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

      8. frac-times N/A

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. associate-+r+ N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. *-commutative N/A

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

      17. *-commutative N/A

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.0

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

   9. Applied rewrites 98.0%

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

   if 5.5e8 < y

   1. Initial program 59.1%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 87.1

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 87.1

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 87.1

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

   9. Applied rewrites 87.1%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 550000000:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22)
   (/ (/ y x) (+ y x))
   (if (<= x -0.072)
     (/ (* x 1.0) (* (+ y x) (+ y x)))
     (if (<= x -2.1e-98) (/ y (fma x x x)) (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = (y / x) / (y + x);
	} else if (x <= -0.072) {
		tmp = (x * 1.0) / ((y + x) * (y + x));
	} else if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -0.072)
		tmp = Float64(Float64(x * 1.0) / Float64(Float64(y + x) * Float64(y + x)));
	elseif (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.072], N[(N[(x * 1.0), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 80.4

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

   7. Applied rewrites 80.4%

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

   if -1.3500000000000001e22 < x < -0.0719999999999999946

   1. Initial program 65.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/r* N/A

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

      8. associate-*r/ N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.0

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 70.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. associate-/l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 70.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 70.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 70.0

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

   9. Applied rewrites 70.0%

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

   if -0.0719999999999999946 < x < -2.09999999999999992e-98

   1. Initial program 82.3%

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

   3. Taylor expanded in y around 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 41.5

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

   5. Applied rewrites 41.5%

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

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-98)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-98)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-98], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999992e-98

   1. Initial program 66.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 89.3

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

      9. lift-+.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. associate-+l+ N/A

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

      12. +-commutative N/A

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

      13. associate-+l+ N/A

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

      14. lower-+.f64 N/A

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

      15. lower-+.f64 89.3

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

   4. Applied rewrites 89.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 99.9

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.9

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 N/A

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

      11. frac-times N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. clear-num N/A

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

      15. un-div-inv N/A

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

      16. lower-/.f64 N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 74.1

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

   11. Applied rewrites 74.1%

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

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      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

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. associate-+l+ N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lower-+.f64 N/A

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

      18. lower-+.f64 N/A

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

      19. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.2

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

   7. Applied rewrites 60.2%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22)
   (/ (/ y x) (+ y x))
   (if (<= x -0.072)
     (/ (* x 1.0) (* (+ y x) (+ y x)))
     (if (<= x -2.1e-98) (/ y (fma x x x)) (/ x (fma y y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = (y / x) / (y + x);
	} else if (x <= -0.072) {
		tmp = (x * 1.0) / ((y + x) * (y + x));
	} else if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -0.072)
		tmp = Float64(Float64(x * 1.0) / Float64(Float64(y + x) * Float64(y + x)));
	elseif (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.072], N[(N[(x * 1.0), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

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

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 100.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 80.4

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   if -1.3500000000000001e22 < x < -0.0719999999999999946

   1. Initial program 65.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 99.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{x}{x + y}}{x + y} \]
   6. Step-by-step derivation

   7. Applied rewrites 70.0%

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{x}{x + y}}{x + y} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{x + y}}{x + y}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{1 \cdot \frac{x}{x + y}}}{x + y} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{1 \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot x}{x + y}}}{x + y} \]

      5. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 \cdot x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{\left(x + y\right) \cdot \left(x + y\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{\left(x + y\right) \cdot \left(x + y\right)} \]

      9. lower-*.f64 70.0

         \[\leadsto \frac{x \cdot 1}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{x \cdot 1}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{x \cdot 1}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \]

      12. lift-+.f64 70.0

          \[\leadsto \frac{x \cdot 1}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{x \cdot 1}{\left(y + x\right) \cdot \color{blue}{\left(x + y\right)}} \]

      14. +-commutative N/A

          \[\leadsto \frac{x \cdot 1}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \]

      15. lift-+.f64 70.0

          \[\leadsto \frac{x \cdot 1}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}} \]

   9. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}} \]

   if -0.0719999999999999946 < x < -2.09999999999999992e-98

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 41.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{x \cdot 1}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22)
   (/ (/ y x) (+ y x))
   (if (<= x -0.072)
     (/ (/ x y) (+ y x))
     (if (<= x -2.1e-98) (/ y (fma x x x)) (/ x (fma y y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = (y / x) / (y + x);
	} else if (x <= -0.072) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -0.072)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.072], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 100.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 80.4

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Applied rewrites 80.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   if -1.3500000000000001e22 < x < -0.0719999999999999946

   1. Initial program 65.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 99.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 69.5

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Applied rewrites 69.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   if -0.0719999999999999946 < x < -2.09999999999999992e-98

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 41.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22)
   (/ (/ y x) x)
   (if (<= x -0.072)
     (/ (/ x y) (+ y x))
     (if (<= x -2.1e-98) (/ y (fma x x x)) (/ x (fma y y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = (y / x) / x;
	} else if (x <= -0.072) {
		tmp = (x / y) / (y + x);
	} else if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -0.072)
		tmp = Float64(Float64(x / y) / Float64(y + x));
	elseif (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.072], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 100.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   6. 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 75.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   7. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.0%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -1.3500000000000001e22 < x < -0.0719999999999999946

   1. Initial program 65.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 99.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 69.5

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Applied rewrites 69.5%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   if -0.0719999999999999946 < x < -2.09999999999999992e-98

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 41.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -0.072:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22)
   (/ (/ y x) x)
   (if (<= x -0.072)
     (/ x (* y y))
     (if (<= x -2.1e-98) (/ y (fma x x x)) (/ x (fma y y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = (y / x) / x;
	} else if (x <= -0.072) {
		tmp = x / (y * y);
	} else if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -0.072)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -0.072], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 100.0

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   6. 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 75.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   7. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.0%

      \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x}} \]

   if -1.3500000000000001e22 < x < -0.0719999999999999946

   1. Initial program 65.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 69.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 69.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -0.0719999999999999946 < x < -2.09999999999999992e-98

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 41.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 55.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-185}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= x -1.35e+22)
     (/ y (* x x))
     (if (<= x -3.5e-185) t_0 (if (<= x 9e-147) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.35e+22) {
		tmp = y / (x * x);
	} else if (x <= -3.5e-185) {
		tmp = t_0;
	} else if (x <= 9e-147) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (x <= (-1.35d+22)) then
        tmp = y / (x * x)
    else if (x <= (-3.5d-185)) then
        tmp = t_0
    else if (x <= 9d-147) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (x <= -1.35e+22) {
		tmp = y / (x * x);
	} else if (x <= -3.5e-185) {
		tmp = t_0;
	} else if (x <= 9e-147) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if x <= -1.35e+22:
		tmp = y / (x * x)
	elif x <= -3.5e-185:
		tmp = t_0
	elif x <= 9e-147:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.5e-185)
		tmp = t_0;
	elseif (x <= 9e-147)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (x <= -1.35e+22)
		tmp = y / (x * x);
	elseif (x <= -3.5e-185)
		tmp = t_0;
	elseif (x <= 9e-147)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+22], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-185], t$95$0, If[LessEqual[x, 9e-147], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3500000000000001e22

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 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 75.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1.3500000000000001e22 < x < -3.4999999999999998e-185 or 8.99999999999999946e-147 < x

   1. Initial program 72.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 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 41.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 41.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]

   if -3.4999999999999998e-185 < x < 8.99999999999999946e-147

   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. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 71.3

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(y + 1\right) + x}} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

      15. lower-+.f64 71.3

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \color{blue}{\left(1 + x\right)}} \]

   4. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{y + \left(1 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      9. lower-*.f64 99.9

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

      12. lower-+.f64 99.9

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{x + y}}}{y + \left(1 + x\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + x}}}{y + \left(1 + x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 85.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   9. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 76.9%

       \[\leadsto \frac{x}{\color{blue}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-98) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.1d-98)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.1e-98:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-98)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.1e-98)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-98], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999992e-98

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      8. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 99.9

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

      3. lower-+.f64 72.8

         \[\leadsto \frac{\frac{y}{\color{blue}{x + 1}}}{x + y} \]

   7. Applied rewrites 72.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{x + 1}}}{x + y} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      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

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      14. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{x + \left(y + 1\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      15. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      16. associate-+l+ N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      17. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      18. lower-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      19. lower-/.f64 99.8

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

      2. +-commutative N/A

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

      3. lower-+.f64 60.2

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   7. Applied rewrites 60.2%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 60.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e-98) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e-98) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e-98)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e-98], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.09999999999999992e-98

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 68.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Applied rewrites 68.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.09999999999999992e-98 < x

   1. Initial program 68.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.0

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 58.0%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 61.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+22) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+22)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e+22], 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 < -1.3500000000000001e22

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 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 75.2

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Applied rewrites 75.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1.3500000000000001e22 < x

   1. Initial program 69.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. 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 57.6

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 37.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

      8. lower-/.f64 85.2

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(y + 1\right) + x}} \]

      13. associate-+l+ N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

      14. lower-+.f64 N/A

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

      15. lower-+.f64 85.2

          \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \color{blue}{\left(1 + x\right)}} \]

   4. Applied rewrites 85.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \left(1 + x\right)}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{y + \left(1 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

      5. associate-/r* N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{y \cdot \frac{\color{blue}{\frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      7. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

      9. lower-*.f64 99.9

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      10. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

      12. lower-+.f64 99.9

          \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

      13. lift-+.f64 N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{x + y}}}{y + \left(1 + x\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

      15. lower-+.f64 99.9

          \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + x}}}{y + \left(1 + x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   8. 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 38.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   9. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \frac{x}{\color{blue}{y}} \]
   11. Step-by-step derivation

   12. Applied rewrites 24.8%

       \[\leadsto \frac{x}{\color{blue}{y}} \]

   if 1 < y

   1. Initial program 59.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 72.2

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Applied rewrites 72.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 26.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
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. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

   6. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\left(x + y\right) + 1}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{\left(x + y\right) + 1} \]

   8. lower-/.f64 86.9

      \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot y}{\left(x + y\right) + 1} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right) + 1}} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(x + y\right)} + 1} \]

   11. associate-+l+ N/A

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

   12. +-commutative N/A

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{\left(y + 1\right) + x}} \]

   13. associate-+l+ N/A

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

   14. lower-+.f64 N/A

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{\color{blue}{y + \left(1 + x\right)}} \]

   15. lower-+.f64 86.8

       \[\leadsto \frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \color{blue}{\left(1 + x\right)}} \]

4. Applied rewrites 86.8%

   \[\leadsto \color{blue}{\frac{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}{y + \left(1 + x\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot y}}{y + \left(1 + x\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{y \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{y \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}}}{y + \left(1 + x\right)} \]

   5. associate-/r* N/A

      \[\leadsto \frac{y \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

   6. lift-/.f64 N/A

      \[\leadsto \frac{y \cdot \frac{\color{blue}{\frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

   7. associate-*r/ N/A

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

   8. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{x + y}}{x + y}}}{y + \left(1 + x\right)} \]

   9. lower-*.f64 99.9

      \[\leadsto \frac{\frac{\color{blue}{y \cdot \frac{x}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{x + y}}}{x + y}}{y + \left(1 + x\right)} \]

   11. +-commutative N/A

       \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

   12. lower-+.f64 99.9

       \[\leadsto \frac{\frac{y \cdot \frac{x}{\color{blue}{y + x}}}{x + y}}{y + \left(1 + x\right)} \]

   13. lift-+.f64 N/A

       \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{x + y}}}{y + \left(1 + x\right)} \]

   14. +-commutative N/A

       \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

   15. lower-+.f64 99.9

       \[\leadsto \frac{\frac{y \cdot \frac{x}{y + x}}{\color{blue}{y + x}}}{y + \left(1 + x\right)} \]

6. Applied rewrites 99.9%

   \[\leadsto \frac{\color{blue}{\frac{y \cdot \frac{x}{y + x}}{y + x}}}{y + \left(1 + x\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
8. 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 47.0

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

9. Applied rewrites 47.0%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

10. Taylor expanded in y around 0

    \[\leadsto \frac{x}{\color{blue}{y}} \]
11. Step-by-step derivation

12. Applied rewrites 25.6%

    \[\leadsto \frac{x}{\color{blue}{y}} \]
13. 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 2024238 
(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))))