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

Percentage Accurate: 68.7% → 99.8%

Time: 15.8s

Alternatives: 26

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: 68.7% 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

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

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(Float64(y + 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 87.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -7 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \left(y + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -7e+170)
     (/ (/ y t_0) (fma y 2.0 x))
     (if (<= x -1.65e-15)
       (/ y (* (+ y x) t_0))
       (/ x (* (+ y x) (* (/ (+ y x) y) (+ y 1.0))))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -7e+170) {
		tmp = (y / t_0) / fma(y, 2.0, x);
	} else if (x <= -1.65e-15) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = x / ((y + x) * (((y + x) / y) * (y + 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -7e+170)
		tmp = Float64(Float64(y / t_0) / fma(y, 2.0, x));
	elseif (x <= -1.65e-15)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(Float64(y + x) / y) * Float64(y + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7e+170], N[(N[(y / t$95$0), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.65e-15], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.00000000000000011e170

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 87.2

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

   9. Simplified 87.2%

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

   if -7.00000000000000011e170 < x < -1.65e-15

   1. Initial program 71.8%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 98.1

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

   6. Applied egg-rr 98.1%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 66.7%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. +-lowering-+.f64 N/A

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

       6. +-lowering-+.f64 N/A

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

       7. *-lft-identity N/A

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

       8. +-lowering-+.f64 89.0

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

   11. Applied egg-rr 89.0%

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

   if -1.65e-15 < x

   1. Initial program 75.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-/l/ N/A

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

      4. times-frac N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

      8. /-lowering-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. associate-/l* N/A

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

      11. clear-num N/A

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

      12. associate-/l/ N/A

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

      13. clear-num N/A

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

      14. *-lowering-*.f64 N/A

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

      15. +-commutative N/A

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

      16. +-lowering-+.f64 N/A

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

      17. associate-/r/ N/A

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 88.6

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

   9. Simplified 88.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \left(y + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ \mathbf{if}\;x \leq -1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y, 2 + \frac{y}{x}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot t\_0}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))))
   (if (<= x -1.2e+26)
     (/ t_0 (fma y (+ 2.0 (/ y x)) x))
     (/ (* x t_0) (* (+ y x) (+ (+ y x) 1.0))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double tmp;
	if (x <= -1.2e+26) {
		tmp = t_0 / fma(y, (2.0 + (y / x)), x);
	} else {
		tmp = (x * t_0) / ((y + x) * ((y + x) + 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	tmp = 0.0
	if (x <= -1.2e+26)
		tmp = Float64(t_0 / fma(y, Float64(2.0 + Float64(y / x)), x));
	else
		tmp = Float64(Float64(x * t_0) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.2e+26], N[(t$95$0 / N[(y * N[(2.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * t$95$0), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.20000000000000002e26

   1. Initial program 57.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 98.6

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot \left(2 + \frac{y}{x}\right)}} \]
   8. 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. accelerator-lowering-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. +-lowering-+.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. /-lowering-/.f64 98.6

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

   9. Simplified 98.6%

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

   10. Taylor expanded in x around inf

       \[\leadsto \frac{\frac{y}{y + \color{blue}{x}}}{\mathsf{fma}\left(y, 2 + \frac{y}{x}, x\right)} \]
   11. Step-by-step derivation

   12. Simplified 98.6%

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

   if -1.20000000000000002e26 < x

   1. Initial program 76.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 95.9

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

   4. Applied egg-rr 95.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{\mathsf{fma}\left(y, 2 + \frac{y}{x}, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-171)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (if (<= y 5.6e+102)
     (* x (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-171) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else if (y <= 5.6e+102) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-171)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	elseif (y <= 5.6e+102)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-171], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.6e+102], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.70000000000000014e-171

   1. Initial program 72.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 63.1

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

   9. Simplified 63.1%

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

   if 2.70000000000000014e-171 < y < 5.60000000000000037e102

   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. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 89.4

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

   4. Applied egg-rr 89.4%

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

   if 5.60000000000000037e102 < y

   1. Initial program 57.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 79.8

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

   7. Simplified 79.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e-171)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (if (<= y 1.26e+47)
     (* y (/ x (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e-171) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else if (y <= 1.26e+47) {
		tmp = y * (x / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e-171)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	elseif (y <= 1.26e+47)
		tmp = Float64(y * Float64(x / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e-171], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.26e+47], N[(y * N[(x / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.60000000000000005e-171

   1. Initial program 72.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 63.1

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

   9. Simplified 63.1%

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

   if 2.60000000000000005e-171 < y < 1.26e47

   1. Initial program 80.5%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 89.2

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

   4. Applied egg-rr 89.2%

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

   if 1.26e47 < y

   1. Initial program 63.1%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 80.3

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

   7. Simplified 80.3%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 1.26 \cdot 10^{+47}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) \cdot \frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.15e+162)
   (/ (/ y x) (* (+ y x) (/ (+ y x) x)))
   (/ (* x (/ y (+ y x))) (* (+ y x) (+ (+ y x) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+162) {
		tmp = (y / x) / ((y + x) * ((y + x) / x));
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.15d+162)) then
        tmp = (y / x) / ((y + x) * ((y + x) / x))
    else
        tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.15e+162) {
		tmp = (y / x) / ((y + x) * ((y + x) / x));
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.15e+162:
		tmp = (y / x) / ((y + x) * ((y + x) / x))
	else:
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.15e+162)
		tmp = Float64(Float64(y / x) / Float64(Float64(y + x) * Float64(Float64(y + x) / x)));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.15e+162)
		tmp = (y / x) / ((y + x) * ((y + x) / x));
	else
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.15e+162], N[(N[(y / x), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14999999999999997e162

   1. Initial program 47.3%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 88.6

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

   9. Simplified 88.6%

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

   if -1.14999999999999997e162 < x

   1. Initial program 75.3%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 96.1

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

   4. Applied egg-rr 96.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{\left(y + x\right) \cdot \frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.9e+172)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (/ (* x (/ y (+ y x))) (* (+ y x) (+ (+ y x) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.9e+172) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.9e+172)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.9e+172], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.9000000000000001e172

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 87.2

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

   9. Simplified 87.2%

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

   if -4.9000000000000001e172 < x

   1. Initial program 74.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 95.3

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

   4. Applied egg-rr 95.3%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.9 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.3e+172)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (* (/ y (+ y x)) (/ x (* (+ y x) (+ (+ y x) 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.3e+172) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else {
		tmp = (y / (y + x)) * (x / ((y + x) * ((y + x) + 1.0)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.3e+172)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.3e+172], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.29999999999999983e172

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 87.2

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

   9. Simplified 87.2%

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

   if -3.29999999999999983e172 < x

   1. Initial program 74.4%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 95.3

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

   4. Applied egg-rr 95.3%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 0.8× speedup

Math

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

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(Float64(y + 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[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.8%

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

   1. times-frac N/A

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

   2. associate-*l/ N/A

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

   3. times-frac N/A

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

   4. *-lowering-*.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. +-lowering-+.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. /-lowering-/.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 10: 99.4% 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 71.8%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

   5. /-lowering-/.f64 N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. /-lowering-/.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. clear-num N/A

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

   4. frac-times N/A

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

   5. *-lft-identity N/A

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

   6. /-lowering-/.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-lowering-+.f64 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. /-lowering-/.f64 N/A

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

   14. +-commutative N/A

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

   15. +-lowering-+.f64 N/A

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

   16. +-commutative N/A

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

   17. +-lowering-+.f64 99.1

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in y around 0

   \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + y \cdot \left(2 + \frac{y}{x}\right)}} \]
8. 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. accelerator-lowering-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. +-lowering-+.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. /-lowering-/.f64 99.1

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

9. Simplified 99.1%

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

Alternative 11: 66.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-171)
   (/ (/ y (+ y (+ x 1.0))) (fma y 2.0 x))
   (if (<= y 8.6e+24)
     (* x (/ y (* (+ x 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-171) {
		tmp = (y / (y + (x + 1.0))) / fma(y, 2.0, x);
	} else if (y <= 8.6e+24) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-171)
		tmp = Float64(Float64(y / Float64(y + Float64(x + 1.0))) / fma(y, 2.0, x));
	elseif (y <= 8.6e+24)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-171], N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.6e+24], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.70000000000000014e-171

   1. Initial program 72.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.0

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 63.1

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

   9. Simplified 63.1%

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

   if 2.70000000000000014e-171 < y < 8.59999999999999975e24

   1. Initial program 84.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 89.5

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

   4. Applied egg-rr 89.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 86.5

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

   7. Simplified 86.5%

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

   if 8.59999999999999975e24 < y

   1. Initial program 62.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 76.8

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

   7. Simplified 76.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;y \leq 8.6 \cdot 10^{+24}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.35 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.35e+171)
     (/ (/ y t_0) (fma y 2.0 x))
     (if (<= x -1.5e-80)
       (/ y (* (+ y x) t_0))
       (/ (* x (/ 1.0 y)) (+ y 1.0))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.35e+171) {
		tmp = (y / t_0) / fma(y, 2.0, x);
	} else if (x <= -1.5e-80) {
		tmp = y / ((y + x) * t_0);
	} else {
		tmp = (x * (1.0 / y)) / (y + 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.35e+171)
		tmp = Float64(Float64(y / t_0) / fma(y, 2.0, x));
	elseif (x <= -1.5e-80)
		tmp = Float64(y / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x * Float64(1.0 / y)) / Float64(y + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.35e+171], N[(N[(y / t$95$0), $MachinePrecision] / N[(y * 2.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-80], N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3499999999999999e171

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 99.8

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{x + 2 \cdot y}} \]
   8. 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. accelerator-lowering-fma.f64 87.2

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

   9. Simplified 87.2%

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

   if -1.3499999999999999e171 < x < -1.50000000000000004e-80

   1. Initial program 77.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 56.9%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. +-lowering-+.f64 N/A

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

       6. +-lowering-+.f64 N/A

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

       7. *-lft-identity N/A

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

       8. +-lowering-+.f64 84.0

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

   11. Applied egg-rr 84.0%

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

   if -1.50000000000000004e-80 < x

   1. Initial program 73.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 84.4

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 57.4

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

   7. Simplified 57.4%

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

      1. associate-/r* N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 58.0

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

   9. Applied egg-rr 58.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+171}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{\mathsf{fma}\left(y, 2, x\right)}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+170)
   (/ (/ y x) (+ y x))
   (if (<= x -4.7e-79)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (/ (* x (/ 1.0 y)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+170) {
		tmp = (y / x) / (y + x);
	} else if (x <= -4.7e-79) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x * (1.0 / y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d+170)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-4.7d-79)) then
        tmp = y / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x * (1.0d0 / y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+170) {
		tmp = (y / x) / (y + x);
	} else if (x <= -4.7e-79) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x * (1.0 / y)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e+170:
		tmp = (y / x) / (y + x)
	elif x <= -4.7e-79:
		tmp = y / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x * (1.0 / y)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+170)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -4.7e-79)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x * Float64(1.0 / y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6e+170)
		tmp = (y / x) / (y + x);
	elseif (x <= -4.7e-79)
		tmp = y / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x * (1.0 / y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+170], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.7e-79], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.99999999999999994e170

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 87.0

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

   7. Simplified 87.0%

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

   if -5.99999999999999994e170 < x < -4.7000000000000002e-79

   1. Initial program 77.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in y around 0

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

   9. Simplified 56.9%

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

       1. associate-/l/ N/A

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

       2. /-lowering-/.f64 N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. +-lowering-+.f64 N/A

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

       6. +-lowering-+.f64 N/A

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

       7. *-lft-identity N/A

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

       8. +-lowering-+.f64 84.0

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

   11. Applied egg-rr 84.0%

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

   if -4.7000000000000002e-79 < x

   1. Initial program 73.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. +-lowering-+.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. +-lowering-+.f64 84.4

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

   4. Applied egg-rr 84.4%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. +-commutative N/A

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

      4. +-lowering-+.f64 57.4

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

   7. Simplified 57.4%

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

      1. associate-/r* N/A

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

      2. associate-*l/ N/A

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

      3. /-lowering-/.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. +-lowering-+.f64 58.0

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

   9. Applied egg-rr 58.0%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{1}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.18e+173)
   (/ (/ y x) (+ y x))
   (if (<= x -6.2e-79)
     (/ y (* (+ y x) (+ y (+ x 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.18e+173) {
		tmp = (y / x) / (y + x);
	} else if (x <= -6.2e-79) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.18d+173)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-6.2d-79)) then
        tmp = y / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.18e+173) {
		tmp = (y / x) / (y + x);
	} else if (x <= -6.2e-79) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.18e+173:
		tmp = (y / x) / (y + x)
	elif x <= -6.2e-79:
		tmp = y / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.18e+173)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -6.2e-79)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.18e+173)
		tmp = (y / x) / (y + x);
	elseif (x <= -6.2e-79)
		tmp = y / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.18e+173], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-79], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.18e173

   1. Initial program 50.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 87.0

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

   7. Simplified 87.0%

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

   if -1.18e173 < x < -6.1999999999999999e-79

   1. Initial program 77.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

      5. /-lowering-/.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

      3. clear-num N/A

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

      4. frac-times N/A

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

      5. *-lft-identity N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-+l+ N/A

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

      10. +-lowering-+.f64 N/A

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

      11. +-lowering-+.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. /-lowering-/.f64 N/A

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 N/A

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

      16. +-commutative N/A

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

      17. +-lowering-+.f64 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   8. Step-by-step derivation

   9. Simplified 56.9%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

       8. +-lowering-+.f64 84.0

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

   11. Applied egg-rr 84.0%

       \[\leadsto \color{blue}{\frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}} \]

   if -6.1999999999999999e-79 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.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. +-lowering-+.f64 58.4

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   7. Simplified 58.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.1e+170)
   (/ (/ y x) (+ y x))
   (if (<= x -2.2e-79) (/ y (* (+ y x) (+ y (+ x 1.0)))) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.1e+170) {
		tmp = (y / x) / (y + x);
	} else if (x <= -2.2e-79) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.1e+170)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -2.2e-79)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.1e+170], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.2e-79], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.09999999999999998e170

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 87.0

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   7. Simplified 87.0%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x + y} \]

   if -2.09999999999999998e170 < x < -2.1999999999999999e-79

   1. Initial program 77.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{x}}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      9. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{\frac{x + y}{x}} \cdot \left(x + y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{x} \cdot \left(x + y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{x} \cdot \left(x + y\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \color{blue}{\left(y + x\right)}} \]

      17. +-lowering-+.f64 98.5

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \left(y + x\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   8. Step-by-step derivation

   9. Simplified 56.9%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

       8. +-lowering-+.f64 84.0

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

   11. Applied egg-rr 84.0%

       \[\leadsto \color{blue}{\frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}} \]

   if -2.1999999999999999e-79 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 57.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{+170}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-79}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e+172)
   (/ (/ y x) x)
   (if (<= x -7e-81) (/ y (* (+ y x) (+ y (+ x 1.0)))) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e+172) {
		tmp = (y / x) / x;
	} else if (x <= -7e-81) {
		tmp = y / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e+172)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -7e-81)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e+172], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -7e-81], N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.49999999999999977e172

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 72.9

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. /-lowering-/.f64 86.8

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -3.49999999999999977e172 < x < -6.99999999999999973e-81

   1. Initial program 77.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}}}{x + y} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x}{x + y} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y}} \]

      3. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{x + y}{x}}} \cdot \frac{\frac{y}{\left(x + y\right) + 1}}{x + y} \]

      4. frac-times N/A

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{\left(x + y\right) + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      5. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + x\right)} + 1}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      9. associate-+l+ N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \color{blue}{\left(x + 1\right)}}}{\frac{x + y}{x} \cdot \left(x + y\right)} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{\frac{x + y}{x} \cdot \left(x + y\right)}} \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{\frac{x + y}{x}} \cdot \left(x + y\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{x} \cdot \left(x + y\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{\color{blue}{y + x}}{x} \cdot \left(x + y\right)} \]

      16. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \color{blue}{\left(y + x\right)}} \]

      17. +-lowering-+.f64 98.5

          \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \color{blue}{\left(y + x\right)}} \]

   6. Applied egg-rr 98.5%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(x + 1\right)}}{\frac{y + x}{x} \cdot \left(y + x\right)}} \]

   7. Taylor expanded in y around 0

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   8. Step-by-step derivation

   9. Simplified 56.9%

      \[\leadsto \frac{\frac{y}{y + \left(x + 1\right)}}{\color{blue}{1} \cdot \left(y + x\right)} \]
   10. Step-by-step derivation

       1. associate-/l/ N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y}{\left(1 \cdot \left(y + x\right)\right) \cdot \left(y + \left(x + 1\right)\right)}} \]

       3. *-commutative N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right) \cdot \left(1 \cdot \left(y + x\right)\right)}} \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \frac{y}{\left(y + \color{blue}{\left(x + 1\right)}\right) \cdot \left(1 \cdot \left(y + x\right)\right)} \]

       7. *-lft-identity N/A

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

       8. +-lowering-+.f64 84.0

          \[\leadsto \frac{y}{\left(y + \left(x + 1\right)\right) \cdot \color{blue}{\left(y + x\right)}} \]

   11. Applied egg-rr 84.0%

       \[\leadsto \color{blue}{\frac{y}{\left(y + \left(x + 1\right)\right) \cdot \left(y + x\right)}} \]

   if -6.99999999999999973e-81 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 57.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{-212}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.9e-212)
   (/ y (* x x))
   (if (<= y 2.25e-161) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.9e-212) {
		tmp = y / (x * x);
	} else if (y <= 2.25e-161) {
		tmp = y / x;
	} else 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 <= (-2.9d-212)) then
        tmp = y / (x * x)
    else if (y <= 2.25d-161) then
        tmp = y / x
    else 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 <= -2.9e-212) {
		tmp = y / (x * x);
	} else if (y <= 2.25e-161) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.9e-212:
		tmp = y / (x * x)
	elif y <= 2.25e-161:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.9e-212)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 2.25e-161)
		tmp = Float64(y / x);
	elseif (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 <= -2.9e-212)
		tmp = y / (x * x);
	elseif (y <= 2.25e-161)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.9e-212], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e-161], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.8999999999999999e-212

   1. Initial program 73.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 35.7

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 35.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -2.8999999999999999e-212 < y < 2.2499999999999998e-161

   1. Initial program 72.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{2} \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + x\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot \color{blue}{\left(x + 1\right)}\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x + x \cdot 1\right)}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot x + \color{blue}{x}\right)} \]

      7. accelerator-lowering-fma.f64 72.1

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 72.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \mathsf{fma}\left(x, x, x\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 88.3

         \[\leadsto \color{blue}{\frac{y}{x}} \]

   8. Simplified 88.3%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 2.2499999999999998e-161 < y < 1

   1. Initial program 86.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 93.1

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot \left(1 + y\right)}} \cdot x \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

      4. +-lowering-+.f64 46.5

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   7. Simplified 46.5%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(y + 1\right)}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 45.7

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 45.7%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 74.4

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+175}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.65e-161)
   (/ y (fma x x x))
   (if (<= y 2.9e+175) (/ x (* (+ y x) (+ y 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.65e-161) {
		tmp = y / fma(x, x, x);
	} else if (y <= 2.9e+175) {
		tmp = x / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.65e-161)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 2.9e+175)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.65e-161], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.9e+175], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.65000000000000014e-161

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 2.65000000000000014e-161 < y < 2.9e175

   1. Initial program 76.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{\frac{x}{x + y}}{x + y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}}{y}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      11. clear-num N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\frac{1}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}}}} \]

      12. associate-/l/ N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \frac{1}{\color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}}} \]

      13. clear-num N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\frac{x + y}{\frac{y}{\left(x + y\right) + 1}}}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}}} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}} \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}} \]

      17. associate-/r/ N/A

          \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\frac{x + y}{y} \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   6. Applied egg-rr 89.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]

      2. +-lowering-+.f64 59.8

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 59.8%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   if 2.9e175 < y

   1. Initial program 59.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 86.1

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 86.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. /-lowering-/.f64 84.4

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 84.4%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 60.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.55e+173)
   (/ (/ y x) x)
   (if (<= x -2.6e-78) (/ y (fma x x x)) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.55e+173) {
		tmp = (y / x) / x;
	} else if (x <= -2.6e-78) {
		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 <= -1.55e+173)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.6e-78)
		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, -1.55e+173], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.6e-78], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.55e173

   1. Initial program 50.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 72.9

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

      3. /-lowering-/.f64 86.8

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 86.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -1.55e173 < x < -2.6000000000000001e-78

   1. Initial program 77.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 55.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 55.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.6000000000000001e-78 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 57.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-161) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-161) {
		tmp = y / x;
	} else 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 <= 2.7d-161) then
        tmp = y / x
    else 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 <= 2.7e-161) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e-161:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-161)
		tmp = Float64(y / x);
	elseif (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 <= 2.7e-161)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-161], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6999999999999999e-161

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{2} \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + x\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot \color{blue}{\left(x + 1\right)}\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x + x \cdot 1\right)}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot x + \color{blue}{x}\right)} \]

      7. accelerator-lowering-fma.f64 43.5

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 43.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \mathsf{fma}\left(x, x, x\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 35.2

         \[\leadsto \color{blue}{\frac{y}{x}} \]

   8. Simplified 35.2%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if 2.6999999999999999e-161 < y < 1

   1. Initial program 86.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 93.1

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 93.1%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot \left(1 + y\right)}} \cdot x \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

      4. +-lowering-+.f64 46.5

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   7. Simplified 46.5%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(y + 1\right)}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 45.7

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 45.7%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 62.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 74.4

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 74.4%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 58.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-161) (/ y (fma x x x)) (/ x (* (+ y x) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-161) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / ((y + x) * (y + 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-161)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-161], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.6999999999999999e-161

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 58.2

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 2.6999999999999999e-161 < y

   1. Initial program 70.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{\frac{x}{x + y}}{x + y}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{x + y} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      3. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot \frac{y}{\left(x + y\right) + 1} \]

      4. times-frac N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. associate-*r/ N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. clear-num N/A

         \[\leadsto x \cdot \color{blue}{\frac{1}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      7. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      9. associate-*l* N/A

         \[\leadsto \frac{x}{\frac{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}}{y}} \]

      10. associate-/l* N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \frac{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}{y}}} \]

      11. clear-num N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\frac{1}{\frac{y}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}}}} \]

      12. associate-/l/ N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \frac{1}{\color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{x + y}}}} \]

      13. clear-num N/A

          \[\leadsto \frac{x}{\left(x + y\right) \cdot \color{blue}{\frac{x + y}{\frac{y}{\left(x + y\right) + 1}}}} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(x + y\right) \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}}} \]

      15. +-commutative N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}} \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot \frac{x + y}{\frac{y}{\left(x + y\right) + 1}}} \]

      17. associate-/r/ N/A

          \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(\frac{x + y}{y} \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   6. Applied egg-rr 88.5%

      \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) \cdot \left(\frac{y + x}{y} \cdot \left(y + \left(x + 1\right)\right)\right)}} \]

   7. Taylor expanded in x around 0

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(1 + y\right)}} \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]

      2. +-lowering-+.f64 68.6

         \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]

   9. Simplified 68.6%

      \[\leadsto \frac{x}{\left(y + x\right) \cdot \color{blue}{\left(y + 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 60.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-78) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-78) {
		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 <= -2.6e-78)
		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, -2.6e-78], 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 < -2.6000000000000001e-78

   1. Initial program 69.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 61.0

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 61.0%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -2.6000000000000001e-78 < x

   1. Initial program 73.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 57.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.9) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.9) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.9)
		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, -2.9], 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 < -2.89999999999999991

   1. Initial program 62.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 64.3

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -2.89999999999999991 < x

   1. Initial program 75.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.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. accelerator-lowering-fma.f64 58.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 34.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -2.85e-95) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.85e-95) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.85d-95)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.85e-95) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.85e-95:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.85e-95)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.85e-95)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.85e-95], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e-95

   1. Initial program 69.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{2} \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x \cdot x\right)} \cdot \left(1 + x\right)} \]

      2. associate-*l* N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \left(x \cdot \left(1 + x\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot \color{blue}{\left(x + 1\right)}\right)} \]

      5. distribute-lft-in N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\left(x \cdot x + x \cdot 1\right)}} \]

      6. *-rgt-identity N/A

         \[\leadsto \frac{x \cdot y}{x \cdot \left(x \cdot x + \color{blue}{x}\right)} \]

      7. accelerator-lowering-fma.f64 49.5

         \[\leadsto \frac{x \cdot y}{x \cdot \color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 49.5%

      \[\leadsto \frac{x \cdot y}{\color{blue}{x \cdot \mathsf{fma}\left(x, x, x\right)}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x}} \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 25.0

         \[\leadsto \color{blue}{\frac{y}{x}} \]

   8. Simplified 25.0%

      \[\leadsto \color{blue}{\frac{y}{x}} \]

   if -2.85e-95 < x

   1. Initial program 73.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 84.3

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 84.3%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot \left(1 + y\right)}} \cdot x \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

      4. +-lowering-+.f64 57.1

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   7. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(y + 1\right)}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 34.8

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 34.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 27.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.65e-15) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-15) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.65d-15)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.65e-15) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.65e-15:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.65e-15)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.65e-15)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.65e-15], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e-15

   1. Initial program 63.6%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   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. Simplified 29.9%

      \[\leadsto \frac{\color{blue}{1} \cdot \frac{x}{x + y}}{x + y} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{x}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 5.6

         \[\leadsto \color{blue}{\frac{1}{x}} \]

   10. Simplified 5.6%

       \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -1.65e-15 < x

   1. Initial program 75.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      10. +-lowering-+.f64 85.2

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   4. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{y \cdot \left(1 + y\right)}} \cdot x \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{y \cdot \left(1 + y\right)}} \cdot x \]

      3. +-commutative N/A

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

      4. +-lowering-+.f64 59.1

         \[\leadsto \frac{1}{y \cdot \color{blue}{\left(y + 1\right)}} \cdot x \]

   7. Simplified 59.1%

      \[\leadsto \color{blue}{\frac{1}{y \cdot \left(y + 1\right)}} \cdot x \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 33.6

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 33.6%

       \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 4.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 71.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.8

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

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. Simplified 41.6%

   \[\leadsto \frac{\color{blue}{1} \cdot \frac{x}{x + y}}{x + y} \]

8. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.3

      \[\leadsto \color{blue}{\frac{1}{x}} \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. 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 2024195 
(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))))