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

Percentage Accurate: 68.8% → 99.8%

Time: 15.1s

Alternatives: 21

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: 68.8% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, \frac{x}{y}, 1\right), 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -6.7e+155)
     (/ (/ y t_0) (+ y x))
     (if (<= x 2.5e-11)
       (* (/ y (+ y x)) (/ x (* t_0 (+ y x))))
       (/ (/ x (+ y x)) (fma y (fma 2.0 (/ x y) 1.0) 1.0))))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= 2.5e-11) {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	} else {
		tmp = (x / (y + x)) / fma(y, fma(2.0, (x / y), 1.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= 2.5e-11)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / fma(y, fma(2.0, Float64(x / y), 1.0), 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.7e+155], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.5e-11], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y * N[(2.0 * N[(x / y), $MachinePrecision] + 1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 93.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.4

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

   9. Applied egg-rr 93.4%

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

   if -6.7e155 < x < 2.50000000000000009e-11

   1. Initial program 68.0%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 97.0

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lower-+.f64 N/A

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

      20. lower-+.f64 97.0

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

   4. Applied egg-rr 97.0%

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

   if 2.50000000000000009e-11 < x

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. clear-num N/A

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

      10. frac-times N/A

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

      11. *-lft-identity N/A

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

      12. lower-/.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 98.6

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

      18. lift-+.f64 N/A

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

   6. Applied egg-rr 98.6%

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

   7. Taylor expanded in y around inf

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

      1. associate-+r+ N/A

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

      2. distribute-lft-in N/A

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

      3. rgt-mult-inverse N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 52.4

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

   9. Simplified 52.4%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\mathsf{fma}\left(y, \mathsf{fma}\left(2, \frac{x}{y}, 1\right), 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{1}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -6.7e+155)
     (/ (/ y t_0) (+ y x))
     (if (<= x -4e+18)
       (* y (/ 1.0 (* t_0 (+ y x))))
       (/ (* x (/ y (+ y x))) (* (+ y x) (+ y 1.0)))))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -4e+18) {
		tmp = y * (1.0 / (t_0 * (y + x)));
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-6.7d+155)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-4d+18)) then
        tmp = y * (1.0d0 / (t_0 * (y + x)))
    else
        tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -4e+18) {
		tmp = y * (1.0 / (t_0 * (y + x)));
	} else {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -6.7e+155:
		tmp = (y / t_0) / (y + x)
	elif x <= -4e+18:
		tmp = y * (1.0 / (t_0 * (y + x)))
	else:
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -4e+18)
		tmp = Float64(y * Float64(1.0 / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -6.7e+155)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -4e+18)
		tmp = y * (1.0 / (t_0 * (y + x)));
	else
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.7e+155], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4e+18], N[(y * N[(1.0 / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 93.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.4

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

   9. Applied egg-rr 93.4%

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

   if -6.7e155 < x < -4e18

   1. Initial program 42.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 42.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-+.f64 N/A

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

      15. clear-num N/A

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

      16. /-rgt-identity N/A

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

      17. lower-/.f64 N/A

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 64.8

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

   9. Applied egg-rr 64.8%

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

   if -4e18 < x

   1. Initial program 72.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 95.7

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

      15. lift-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. associate-+l+ N/A

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

      18. +-commutative N/A

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

      19. associate-+l+ N/A

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

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-+.f64 83.7

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

   7. Simplified 83.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{1}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-179}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -5.6e+102)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.7e-179)
       (* y (/ x (* t_0 (* (+ y x) (+ y x)))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -5.6e+102) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.7e-179) {
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-5.6d+102)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.7d-179)) then
        tmp = y * (x / (t_0 * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -5.6e+102) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.7e-179) {
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -5.6e+102:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.7e-179:
		tmp = y * (x / (t_0 * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -5.6e+102)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.7e-179)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -5.6e+102)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.7e-179)
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+102], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-179], N[(y * N[(x / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000037e102

   1. Initial program 48.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 80.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 80.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 80.6

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

   9. Applied egg-rr 80.6%

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

   if -5.60000000000000037e102 < x < -1.6999999999999999e-179

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 87.5

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. associate-+l+ N/A

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

      16. lower-+.f64 N/A

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

      17. lower-+.f64 87.5

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

   4. Applied egg-rr 87.5%

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

   if -1.6999999999999999e-179 < x

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 57.9

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

   7. Simplified 57.9%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-179}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + \left(1 + x\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{t\_0 \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -6.7e+155)
     (/ (/ y t_0) (+ y x))
     (* (/ y (+ y x)) (/ x (* t_0 (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-6.7d+155)) then
        tmp = (y / t_0) / (y + x)
    else
        tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -6.7e+155:
		tmp = (y / t_0) / (y + x)
	else:
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(t_0 * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -6.7e+155)
		tmp = (y / t_0) / (y + x);
	else
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.7e+155], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 93.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.4

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

   9. Applied egg-rr 93.4%

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

   if -6.7e155 < x

   1. Initial program 68.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 94.2

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

      14. lift-+.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. associate-+l+ N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lower-+.f64 N/A

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

      20. lower-+.f64 94.2

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

   4. Applied egg-rr 94.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-/r* N/A

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

   10. associate-*r/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied egg-rr 99.9%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. frac-2neg N/A

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

   4. lift-+.f64 N/A

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

   5. frac-2neg N/A

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

   6. frac-times N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. frac-times N/A

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

   10. clear-num N/A

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

   11. frac-times N/A

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

   12. *-rgt-identity N/A

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

   13. clear-num N/A

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

   14. lift-/.f64 N/A

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

   15. div-inv N/A

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

   16. div-inv N/A

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

   17. clear-num N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 7: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. times-frac N/A

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

   7. associate-*l/ N/A

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

   8. lift-*.f64 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 8: 66.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-166)
   (/ (/ y (+ y (+ 1.0 x))) (+ y x))
   (if (<= y 8.5e+25)
     (* x (/ y (* (+ 1.0 x) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y x)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-166) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (y <= 8.5e+25) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) / (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 (y <= 2.4d-166) then
        tmp = (y / (y + (1.0d0 + x))) / (y + x)
    else if (y <= 8.5d+25) then
        tmp = x * (y / ((1.0d0 + x) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + x)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-166) {
		tmp = (y / (y + (1.0 + x))) / (y + x);
	} else if (y <= 8.5e+25) {
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + x)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-166:
		tmp = (y / (y + (1.0 + x))) / (y + x)
	elif y <= 8.5e+25:
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + x)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-166)
		tmp = Float64(Float64(y / Float64(y + Float64(1.0 + x))) / Float64(y + x));
	elseif (y <= 8.5e+25)
		tmp = Float64(x * Float64(y / Float64(Float64(1.0 + x) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-166)
		tmp = (y / (y + (1.0 + x))) / (y + x);
	elseif (y <= 8.5e+25)
		tmp = x * (y / ((1.0 + x) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + x)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-166], N[(N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+25], N[(x * N[(y / N[(N[(1.0 + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.3999999999999999e-166

   1. Initial program 68.6%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 54.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 54.9

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 54.9

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

   9. Applied egg-rr 54.9%

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

   if 2.3999999999999999e-166 < y < 8.5000000000000007e25

   1. Initial program 82.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 91.3

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. associate-+l+ N/A

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

      16. lower-+.f64 N/A

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

      17. lower-+.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   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. lower-+.f64 89.7

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

   7. Simplified 89.7%

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

   if 8.5000000000000007e25 < y

   1. Initial program 56.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 80.6%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-166}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;x \cdot \frac{y}{\left(1 + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.5)
   (/ (/ y x) x)
   (if (<= y 3.1e-174)
     (/ y (fma x x x))
     (if (<= y 4e+73) (/ x (fma y y y)) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.5) {
		tmp = (y / x) / x;
	} else if (y <= 3.1e-174) {
		tmp = y / fma(x, x, x);
	} else if (y <= 4e+73) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.5)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 3.1e-174)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 4e+73)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.5], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 3.1e-174], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+73], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.5

   1. Initial program 68.4%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 10.7

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

   5. Simplified 10.7%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. *-inverses N/A

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

      4. *-lft-identity N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 19.8

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

   7. Applied egg-rr 19.8%

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

   if -2.5 < y < 3.0999999999999999e-174

   1. Initial program 68.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 79.4

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

   5. Simplified 79.4%

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

   if 3.0999999999999999e-174 < y < 3.99999999999999993e73

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 49.4

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

   5. Simplified 49.4%

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

   if 3.99999999999999993e73 < y

   1. Initial program 52.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.9

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

   5. Simplified 74.9%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 80.2

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

   7. Applied egg-rr 80.2%

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

Alternative 10: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= x -6.7e+155)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.12e-117)
       (/ y (* t_0 (+ y x)))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.12e-117) {
		tmp = y / (t_0 * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (x <= (-6.7d+155)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.12d-117)) then
        tmp = y / (t_0 * (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.12e-117) {
		tmp = y / (t_0 * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if x <= -6.7e+155:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.12e-117:
		tmp = y / (t_0 * (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.12e-117)
		tmp = Float64(y / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (x <= -6.7e+155)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.12e-117)
		tmp = y / (t_0 * (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.7e+155], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.12e-117], N[(y / N[(t$95$0 * N[(y + x), $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 < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 93.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-+.f64 N/A

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

      6. lower-/.f64 93.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 93.4

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

   9. Applied egg-rr 93.4%

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

   if -6.7e155 < x < -1.12e-117

   1. Initial program 56.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 45.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.12e-117 < x

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.0

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

   7. Simplified 60.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.7e+155)
   (/ (/ y x) (+ y x))
   (if (<= x -1.12e-117)
     (/ y (* (+ y (+ 1.0 x)) (+ y x)))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.12e-117) {
		tmp = y / ((y + (1.0 + x)) * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6.7d+155)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1.12d-117)) then
        tmp = y / ((y + (1.0d0 + x)) * (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.12e-117) {
		tmp = y / ((y + (1.0 + x)) * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.7e+155:
		tmp = (y / x) / (y + x)
	elif x <= -1.12e-117:
		tmp = y / ((y + (1.0 + x)) * (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1.12e-117)
		tmp = Float64(y / Float64(Float64(y + Float64(1.0 + x)) * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.7e+155)
		tmp = (y / x) / (y + x);
	elseif (x <= -1.12e-117)
		tmp = y / ((y + (1.0 + x)) * (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.7e+155], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.12e-117], N[(y / N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $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 < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.0

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

   7. Simplified 93.0%

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

   if -6.7e155 < x < -1.12e-117

   1. Initial program 56.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 45.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.12e-117 < x

   1. Initial program 72.2%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 60.0

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

   7. Simplified 60.0%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.7e+155)
   (/ (/ y x) (+ y x))
   (if (<= x -1.12e-117) (/ y (* (+ y (+ 1.0 x)) (+ y x))) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.12e-117) {
		tmp = y / ((y + (1.0 + x)) * (y + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1.12e-117)
		tmp = Float64(y / Float64(Float64(y + Float64(1.0 + x)) * Float64(y + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.7e+155], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.12e-117], N[(y / N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7e155

   1. Initial program 60.1%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 93.0

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

   7. Simplified 93.0%

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

   if -6.7e155 < x < -1.12e-117

   1. Initial program 56.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 45.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.12e-117 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 58.3

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

   5. Simplified 58.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6.7e+155)
   (/ (/ y x) x)
   (if (<= x -1.12e-117) (/ y (* (+ y (+ 1.0 x)) (+ y x))) (/ x (fma y y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.7e+155) {
		tmp = (y / x) / x;
	} else if (x <= -1.12e-117) {
		tmp = y / ((y + (1.0 + x)) * (y + x));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.7e+155)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.12e-117)
		tmp = Float64(y / Float64(Float64(y + Float64(1.0 + x)) * Float64(y + x)));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.7e+155], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.12e-117], N[(y / N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.7e155

   1. Initial program 60.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{{x}^{3}}} \]
   4. Step-by-step derivation

      1. cube-mult N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 60.1

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

   5. Simplified 60.1%

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

      1. lift-*.f64 N/A

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

      2. times-frac N/A

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

      3. *-inverses N/A

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

      4. *-lft-identity N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 92.9

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

   7. Applied egg-rr 92.9%

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

   if -6.7e155 < x < -1.12e-117

   1. Initial program 56.9%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. times-frac N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/r* N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 45.4%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 N/A

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

      8. associate-/l/ N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 66.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 66.8

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

   9. Applied egg-rr 66.8%

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

   if -1.12e-117 < x

   1. Initial program 72.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 58.3

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

   5. Simplified 58.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.12 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y 3.1e-174)
     t_0
     (if (<= y 2.9e-13) (/ x y) (if (<= y 8.5e+25) t_0 (/ x (* y y)))))))

C

double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= 3.1e-174) {
		tmp = t_0;
	} else if (y <= 2.9e-13) {
		tmp = x / y;
	} else if (y <= 8.5e+25) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= 3.1d-174) then
        tmp = t_0
    else if (y <= 2.9d-13) then
        tmp = x / y
    else if (y <= 8.5d+25) then
        tmp = t_0
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= 3.1e-174) {
		tmp = t_0;
	} else if (y <= 2.9e-13) {
		tmp = x / y;
	} else if (y <= 8.5e+25) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= 3.1e-174:
		tmp = t_0
	elif y <= 2.9e-13:
		tmp = x / y
	elif y <= 8.5e+25:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= 3.1e-174)
		tmp = t_0;
	elseif (y <= 2.9e-13)
		tmp = Float64(x / y);
	elseif (y <= 8.5e+25)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= 3.1e-174)
		tmp = t_0;
	elseif (y <= 2.9e-13)
		tmp = x / y;
	elseif (y <= 8.5e+25)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.1e-174], t$95$0, If[LessEqual[y, 2.9e-13], N[(x / y), $MachinePrecision], If[LessEqual[y, 8.5e+25], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0999999999999999e-174 or 2.8999999999999998e-13 < y < 8.5000000000000007e25

   1. Initial program 68.8%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 39.0

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

   5. Simplified 39.0%

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

   if 3.0999999999999999e-174 < y < 2.8999999999999998e-13

   1. Initial program 84.5%

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

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 96.1

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. associate-+l+ N/A

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

      14. +-commutative N/A

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

      15. associate-+l+ N/A

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

      16. lower-+.f64 N/A

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

      17. lower-+.f64 96.1

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

   4. Applied egg-rr 96.1%

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

   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. lower-+.f64 95.6

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

   7. Simplified 95.6%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 47.8

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

   10. Simplified 47.8%

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

   if 8.5000000000000007e25 < y

   1. Initial program 56.1%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 75.4

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

   5. Simplified 75.4%

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

Alternative 15: 57.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+73}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-174)
   (/ y (fma x x x))
   (if (<= y 4e+73) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-174) {
		tmp = y / fma(x, x, x);
	} else if (y <= 4e+73) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-174)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 4e+73)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-174], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+73], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.0999999999999999e-174

   1. Initial program 68.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 53.5

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

   5. Simplified 53.5%

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

   if 3.0999999999999999e-174 < y < 3.99999999999999993e73

   1. Initial program 81.3%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 49.4

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

   5. Simplified 49.4%

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

   if 3.99999999999999993e73 < y

   1. Initial program 52.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 74.9

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

   5. Simplified 74.9%

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 80.2

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

   7. Applied egg-rr 80.2%

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

Alternative 16: 56.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-174) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-174) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-174)
		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[y, 3.1e-174], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.0999999999999999e-174

   1. Initial program 68.6%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. lower-fma.f64 53.5

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 53.5%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 3.0999999999999999e-174 < y

   1. Initial program 65.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 63.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 63.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{+24}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e+24) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e+24) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e+24)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.1e+24], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000001e24

   1. Initial program 51.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. lower-*.f64 59.7

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1.10000000000000001e24 < x

   1. Initial program 72.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. lower-fma.f64 58.1

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 37.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      7. associate-/l* N/A

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

      10. lower-/.f64 83.6

          \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

      11. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

      13. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \cdot x \]

      14. +-commutative N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \cdot x \]

      15. associate-+l+ N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      16. lower-+.f64 N/A

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

      17. lower-+.f64 83.6

          \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \cdot x \]

   4. Applied egg-rr 83.6%

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot x} \]

   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. lower-+.f64 75.0

         \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

   7. Simplified 75.0%

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   9. Step-by-step derivation

      1. lower-/.f64 26.8

         \[\leadsto \color{blue}{\frac{x}{y}} \]

   10. Simplified 26.8%

       \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 56.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. lower-*.f64 69.9

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 27.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   7. associate-/l* N/A

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \cdot x} \]

   10. lower-/.f64 80.8

       \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \cdot x \]

   11. lift-+.f64 N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \cdot x \]

   12. lift-+.f64 N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \cdot x \]

   13. associate-+l+ N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \cdot x \]

   14. +-commutative N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(y + 1\right) + x\right)}} \cdot x \]

   15. associate-+l+ N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

   16. lower-+.f64 N/A

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(y + \left(1 + x\right)\right)}} \cdot x \]

   17. lower-+.f64 80.8

       \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \color{blue}{\left(1 + x\right)}\right)} \cdot x \]

4. Applied egg-rr 80.8%

   \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(y + \left(1 + x\right)\right)} \cdot x} \]

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. lower-+.f64 69.3

      \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

7. Simplified 69.3%

   \[\leadsto \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + 1\right)}} \cdot x \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Step-by-step derivation

   1. lower-/.f64 25.6

      \[\leadsto \color{blue}{\frac{x}{y}} \]

10. Simplified 25.6%

    \[\leadsto \color{blue}{\frac{x}{y}} \]
11. Add Preprocessing

Alternative 20: 4.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 y))

C

double code(double x, double y) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y) {
	return 1.0 / y;
}

Python

def code(x, y):
	return 1.0 / y

Julia

function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{1}}{x + y} \]
6. Step-by-step derivation

7. Simplified 48.2%

   \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \color{blue}{1}}{x + y} \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{y}} \]
9. Step-by-step derivation

   1. lower-/.f64 4.2

      \[\leadsto \color{blue}{\frac{1}{y}} \]

10. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{1}{y}} \]
11. Add Preprocessing

Alternative 21: 3.5% accurate, 39.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{\left(x + y\right)} + 1\right)} \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

   6. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   7. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   11. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{x + y} \]

   2. +-commutative N/A

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

   3. lower-+.f64 54.4

      \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{x + y} \]

7. Simplified 54.4%

   \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{x + y} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1} \]
9. Step-by-step derivation

10. Simplified 3.3%

    \[\leadsto \color{blue}{1} \]
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 2024207 
(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))))