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

Percentage Accurate: 70.0% → 99.8%

Time: 13.5s

Alternatives: 19

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: 70.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.3%

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

   1. times-frac N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   12. +-lowering-+.f64 99.9

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 94.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \mathsf{fma}\left(x, 2 + \frac{1}{y}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.6e+151)
   (/ (* x (/ y (+ y x))) (* (+ y x) (+ (+ y x) 1.0)))
   (/ (/ x (+ y x)) (+ y (fma x (+ 2.0 (/ 1.0 y)) 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.6e+151) {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + fma(x, (2.0 + (1.0 / y)), 1.0));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.6e+151)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + fma(x, Float64(2.0 + Float64(1.0 / y)), 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.6e+151], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * N[(2.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.6000000000000001e151

   1. Initial program 71.5%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 97.5

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

   4. Applied egg-rr 97.5%

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

   if 7.6000000000000001e151 < y

   1. Initial program 60.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. frac-times N/A

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

      4. *-lft-identity N/A

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

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

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

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

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

      14. +-commutative N/A

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

      15. +-lowering-+.f64 100.0

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-+l+ N/A

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

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

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

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

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

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

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

      6. /-lowering-/.f64 90.3

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

   9. Simplified 90.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+151}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \mathsf{fma}\left(x, 2 + \frac{1}{y}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e-12)
   (/ (* x (/ y (* (+ y x) (+ y x)))) (+ (+ y x) 1.0))
   (if (<= x 2.4e-32)
     (/ (* x (/ y (+ y x))) (* (+ y x) (+ y 1.0)))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-12) {
		tmp = (x * (y / ((y + x) * (y + x)))) / ((y + x) + 1.0);
	} else if (x <= 2.4e-32) {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.2d-12)) then
        tmp = (x * (y / ((y + x) * (y + x)))) / ((y + x) + 1.0d0)
    else if (x <= 2.4d-32) then
        tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-12) {
		tmp = (x * (y / ((y + x) * (y + x)))) / ((y + x) + 1.0);
	} else if (x <= 2.4e-32) {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e-12:
		tmp = (x * (y / ((y + x) * (y + x)))) / ((y + x) + 1.0)
	elif x <= 2.4e-32:
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e-12)
		tmp = Float64(Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + x)))) / Float64(Float64(y + x) + 1.0));
	elseif (x <= 2.4e-32)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e-12)
		tmp = (x * (y / ((y + x) * (y + x)))) / ((y + x) + 1.0);
	elseif (x <= 2.4e-32)
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e-12], N[(N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-32], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999994e-12

   1. Initial program 69.2%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 95.8

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

   4. Applied egg-rr 95.8%

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

   if -1.19999999999999994e-12 < x < 2.4000000000000001e-32

   1. Initial program 72.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 99.9%

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

   if 2.4000000000000001e-32 < x

   1. Initial program 67.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 28.1

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

   7. Simplified 28.1%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e-9)
   (/ y (* (+ y x) (+ x (+ y 1.0))))
   (if (<= x 2.4e-32)
     (/ (* x (/ y (+ y x))) (* (+ y x) (+ y 1.0)))
     (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-9) {
		tmp = y / ((y + x) * (x + (y + 1.0)));
	} else if (x <= 2.4e-32) {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.2d-9)) then
        tmp = y / ((y + x) * (x + (y + 1.0d0)))
    else if (x <= 2.4d-32) then
        tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-9) {
		tmp = y / ((y + x) * (x + (y + 1.0)));
	} else if (x <= 2.4e-32) {
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e-9:
		tmp = y / ((y + x) * (x + (y + 1.0)))
	elif x <= 2.4e-32:
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e-9)
		tmp = Float64(y / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	elseif (x <= 2.4e-32)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.2e-9)
		tmp = y / ((y + x) * (x + (y + 1.0)));
	elseif (x <= 2.4e-32)
		tmp = (x * (y / (y + x))) / ((y + x) * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e-9], N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-32], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2e-9

   1. Initial program 69.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around inf

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

   7. Simplified 79.3%

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

      1. +-commutative N/A

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

      2. *-rgt-identity N/A

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

      3. associate-/l/ N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

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

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

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

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

      10. +-commutative N/A

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

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

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

      12. associate-+l+ N/A

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

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

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

      14. +-lowering-+.f64 90.3

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

   9. Applied egg-rr 90.3%

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

   if -1.2e-9 < x < 2.4000000000000001e-32

   1. Initial program 72.6%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.9

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

   4. Applied egg-rr 99.9%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 99.9%

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

   if 2.4000000000000001e-32 < x

   1. Initial program 67.5%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 28.1

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

   7. Simplified 28.1%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+124}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-167)
   (/ y (* x (+ x (+ y 1.0))))
   (if (<= y 4e+124)
     (* x (/ y (* (+ (+ y x) 1.0) (* (+ y x) (+ y x)))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-167) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 4e+124) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-167) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else if (y <= 4d+124) then
        tmp = x * (y / (((y + x) + 1.0d0) * ((y + x) * (y + x))))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-167) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 4e+124) {
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.8e-167:
		tmp = y / (x * (x + (y + 1.0)))
	elif y <= 4e+124:
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-167)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (y <= 4e+124)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-167)
		tmp = y / (x * (x + (y + 1.0)));
	elseif (y <= 4e+124)
		tmp = x * (y / (((y + x) + 1.0) * ((y + x) * (y + x))));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.8e-167], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e+124], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8e-167

   1. Initial program 68.7%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 57.9

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

   7. Simplified 57.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+l+ N/A

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

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

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

      7. +-lowering-+.f64 62.5

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

   9. Applied egg-rr 62.5%

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

   if 1.8e-167 < y < 3.99999999999999979e124

   1. Initial program 82.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 89.7

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

   4. Applied egg-rr 89.7%

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

   if 3.99999999999999979e124 < y

   1. Initial program 57.2%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 84.6

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

   7. Simplified 84.6%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+124}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + x\right) + 1\\ \mathbf{if}\;y \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \frac{x}{t\_0 \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (+ y x) 1.0)))
   (if (<= y 1.9e-167)
     (/ y (* x (+ x (+ y 1.0))))
     (if (<= y 4.7e+32)
       (* y (/ x (* t_0 (* (+ y x) (+ y x)))))
       (if (<= y 4.1e+155)
         (/ x (* (+ y x) t_0))
         (/ (/ x (+ y 1.0)) (+ y x)))))))

C

double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.9e-167) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 4.7e+32) {
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	} else if (y <= 4.1e+155) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) + 1.0d0
    if (y <= 1.9d-167) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else if (y <= 4.7d+32) then
        tmp = y * (x / (t_0 * ((y + x) * (y + x))))
    else if (y <= 4.1d+155) then
        tmp = x / ((y + x) * t_0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) + 1.0;
	double tmp;
	if (y <= 1.9e-167) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 4.7e+32) {
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	} else if (y <= 4.1e+155) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) + 1.0
	tmp = 0
	if y <= 1.9e-167:
		tmp = y / (x * (x + (y + 1.0)))
	elif y <= 4.7e+32:
		tmp = y * (x / (t_0 * ((y + x) * (y + x))))
	elif y <= 4.1e+155:
		tmp = x / ((y + x) * t_0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) + 1.0)
	tmp = 0.0
	if (y <= 1.9e-167)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (y <= 4.7e+32)
		tmp = Float64(y * Float64(x / Float64(t_0 * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 4.1e+155)
		tmp = Float64(x / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) + 1.0;
	tmp = 0.0;
	if (y <= 1.9e-167)
		tmp = y / (x * (x + (y + 1.0)));
	elseif (y <= 4.7e+32)
		tmp = y * (x / (t_0 * ((y + x) * (y + x))));
	elseif (y <= 4.1e+155)
		tmp = x / ((y + x) * t_0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[y, 1.9e-167], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.7e+32], N[(y * N[(x / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.1e+155], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.89999999999999984e-167

   1. Initial program 68.7%

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

      1. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 88.3

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

   4. Applied egg-rr 88.3%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 57.9

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

   7. Simplified 57.9%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+l+ N/A

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

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

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

      7. +-lowering-+.f64 62.5

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

   9. Applied egg-rr 62.5%

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

   if 1.89999999999999984e-167 < y < 4.70000000000000023e32

   1. Initial program 89.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. +-lowering-+.f64 94.5

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

   4. Applied egg-rr 94.5%

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

   if 4.70000000000000023e32 < y < 4.0999999999999998e155

   1. Initial program 57.9%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 92.5

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

   6. Applied egg-rr 92.5%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 87.5%

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

   if 4.0999999999999998e155 < y

   1. Initial program 64.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 92.9

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

   7. Simplified 92.9%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+32}:\\ \;\;\;\;y \cdot \frac{x}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.9e+155)
   (/ (* x (/ y (+ y x))) (* (+ y x) (+ (+ y x) 1.0)))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.9e+155) {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.9d+155) then
        tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.9e+155) {
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.9e+155:
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.9e+155)
		tmp = Float64(Float64(x * Float64(y / Float64(y + x))) / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.9e+155)
		tmp = (x * (y / (y + x))) / ((y + x) * ((y + x) + 1.0));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.9e+155], N[(N[(x * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.8999999999999998e155

   1. Initial program 70.9%

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. times-frac N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 97.5

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

   4. Applied egg-rr 97.5%

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

   if 3.8999999999999998e155 < y

   1. Initial program 64.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 92.9

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

   7. Simplified 92.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{x \cdot \frac{y}{y + x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-140)
   (/ y (* x (+ x (+ y 1.0))))
   (if (<= y 6.8e+155)
     (/ x (* (+ y x) (+ (+ y x) 1.0)))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 6.8e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.2d-140) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else if (y <= 6.8d+155) then
        tmp = x / ((y + x) * ((y + x) + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 6.8e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e-140:
		tmp = y / (x * (x + (y + 1.0)))
	elif y <= 6.8e+155:
		tmp = x / ((y + x) * ((y + x) + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-140)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (y <= 6.8e+155)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e-140)
		tmp = y / (x * (x + (y + 1.0)));
	elseif (y <= 6.8e+155)
		tmp = x / ((y + x) * ((y + x) + 1.0));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e-140], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+155], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000001e-140

   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. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 88.2

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 58.4

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

   7. Simplified 58.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+l+ N/A

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

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

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

      7. +-lowering-+.f64 62.9

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

   9. Applied egg-rr 62.9%

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

   if 7.2000000000000001e-140 < y < 6.8000000000000002e155

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 95.2

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 83.2%

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

   if 6.8000000000000002e155 < y

   1. Initial program 64.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. +-commutative N/A

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

      3. +-lowering-+.f64 92.9

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

   7. Simplified 92.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e-140)
   (/ y (* x (+ x (+ y 1.0))))
   (if (<= y 3.5e+155) (/ x (* (+ y x) (+ (+ y x) 1.0))) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 3.5e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.8d-140) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else if (y <= 3.5d+155) then
        tmp = x / ((y + x) * ((y + x) + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 3.5e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e-140:
		tmp = y / (x * (x + (y + 1.0)))
	elif y <= 3.5e+155:
		tmp = x / ((y + x) * ((y + x) + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e-140)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (y <= 3.5e+155)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e-140)
		tmp = y / (x * (x + (y + 1.0)));
	elseif (y <= 3.5e+155)
		tmp = x / ((y + x) * ((y + x) + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e-140], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.5e+155], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.80000000000000038e-140

   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. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. +-lowering-+.f64 88.2

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

   4. Applied egg-rr 88.2%

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

   5. Taylor expanded in y around 0

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

      1. /-lowering-/.f64 58.4

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

   7. Simplified 58.4%

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

      1. associate-/l/ N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. associate-+l+ N/A

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

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

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

      7. +-lowering-+.f64 62.9

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

   9. Applied egg-rr 62.9%

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

   if 7.80000000000000038e-140 < y < 3.49999999999999985e155

   1. Initial program 75.0%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 99.8

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

   4. Applied egg-rr 99.8%

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

      1. associate-/l* N/A

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

      2. associate-/r* N/A

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

      3. times-frac N/A

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

      4. *-commutative N/A

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

      5. frac-times N/A

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

      6. associate-/r* N/A

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

      7. frac-times N/A

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

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

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

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

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

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

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

      11. +-commutative N/A

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

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

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

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

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

      14. +-commutative N/A

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

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

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

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

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

      17. +-commutative N/A

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

      18. +-lowering-+.f64 95.2

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

   6. Applied egg-rr 95.2%

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

   7. Taylor expanded in y around inf

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

   9. Simplified 83.2%

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

   if 3.49999999999999985e155 < y

   1. Initial program 64.4%

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

      1. times-frac N/A

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

      2. *-commutative N/A

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

      3. associate-/r* N/A

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

      4. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-lowering-+.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 92.9

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

   7. Simplified 92.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-139}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.05e-139)
   (/ y (* x (+ x (+ y 1.0))))
   (if (<= y 3.9e+155) (/ x (* (+ y x) (+ (+ y x) 1.0))) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-139) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 3.9e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.05d-139) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else if (y <= 3.9d+155) then
        tmp = x / ((y + x) * ((y + x) + 1.0d0))
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.05e-139) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (y <= 3.9e+155) {
		tmp = x / ((y + x) * ((y + x) + 1.0));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.05e-139:
		tmp = y / (x * (x + (y + 1.0)))
	elif y <= 3.9e+155:
		tmp = x / ((y + x) * ((y + x) + 1.0))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.05e-139)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (y <= 3.9e+155)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(y + x) + 1.0)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.05e-139)
		tmp = y / (x * (x + (y + 1.0)));
	elseif (y <= 3.9e+155)
		tmp = x / ((y + x) * ((y + x) + 1.0));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.05e-139], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+155], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.05000000000000004e-139

   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. *-commutative N/A

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

      2. times-frac N/A

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

      3. associate-*r/ N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right) + 1}} \]

      11. +-lowering-+.f64 88.2

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right)} + 1} \]

   4. Applied egg-rr 88.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 58.4

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]

   7. Simplified 58.4%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-lowering-+.f64 62.9

         \[\leadsto \frac{y}{x \cdot \left(x + \color{blue}{\left(y + 1\right)}\right)} \]

   9. Applied egg-rr 62.9%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}} \]

   if 1.05000000000000004e-139 < y < 3.8999999999999998e155

   1. Initial program 75.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.8

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]
   5. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{\frac{x}{x + y}}{x + y}} \]

      2. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. times-frac N/A

         \[\leadsto \color{blue}{\frac{y \cdot x}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. frac-times N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\left(x + y\right) + 1}} \]

      6. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{\left(x + y\right) + 1} \]

      7. frac-times N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y} \cdot x}}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\color{blue}{\frac{y}{x + y}} \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}} \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\color{blue}{y + x}} \cdot x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      14. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\color{blue}{\left(y + x\right)} \cdot \left(\left(x + y\right) + 1\right)} \]

      16. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\left(y + x\right) \cdot \color{blue}{\left(\left(x + y\right) + 1\right)}} \]

      17. +-commutative N/A

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      18. +-lowering-+.f64 95.2

          \[\leadsto \frac{\frac{y}{y + x} \cdot x}{\left(y + x\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   6. Applied egg-rr 95.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + x} \cdot x}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

   7. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)} \]
   8. Step-by-step derivation

   9. Simplified 83.2%

      \[\leadsto \frac{\color{blue}{x}}{\left(y + x\right) \cdot \left(\left(y + x\right) + 1\right)} \]

   if 3.8999999999999998e155 < y

   1. Initial program 64.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 80.6

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. /-lowering-/.f64 92.8

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.1e-31)
   (/ y (* x (+ x (+ y 1.0))))
   (if (<= x 2.4e-32) (/ x (fma y y y)) (/ (/ x y) y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.1e-31) {
		tmp = y / (x * (x + (y + 1.0)));
	} else if (x <= 2.4e-32) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.1e-31)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	elseif (x <= 2.4e-32)
		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, -1.1e-31], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-32], 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 < -1.10000000000000005e-31

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\left(x + y\right) + 1}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right) + 1}} \]

      11. +-lowering-+.f64 95.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right)} + 1} \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 77.2

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]

   7. Simplified 77.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-lowering-+.f64 84.0

         \[\leadsto \frac{y}{x \cdot \left(x + \color{blue}{\left(y + 1\right)}\right)} \]

   9. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}} \]

   if -1.10000000000000005e-31 < x < 2.4000000000000001e-32

   1. Initial program 72.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 81.3

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 2.4000000000000001e-32 < x

   1. Initial program 67.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 26.0

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 26.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   6. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]

      3. /-lowering-/.f64 27.5

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   7. Applied egg-rr 27.5%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{elif}\;y \leq 122000000000:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-140)
   (/ y (fma x x x))
   (if (<= y 122000000000.0) (/ x (fma y y y)) (/ x (* y (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-140) {
		tmp = y / fma(x, x, x);
	} else if (y <= 122000000000.0) {
		tmp = x / fma(y, y, y);
	} else {
		tmp = x / (y * (y + x));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-140)
		tmp = Float64(y / fma(x, x, x));
	elseif (y <= 122000000000.0)
		tmp = Float64(x / fma(y, y, y));
	else
		tmp = Float64(x / Float64(y * Float64(y + x)));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-140], N[(y / N[(x * x + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 122000000000.0], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.9999999999999999e-140

   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. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. accelerator-lowering-fma.f64 58.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if 7.9999999999999999e-140 < y < 1.22e11

   1. Initial program 91.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 35.8

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 35.8%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]

   if 1.22e11 < y

   1. Initial program 61.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

      12. +-lowering-+.f64 99.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. Taylor expanded in y around inf

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 75.4

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]

   7. Simplified 75.4%

      \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{x + y} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot y}} \]

      3. +-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{\left(y + x\right)} \cdot y} \]

      4. *-commutative N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(y + x\right)}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot \left(y + x\right)}} \]

      6. +-lowering-+.f64 80.9

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + x\right)}} \]

   9. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + x\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.6 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.6e-32) (/ y (* x (+ x (+ y 1.0)))) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.6e-32) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.6e-32)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.6e-32], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6000000000000001e-32

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. times-frac N/A

         \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\left(x + y\right) + 1}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}}{\left(x + y\right) + 1} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1} \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right) + 1}} \]

      11. +-lowering-+.f64 95.9

          \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right)} + 1} \]

   4. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 77.2

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]

   7. Simplified 77.2%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
   8. Step-by-step derivation

      1. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{\left(\left(x + y\right) + 1\right) \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(\left(x + y\right) + 1\right)}} \]

      5. associate-+l+ N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

      7. +-lowering-+.f64 84.0

         \[\leadsto \frac{y}{x \cdot \left(x + \color{blue}{\left(y + 1\right)}\right)} \]

   9. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}} \]

   if -1.6000000000000001e-32 < x

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 59.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 60.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{\mathsf{fma}\left(x, x, x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.5e-32) (/ y (fma x x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.5e-32) {
		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 <= -7.5e-32)
		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, -7.5e-32], 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 < -7.49999999999999953e-32

   1. Initial program 69.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(1 + x\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{y}{x \cdot \color{blue}{\left(x + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x + x \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{y}{x \cdot x + \color{blue}{x}} \]

      5. accelerator-lowering-fma.f64 78.3

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, x\right)}} \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, x, x\right)}} \]

   if -7.49999999999999953e-32 < x

   1. Initial program 70.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 59.2

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(y, y, y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7e+22) (/ y (* x x)) (/ x (fma y y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / fma(y, y, y);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7e+22)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(x / fma(y, y, y));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7e+22], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7e22

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 85.6

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -7e22 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y + y \cdot 1}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{x}{y \cdot y + \color{blue}{y}} \]

      5. accelerator-lowering-fma.f64 59.5

         \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(y, y, y\right)}} \]

   5. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(y, y, y\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 48.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+22}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8.5e+22) (/ y (* x x)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.5e+22) {
		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 <= (-8.5d+22)) 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 <= -8.5e+22) {
		tmp = y / (x * x);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.5e+22:
		tmp = y / (x * x)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.5e+22)
		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 <= -8.5e+22)
		tmp = y / (x * x);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.5e+22], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999979e22

   1. Initial program 67.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

      3. *-lowering-*.f64 85.6

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   5. Simplified 85.6%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -8.49999999999999979e22 < x

   1. Initial program 71.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

      2. unpow2 N/A

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

      3. *-lowering-*.f64 43.5

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   5. Simplified 43.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.1% 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 70.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]

   2. unpow2 N/A

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   3. *-lowering-*.f64 37.9

      \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

5. Simplified 37.9%

   \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
6. Add Preprocessing

Alternative 18: 5.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y + x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (+ y x)))

C

double code(double x, double y) {
	return 1.0 / (y + x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y + x)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (y + x);
}

Python

def code(x, y):
	return 1.0 / (y + x)

Julia

function code(x, y)
	return Float64(1.0 / Float64(y + x))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (y + x);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. times-frac N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   3. associate-/r* N/A

      \[\leadsto \frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   4. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}}{x + y} \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) + 1}} \cdot \frac{x}{x + y}}{x + y} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} + 1} \cdot \frac{x}{x + y}}{x + y} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{\frac{x}{x + y}}}{x + y} \]

   11. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{\color{blue}{x + y}}}{x + y} \]

   12. +-lowering-+.f64 99.9

       \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{\color{blue}{x + y}} \]

4. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1} \cdot \frac{x}{x + y}}{x + y}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{1}}{x + y} \]
6. Step-by-step derivation

7. Simplified 52.7%

   \[\leadsto \frac{\frac{y}{\left(x + y\right) + 1} \cdot \color{blue}{1}}{x + y} \]

8. Taylor expanded in y around inf

   \[\leadsto \frac{\color{blue}{1}}{x + y} \]
9. Step-by-step derivation

10. Simplified 5.3%

    \[\leadsto \frac{\color{blue}{1}}{x + y} \]

11. Final simplification 5.3%

    \[\leadsto \frac{1}{y + x} \]
12. Add Preprocessing

Alternative 19: 4.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 70.3%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. times-frac N/A

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\left(x + y\right) + 1}} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}}{\left(x + y\right) + 1} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \frac{\color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1} \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot x}{\left(x + y\right) + 1} \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right) + 1}} \]

   11. +-lowering-+.f64 89.5

       \[\leadsto \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\color{blue}{\left(x + y\right)} + 1} \]

4. Applied egg-rr 89.5%

   \[\leadsto \color{blue}{\frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot x}{\left(x + y\right) + 1}} \]

5. Taylor expanded in y around 0

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 51.8

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]

7. Simplified 51.8%

   \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{\left(x + y\right) + 1} \]

8. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Step-by-step derivation

   1. /-lowering-/.f64 4.3

      \[\leadsto \color{blue}{\frac{1}{x}} \]

10. Simplified 4.3%

    \[\leadsto \color{blue}{\frac{1}{x}} \]
11. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024199 
(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))))