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

Percentage Accurate: 68.2% → 99.8%

Time: 7.9s

Alternatives: 16

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: 68.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-/l/ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f64 N/A

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

   8. times-frac N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1 \cdot y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq 1850000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-160)
   (/ (* 1.0 y) (* (+ 1.0 (+ x y)) (+ x y)))
   (if (<= x 1850000000000.0)
     (/ x (fma y y y))
     (* (/ x (+ y x)) (pow y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-160) {
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	} else if (x <= 1850000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / (y + x)) * pow(y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-160)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y)));
	elseif (x <= 1850000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (y ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-160], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1850000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.70000000000000011e-160

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 75.1

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 75.1

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

   9. Applied rewrites 75.1%

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

   if -1.70000000000000011e-160 < x < 1.85e12

   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 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 83.7

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

   5. Applied rewrites 83.7%

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

   if 1.85e12 < x

   1. Initial program 66.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 27.8

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

   7. Applied rewrites 27.8%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1 \cdot y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq 1850000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1850000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-92)
   (/ y (fma x x x))
   (if (<= x 1850000000000.0)
     (/ x (fma y y y))
     (* (/ x (+ y x)) (pow y -1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-92) {
		tmp = y / fma(x, x, x);
	} else if (x <= 1850000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / (y + x)) * pow(y, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-92)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 1850000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (y ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.1e-92], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1850000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[y, -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1000000000000002e-92

   1. Initial program 68.1%

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

   3. 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 72.1

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

   5. Applied rewrites 72.1%

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

   if -4.1000000000000002e-92 < x < 1.85e12

   1. Initial program 70.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 83.3

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

   5. Applied rewrites 83.3%

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

   if 1.85e12 < x

   1. Initial program 66.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 27.8

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

   7. Applied rewrites 27.8%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 1850000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {y}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1 \cdot y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-160)
   (/ (* 1.0 y) (* (+ 1.0 (+ x y)) (+ x y)))
   (* (/ x (+ y x)) (pow (+ 1.0 y) -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-160) {
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	} else {
		tmp = (x / (y + x)) * pow((1.0 + y), -1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-160)) then
        tmp = (1.0d0 * y) / ((1.0d0 + (x + y)) * (x + y))
    else
        tmp = (x / (y + x)) * ((1.0d0 + y) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-160) {
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	} else {
		tmp = (x / (y + x)) * Math.pow((1.0 + y), -1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-160:
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y))
	else:
		tmp = (x / (y + x)) * math.pow((1.0 + y), -1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-160)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) * (Float64(1.0 + y) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-160)
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	else
		tmp = (x / (y + x)) * ((1.0 + y) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-160], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[Power[N[(1.0 + y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000011e-160

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 75.1

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 75.1

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

   9. Applied rewrites 75.1%

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

   if -1.70000000000000011e-160 < x

   1. Initial program 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 63.6

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

   7. Applied rewrites 63.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1 \cdot y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + x} \cdot {\left(1 + y\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(1 + \left(y + x\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x - x \cdot \frac{\mathsf{fma}\left(3, x, 1\right)}{y}}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.2e+144)
   (* (/ y (+ y x)) (/ x (* (+ 1.0 (+ y x)) (+ y x))))
   (/ (/ (- x (* x (/ (fma 3.0 x 1.0) y))) y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.2e+144) {
		tmp = (y / (y + x)) * (x / ((1.0 + (y + x)) * (y + x)));
	} else {
		tmp = ((x - (x * (fma(3.0, x, 1.0) / y))) / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.2e+144)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(1.0 + Float64(y + x)) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x - Float64(x * Float64(fma(3.0, x, 1.0) / y))) / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.2e+144], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - N[(x * N[(N[(3.0 * x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999998e144

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.1

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 96.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 5.1999999999999998e144 < y

   1. Initial program 55.7%

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

   3. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 90.5%

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

Alternative 6: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-160}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+96}:\\ \;\;\;\;\frac{x \cdot y}{\left(y + x\right) \cdot \left(t\_0 \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y 2.65e-160)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= y 1.4e+96)
       (/ (* x y) (* (+ y x) (* t_0 (+ y x))))
       (* (/ (/ x (+ y x)) t_0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 2.65e-160) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 1.4e+96) {
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 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 = 1.0d0 + (y + x)
    if (y <= 2.65d-160) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (y <= 1.4d+96) then
        tmp = (x * y) / ((y + x) * (t_0 * (y + x)))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 2.65e-160) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 1.4e+96) {
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= 2.65e-160:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif y <= 1.4e+96:
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= 2.65e-160)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (y <= 1.4e+96)
		tmp = Float64(Float64(x * y) / Float64(Float64(y + x) * Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= 2.65e-160)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (y <= 1.4e+96)
		tmp = (x * y) / ((y + x) * (t_0 * (y + x)));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.65e-160], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+96], N[(N[(x * y), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6500000000000001e-160

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 58.1%

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

   if 2.6500000000000001e-160 < y < 1.4e96

   1. Initial program 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 85.0

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 85.0

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 85.0

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 85.0

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

   4. Applied rewrites 85.0%

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

   if 1.4e96 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.8%

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

Alternative 7: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq 2.65 \cdot 10^{-160}:\\ \;\;\;\;1 \cdot \frac{\frac{y}{t\_0}}{y + x}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+96}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y 2.65e-160)
     (* 1.0 (/ (/ y t_0) (+ y x)))
     (if (<= y 1.4e+96)
       (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0)))
       (* (/ (/ x (+ y x)) t_0) 1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 2.65e-160) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 1.4e+96) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / t_0) * 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 = 1.0d0 + (y + x)
    if (y <= 2.65d-160) then
        tmp = 1.0d0 * ((y / t_0) / (y + x))
    else if (y <= 1.4d+96) then
        tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 2.65e-160) {
		tmp = 1.0 * ((y / t_0) / (y + x));
	} else if (y <= 1.4e+96) {
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= 2.65e-160:
		tmp = 1.0 * ((y / t_0) / (y + x))
	elif y <= 1.4e+96:
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= 2.65e-160)
		tmp = Float64(1.0 * Float64(Float64(y / t_0) / Float64(y + x)));
	elseif (y <= 1.4e+96)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= 2.65e-160)
		tmp = 1.0 * ((y / t_0) / (y + x));
	elseif (y <= 1.4e+96)
		tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.65e-160], N[(1.0 * N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+96], 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], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6500000000000001e-160

   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

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 58.1%

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

   if 2.6500000000000001e-160 < y < 1.4e96

   1. Initial program 84.9%

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

   if 1.4e96 < 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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.7

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 85.8%

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

Alternative 8: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{t\_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))))
   (if (<= y 5.2e+144)
     (* (/ y (+ y x)) (/ x (* t_0 (+ y x))))
     (* (/ (/ x (+ y x)) t_0) 1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 5.2e+144) {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 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 = 1.0d0 + (y + x)
    if (y <= 5.2d+144) then
        tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
    else
        tmp = ((x / (y + x)) / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double tmp;
	if (y <= 5.2e+144) {
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	} else {
		tmp = ((x / (y + x)) / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	tmp = 0
	if y <= 5.2e+144:
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)))
	else:
		tmp = ((x / (y + x)) / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	tmp = 0.0
	if (y <= 5.2e+144)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(t_0 * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	tmp = 0.0;
	if (y <= 5.2e+144)
		tmp = (y / (y + x)) * (x / (t_0 * (y + x)));
	else
		tmp = ((x / (y + x)) / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e+144], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999998e144

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 96.1

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 96.1

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 96.1

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

      22. lift-+.f64 N/A

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

      23. +-commutative N/A

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

      24. lower-+.f64 96.1

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

   4. Applied rewrites 96.1%

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

   if 5.1999999999999998e144 < y

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 90.1%

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

Alternative 9: 93.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left(y + x\right)\\ t_1 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{y}{t\_0 \cdot \left(y + x\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1}{t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (+ y x))) (t_1 (/ x (+ y x))))
   (if (<= y 5.2e+144) (* (/ y (* t_0 (+ y x))) t_1) (* (/ t_1 t_0) 1.0))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double t_1 = x / (y + x);
	double tmp;
	if (y <= 5.2e+144) {
		tmp = (y / (t_0 * (y + x))) * t_1;
	} else {
		tmp = (t_1 / t_0) * 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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (y + x)
    t_1 = x / (y + x)
    if (y <= 5.2d+144) then
        tmp = (y / (t_0 * (y + x))) * t_1
    else
        tmp = (t_1 / t_0) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (y + x);
	double t_1 = x / (y + x);
	double tmp;
	if (y <= 5.2e+144) {
		tmp = (y / (t_0 * (y + x))) * t_1;
	} else {
		tmp = (t_1 / t_0) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (y + x)
	t_1 = x / (y + x)
	tmp = 0
	if y <= 5.2e+144:
		tmp = (y / (t_0 * (y + x))) * t_1
	else:
		tmp = (t_1 / t_0) * 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(y + x))
	t_1 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 5.2e+144)
		tmp = Float64(Float64(y / Float64(t_0 * Float64(y + x))) * t_1);
	else
		tmp = Float64(Float64(t_1 / t_0) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (y + x);
	t_1 = x / (y + x);
	tmp = 0.0;
	if (y <= 5.2e+144)
		tmp = (y / (t_0 * (y + x))) * t_1;
	else
		tmp = (t_1 / t_0) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5.2e+144], N[(N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(t$95$1 / t$95$0), $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y < 5.1999999999999998e144

   1. Initial program 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. lift-+.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. lift-+.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 N/A

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

      22. lower-/.f64 96.0

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

      23. lift-+.f64 N/A

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

      24. +-commutative N/A

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

   4. Applied rewrites 96.0%

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

   if 5.1999999999999998e144 < y

   1. Initial program 55.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.6

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.6

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.6

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 90.1%

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

Alternative 10: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

Alternative 11: 64.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-160}:\\ \;\;\;\;\frac{1 \cdot y}{\left(1 + \left(x + y\right)\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{1 + \left(y + x\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.7e-160)
   (/ (* 1.0 y) (* (+ 1.0 (+ x y)) (+ x y)))
   (* (/ (/ x (+ y x)) (+ 1.0 (+ y x))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-160) {
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	} else {
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.7d-160)) then
        tmp = (1.0d0 * y) / ((1.0d0 + (x + y)) * (x + y))
    else
        tmp = ((x / (y + x)) / (1.0d0 + (y + x))) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.7e-160) {
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	} else {
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.7e-160:
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y))
	else:
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.7e-160)
		tmp = Float64(Float64(1.0 * y) / Float64(Float64(1.0 + Float64(x + y)) * Float64(x + y)));
	else
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(1.0 + Float64(y + x))) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.7e-160)
		tmp = (1.0 * y) / ((1.0 + (x + y)) * (x + y));
	else
		tmp = ((x / (y + x)) / (1.0 + (y + x))) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.7e-160], N[(N[(1.0 * y), $MachinePrecision] / N[(N[(1.0 + N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.70000000000000011e-160

   1. Initial program 70.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.8

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 68.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 75.1

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 75.1

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

      14. lift-+.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 75.1

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

   9. Applied rewrites 75.1%

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

   if -1.70000000000000011e-160 < x

   1. Initial program 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift-*.f64 N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-/.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around 0

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

   9. Applied rewrites 64.4%

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

Alternative 12: 59.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;x \leq 2.06 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-92)
   (/ y (fma x x x))
   (if (<= x 2.06e-78) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-92) {
		tmp = y / fma(x, x, x);
	} else if (x <= 2.06e-78) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-92)
		tmp = Float64(y / fma(x, x, x));
	elseif (x <= 2.06e-78)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.1e-92], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.06e-78], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1000000000000002e-92

   1. Initial program 68.1%

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

   3. 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 72.1

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

   5. Applied rewrites 72.1%

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

   if -4.1000000000000002e-92 < x < 2.06000000000000008e-78

   1. Initial program 67.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if 2.06000000000000008e-78 < x

   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. Taylor expanded in y around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 37.1

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

   5. Applied rewrites 37.1%

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

Alternative 13: 59.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-92) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-92) {
		tmp = y / fma(x, x, x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-92)
		tmp = Float64(y / fma(x, x, x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.1e-92], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1000000000000002e-92

   1. Initial program 68.1%

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

   3. 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 72.1

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

   5. Applied rewrites 72.1%

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

   if -4.1000000000000002e-92 < x

   1. Initial program 69.3%

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

   3. Taylor expanded in 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 65.5

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

   5. Applied rewrites 65.5%

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

Alternative 14: 60.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -950000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -950000.0) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -950000.0) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -950000.0)
		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, -950000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e5

   1. Initial program 63.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 73.1%

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

   if -9.5e5 < x

   1. Initial program 71.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 62.6

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

   5. Applied rewrites 62.6%

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

Alternative 15: 47.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.07) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.07) {
		tmp = y / (x * x);
	} 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 (x <= (-0.07d0)) then
        tmp = y / (x * x)
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.07) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.07:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.07)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.07)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.07], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.070000000000000007

   1. Initial program 63.1%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 73.1%

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

   if -0.070000000000000007 < x

   1. Initial program 71.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 99.9

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

      15. lift-+.f64 N/A

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

      16. +-commutative N/A

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

      17. lower-+.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lower-+.f64 99.9

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

      21. lift-+.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   6. 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 40.5

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

   7. Applied rewrites 40.5%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.07:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 36.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lift-+.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-/.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 99.8

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

   18. lift-+.f64 N/A

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

   19. +-commutative N/A

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

   20. lower-+.f64 99.8

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

   21. lift-+.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
6. 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 33.9

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

7. Applied rewrites 33.9%

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

8. Final simplification 33.9%

   \[\leadsto \frac{x}{y \cdot y} \]
9. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024320 
(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))))