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

Percentage Accurate: 68.4% → 99.3%

Time: 16.0s

Alternatives: 19

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.4% 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: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (* (+ x y) (/ (+ y (+ x 1.0)) y))) (+ x y)))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / ((x + y) * ((y + (x + 1.0d0)) / y))) / (x + y)
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / ((x + y) * ((y + (x + 1.0)) / y))) / (x + y)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(Float64(x + y) * Float64(Float64(y + Float64(x + 1.0)) / y))) / Float64(x + y))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / ((x + y) * ((y + (x + 1.0)) / y))) / (x + y);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]

Derivation

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. associate-*l* 64.4%

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

   2. times-frac 91.8%

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

   3. +-commutative 91.8%

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

   4. +-commutative 91.8%

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

   5. associate-+r+ 91.8%

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

   6. +-commutative 91.8%

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

   7. associate-+l+ 91.8%

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

4. Applied egg-rr 91.8%

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

   1. clear-num 91.5%

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

   2. inv-pow 91.5%

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

   3. +-commutative 91.5%

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

6. Applied egg-rr 91.5%

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

   1. unpow-1 91.5%

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

   2. associate-/l* 99.4%

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

8. Simplified 99.4%

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

   1. associate-*l/ 99.4%

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

   2. un-div-inv 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 94.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 2.12 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-5)
   (/ (/ x (* (+ x y) (/ (+ x 1.0) y))) (+ x y))
   (if (<= y 2.12e+92)
     (* x (/ y (* (* (+ x y) (+ x y)) (+ x (+ y 1.0)))))
     (/ (/ x (+ x y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 2.12e+92) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-5) then
        tmp = (x / ((x + y) * ((x + 1.0d0) / y))) / (x + y)
    else if (y <= 2.12d+92) then
        tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0d0))))
    else
        tmp = (x / (x + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 2.12e+92) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.3e-5:
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y)
	elif y <= 2.12e+92:
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))))
	else:
		tmp = (x / (x + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-5)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(Float64(x + 1.0) / y))) / Float64(x + y));
	elseif (y <= 2.12e+92)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-5)
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	elseif (y <= 2.12e+92)
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	else
		tmp = (x / (x + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.3e-5], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.12e+92], N[(x * N[(y / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.29999999999999992e-5

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. times-frac 93.7%

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

      3. +-commutative 93.7%

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

      4. +-commutative 93.7%

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

      5. associate-+r+ 93.7%

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

      6. +-commutative 93.7%

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

      7. associate-+l+ 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. clear-num 93.4%

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

      2. inv-pow 93.4%

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

      3. +-commutative 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. unpow-1 93.4%

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

      2. associate-/l* 99.4%

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

   8. Simplified 99.4%

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

      1. associate-*l/ 99.4%

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

      2. un-div-inv 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in y around 0 85.7%

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

       1. +-commutative 85.7%

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

   13. Simplified 85.7%

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

   if 1.29999999999999992e-5 < y < 2.11999999999999999e92

   1. Initial program 77.3%

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

      1. associate-/l* 86.7%

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

      2. associate-+l+ 86.7%

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

   3. Simplified 86.7%

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

   if 2.11999999999999999e92 < y

   1. Initial program 52.8%

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

      1. associate-*l* 52.8%

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

      2. times-frac 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. associate-+r+ 85.3%

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

      6. +-commutative 85.3%

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

      7. associate-+l+ 85.3%

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

   4. Applied egg-rr 85.3%

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

      1. clear-num 85.3%

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

      2. inv-pow 85.3%

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

      3. +-commutative 85.3%

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

   6. Applied egg-rr 85.3%

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

      1. unpow-1 85.3%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-*l/ 99.3%

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

      2. un-div-inv 99.3%

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

   10. Applied egg-rr 99.3%

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

   11. Taylor expanded in y around inf 84.9%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 2.12 \cdot 10^{+92}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-5)
   (/ (/ x (* (+ x y) (/ (+ x 1.0) y))) (+ x y))
   (if (<= y 1.6e+154)
     (* x (/ (/ y (* (+ x y) (+ y (+ x 1.0)))) (+ x y)))
     (/ (/ x (+ x y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 1.6e+154) {
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-5) then
        tmp = (x / ((x + y) * ((x + 1.0d0) / y))) / (x + y)
    else if (y <= 1.6d+154) then
        tmp = x * ((y / ((x + y) * (y + (x + 1.0d0)))) / (x + y))
    else
        tmp = (x / (x + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 1.6e+154) {
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.3e-5:
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y)
	elif y <= 1.6e+154:
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y))
	else:
		tmp = (x / (x + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-5)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(Float64(x + 1.0) / y))) / Float64(x + y));
	elseif (y <= 1.6e+154)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))) / Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-5)
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	elseif (y <= 1.6e+154)
		tmp = x * ((y / ((x + y) * (y + (x + 1.0)))) / (x + y));
	else
		tmp = (x / (x + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.3e-5], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+154], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.29999999999999992e-5

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. times-frac 93.7%

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

      3. +-commutative 93.7%

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

      4. +-commutative 93.7%

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

      5. associate-+r+ 93.7%

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

      6. +-commutative 93.7%

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

      7. associate-+l+ 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. clear-num 93.4%

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

      2. inv-pow 93.4%

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

      3. +-commutative 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. unpow-1 93.4%

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

      2. associate-/l* 99.4%

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

   8. Simplified 99.4%

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

      1. associate-*l/ 99.4%

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

      2. un-div-inv 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in y around 0 85.7%

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

       1. +-commutative 85.7%

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

   13. Simplified 85.7%

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

   if 1.29999999999999992e-5 < y < 1.6e154

   1. Initial program 57.1%

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

      1. associate-/l* 67.9%

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

      2. associate-+l+ 67.9%

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

   3. Simplified 67.9%

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

      1. *-un-lft-identity 67.9%

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

      2. associate-+r+ 67.9%

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

      3. associate-*l* 67.9%

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

      4. times-frac 92.3%

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

      5. +-commutative 92.3%

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

      6. +-commutative 92.3%

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

      7. associate-+r+ 92.3%

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

      8. +-commutative 92.3%

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

      9. associate-+l+ 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. associate-*l/ 92.4%

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

      2. *-lft-identity 92.4%

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

      3. +-commutative 92.4%

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

   8. Simplified 92.4%

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

   if 1.6e154 < y

   1. Initial program 62.3%

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

      1. associate-*l* 62.3%

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

      2. times-frac 81.6%

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

      3. +-commutative 81.6%

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

      4. +-commutative 81.6%

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

      5. associate-+r+ 81.6%

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

      6. +-commutative 81.6%

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

      7. associate-+l+ 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. clear-num 81.6%

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

      2. inv-pow 81.6%

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

      3. +-commutative 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. unpow-1 81.6%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-*l/ 99.1%

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

      2. un-div-inv 99.1%

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

   10. Applied egg-rr 99.1%

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

   11. Taylor expanded in y around inf 92.2%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y -1.05e+142)
     (/ (/ x (* (+ x y) (/ (+ x 1.0) y))) (+ x y))
     (if (<= y 1.35e+154)
       (* t_0 (/ y (* (+ x y) (+ y (+ x 1.0)))))
       (/ t_0 (+ x y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -1.05e+142) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = t_0 / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= (-1.05d+142)) then
        tmp = (x / ((x + y) * ((x + 1.0d0) / y))) / (x + y)
    else if (y <= 1.35d+154) then
        tmp = t_0 * (y / ((x + y) * (y + (x + 1.0d0))))
    else
        tmp = t_0 / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= -1.05e+142) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 1.35e+154) {
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	} else {
		tmp = t_0 / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= -1.05e+142:
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y)
	elif y <= 1.35e+154:
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))))
	else:
		tmp = t_0 / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= -1.05e+142)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(Float64(x + 1.0) / y))) / Float64(x + y));
	elseif (y <= 1.35e+154)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(t_0 / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= -1.05e+142)
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	elseif (y <= 1.35e+154)
		tmp = t_0 * (y / ((x + y) * (y + (x + 1.0))));
	else
		tmp = t_0 / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.05e+142], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+154], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.05e142

   1. Initial program 56.4%

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

      1. associate-*l* 56.4%

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

      2. times-frac 70.6%

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

      3. +-commutative 70.6%

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

      4. +-commutative 70.6%

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

      5. associate-+r+ 70.6%

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

      6. +-commutative 70.6%

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

      7. associate-+l+ 70.6%

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

   4. Applied egg-rr 70.6%

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

      1. clear-num 70.6%

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

      2. inv-pow 70.6%

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

      3. +-commutative 70.6%

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

   6. Applied egg-rr 70.6%

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

      1. unpow-1 70.6%

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

      2. associate-/l* 99.5%

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

   8. Simplified 99.5%

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

      1. associate-*l/ 99.6%

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

      2. un-div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in y around 0 51.7%

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

       1. +-commutative 51.7%

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

   13. Simplified 51.7%

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

   if -1.05e142 < y < 1.35000000000000003e154

   1. Initial program 66.2%

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

      1. associate-*l* 66.2%

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

      2. times-frac 97.4%

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

      3. +-commutative 97.4%

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

      4. +-commutative 97.4%

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

      5. associate-+r+ 97.4%

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

      6. +-commutative 97.4%

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

      7. associate-+l+ 97.4%

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

   4. Applied egg-rr 97.4%

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

   if 1.35000000000000003e154 < y

   1. Initial program 62.3%

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

      1. associate-*l* 62.3%

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

      2. times-frac 81.6%

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

      3. +-commutative 81.6%

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

      4. +-commutative 81.6%

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

      5. associate-+r+ 81.6%

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

      6. +-commutative 81.6%

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

      7. associate-+l+ 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. clear-num 81.6%

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

      2. inv-pow 81.6%

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

      3. +-commutative 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. unpow-1 81.6%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-*l/ 99.1%

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

      2. un-div-inv 99.1%

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

   10. Applied egg-rr 99.1%

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

   11. Taylor expanded in y around inf 92.2%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+136}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-5)
   (/ (/ x (* (+ x y) (/ (+ x 1.0) y))) (+ x y))
   (if (<= y 2.3e+136)
     (* (/ y (* (+ x y) (+ y (+ x 1.0)))) (/ x y))
     (/ (/ x (+ x y)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 2.3e+136) {
		tmp = (y / ((x + y) * (y + (x + 1.0)))) * (x / y);
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-5) then
        tmp = (x / ((x + y) * ((x + 1.0d0) / y))) / (x + y)
    else if (y <= 2.3d+136) then
        tmp = (y / ((x + y) * (y + (x + 1.0d0)))) * (x / y)
    else
        tmp = (x / (x + y)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-5) {
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	} else if (y <= 2.3e+136) {
		tmp = (y / ((x + y) * (y + (x + 1.0)))) * (x / y);
	} else {
		tmp = (x / (x + y)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.9e-5:
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y)
	elif y <= 2.3e+136:
		tmp = (y / ((x + y) * (y + (x + 1.0)))) * (x / y)
	else:
		tmp = (x / (x + y)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-5)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(Float64(x + 1.0) / y))) / Float64(x + y));
	elseif (y <= 2.3e+136)
		tmp = Float64(Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))) * Float64(x / y));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-5)
		tmp = (x / ((x + y) * ((x + 1.0) / y))) / (x + y);
	elseif (y <= 2.3e+136)
		tmp = (y / ((x + y) * (y + (x + 1.0)))) * (x / y);
	else
		tmp = (x / (x + y)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.9e-5], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e+136], N[(N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.9000000000000001e-5

   1. Initial program 66.3%

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

      1. associate-*l* 66.3%

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

      2. times-frac 93.7%

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

      3. +-commutative 93.7%

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

      4. +-commutative 93.7%

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

      5. associate-+r+ 93.7%

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

      6. +-commutative 93.7%

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

      7. associate-+l+ 93.7%

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

   4. Applied egg-rr 93.7%

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

      1. clear-num 93.4%

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

      2. inv-pow 93.4%

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

      3. +-commutative 93.4%

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

   6. Applied egg-rr 93.4%

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

      1. unpow-1 93.4%

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

      2. associate-/l* 99.4%

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

   8. Simplified 99.4%

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

      1. associate-*l/ 99.4%

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

      2. un-div-inv 99.4%

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

   10. Applied egg-rr 99.4%

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

   11. Taylor expanded in y around 0 85.7%

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

       1. +-commutative 85.7%

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

   13. Simplified 85.7%

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

   if 1.9000000000000001e-5 < y < 2.3e136

   1. Initial program 58.3%

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

      1. associate-*l* 58.3%

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

      2. times-frac 91.2%

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

      3. +-commutative 91.2%

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

      4. +-commutative 91.2%

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

      5. associate-+r+ 91.2%

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

      6. +-commutative 91.2%

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

      7. associate-+l+ 91.2%

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

   4. Applied egg-rr 91.2%

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

   5. Taylor expanded in x around 0 82.4%

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

   if 2.3e136 < y

   1. Initial program 60.8%

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

      1. associate-*l* 60.8%

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

      2. times-frac 83.7%

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

      3. +-commutative 83.7%

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

      4. +-commutative 83.7%

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

      5. associate-+r+ 83.7%

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

      6. +-commutative 83.7%

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

      7. associate-+l+ 83.7%

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

   4. Applied egg-rr 83.7%

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

      1. clear-num 83.7%

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

      2. inv-pow 83.7%

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

      3. +-commutative 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. unpow-1 83.7%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. associate-*l/ 99.2%

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

      2. un-div-inv 99.2%

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

   10. Applied egg-rr 99.2%

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

   11. Taylor expanded in y around inf 92.1%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{x}{\left(x + y\right) \cdot \frac{x + 1}{y}}}{x + y}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+136}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-58} \lor \neg \left(x \leq -2.6 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e+29)
   (/ (/ y x) (+ x y))
   (if (or (<= x -1.55e-58) (not (<= x -2.6e-152)))
     (/ (/ x y) (+ y 1.0))
     (/ y (* x (+ y (+ x 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+29) {
		tmp = (y / x) / (x + y);
	} else if ((x <= -1.55e-58) || !(x <= -2.6e-152)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-9.5d+29)) then
        tmp = (y / x) / (x + y)
    else if ((x <= (-1.55d-58)) .or. (.not. (x <= (-2.6d-152)))) then
        tmp = (x / y) / (y + 1.0d0)
    else
        tmp = y / (x * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e+29) {
		tmp = (y / x) / (x + y);
	} else if ((x <= -1.55e-58) || !(x <= -2.6e-152)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (y + (x + 1.0)));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e+29:
		tmp = (y / x) / (x + y)
	elif (x <= -1.55e-58) or not (x <= -2.6e-152):
		tmp = (x / y) / (y + 1.0)
	else:
		tmp = y / (x * (y + (x + 1.0)))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e+29)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif ((x <= -1.55e-58) || !(x <= -2.6e-152))
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(x * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e+29)
		tmp = (y / x) / (x + y);
	elseif ((x <= -1.55e-58) || ~((x <= -2.6e-152)))
		tmp = (x / y) / (y + 1.0);
	else
		tmp = y / (x * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -9.5e+29], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.55e-58], N[Not[LessEqual[x, -2.6e-152]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.5000000000000003e29

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. times-frac 82.4%

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

      3. +-commutative 82.4%

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

      4. +-commutative 82.4%

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

      5. associate-+r+ 82.4%

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

      6. +-commutative 82.4%

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

      7. associate-+l+ 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. +-commutative 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

      2. associate-/l* 99.5%

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

   8. Simplified 99.5%

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

      1. associate-*l/ 99.6%

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

      2. un-div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around inf 78.9%

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

   if -9.5000000000000003e29 < x < -1.55e-58 or -2.60000000000000013e-152 < x

   1. Initial program 64.3%

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

      1. associate-*l* 64.3%

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

      2. times-frac 93.8%

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

      3. +-commutative 93.8%

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

      4. +-commutative 93.8%

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

      5. associate-+r+ 93.8%

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

      6. +-commutative 93.8%

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

      7. associate-+l+ 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in x around 0 59.8%

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

      1. associate-/r* 61.2%

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

      2. +-commutative 61.2%

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

   7. Simplified 61.2%

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

   if -1.55e-58 < x < -2.60000000000000013e-152

   1. Initial program 78.0%

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

      1. associate-/l* 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 78.0%

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

      2. associate-+r+ 78.0%

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

      3. associate-/r* 78.0%

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

      4. clear-num 78.0%

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

      5. associate-+r+ 78.0%

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

      6. +-commutative 78.0%

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

      7. associate-+l+ 78.0%

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

      8. *-commutative 78.0%

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

      9. associate-/l* 99.8%

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

      10. pow2 99.8%

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

      11. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 47.1%

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

      1. +-commutative 47.1%

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

      2. clear-num 47.2%

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

      3. *-un-lft-identity 47.2%

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

   9. Applied egg-rr 47.2%

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

       1. *-lft-identity 47.2%

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

       2. associate-/l/ 47.3%

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

       3. *-commutative 47.3%

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

       4. +-commutative 47.3%

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

       5. +-commutative 47.3%

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

       6. +-commutative 47.3%

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

   11. Simplified 47.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-58} \lor \neg \left(x \leq -2.6 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+30} \lor \neg \left(x \leq -1.75 \cdot 10^{-57}\right) \land x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (or (<= x -3.6e+30) (and (not (<= x -1.75e-57)) (<= x -2.6e-152)))
   (/ y (* x (+ x 1.0)))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e+30) || (!(x <= -1.75e-57) && (x <= -2.6e-152))) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-3.6d+30)) .or. (.not. (x <= (-1.75d-57))) .and. (x <= (-2.6d-152))) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if ((x <= -3.6e+30) || (!(x <= -1.75e-57) && (x <= -2.6e-152))) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if (x <= -3.6e+30) or (not (x <= -1.75e-57) and (x <= -2.6e-152)):
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if ((x <= -3.6e+30) || (!(x <= -1.75e-57) && (x <= -2.6e-152)))
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -3.6e+30) || (~((x <= -1.75e-57)) && (x <= -2.6e-152)))
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[Or[LessEqual[x, -3.6e+30], And[N[Not[LessEqual[x, -1.75e-57]], $MachinePrecision], LessEqual[x, -2.6e-152]]], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6000000000000002e30 or -1.74999999999999996e-57 < x < -2.60000000000000013e-152

   1. Initial program 64.7%

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

      1. associate-/l* 74.8%

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

      2. associate-+l+ 74.8%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in y around 0 64.3%

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

   if -3.6000000000000002e30 < x < -1.74999999999999996e-57 or -2.60000000000000013e-152 < x

   1. Initial program 64.3%

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

      1. associate-*l* 64.3%

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

      2. times-frac 93.8%

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

      3. +-commutative 93.8%

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

      4. +-commutative 93.8%

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

      5. associate-+r+ 93.8%

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

      6. +-commutative 93.8%

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

      7. associate-+l+ 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in x around 0 59.8%

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

      1. associate-/r* 61.2%

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

      2. +-commutative 61.2%

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

   7. Simplified 61.2%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{+30} \lor \neg \left(x \leq -1.75 \cdot 10^{-57}\right) \land x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-59} \lor \neg \left(x \leq -2.6 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e+30)
   (/ (/ y x) (+ x y))
   (if (or (<= x -8.2e-59) (not (<= x -2.6e-152)))
     (/ (/ x y) (+ y 1.0))
     (/ y (* x (+ x 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+30) {
		tmp = (y / x) / (x + y);
	} else if ((x <= -8.2e-59) || !(x <= -2.6e-152)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d+30)) then
        tmp = (y / x) / (x + y)
    else if ((x <= (-8.2d-59)) .or. (.not. (x <= (-2.6d-152)))) then
        tmp = (x / y) / (y + 1.0d0)
    else
        tmp = y / (x * (x + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e+30) {
		tmp = (y / x) / (x + y);
	} else if ((x <= -8.2e-59) || !(x <= -2.6e-152)) {
		tmp = (x / y) / (y + 1.0);
	} else {
		tmp = y / (x * (x + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.35e+30:
		tmp = (y / x) / (x + y)
	elif (x <= -8.2e-59) or not (x <= -2.6e-152):
		tmp = (x / y) / (y + 1.0)
	else:
		tmp = y / (x * (x + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.35e+30)
		tmp = Float64(Float64(y / x) / Float64(x + y));
	elseif ((x <= -8.2e-59) || !(x <= -2.6e-152))
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	else
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e+30)
		tmp = (y / x) / (x + y);
	elseif ((x <= -8.2e-59) || ~((x <= -2.6e-152)))
		tmp = (x / y) / (y + 1.0);
	else
		tmp = y / (x * (x + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.35e+30], N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -8.2e-59], N[Not[LessEqual[x, -2.6e-152]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3499999999999999e30

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. times-frac 82.4%

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

      3. +-commutative 82.4%

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

      4. +-commutative 82.4%

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

      5. associate-+r+ 82.4%

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

      6. +-commutative 82.4%

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

      7. associate-+l+ 82.4%

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

   4. Applied egg-rr 82.4%

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

      1. clear-num 82.4%

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

      2. inv-pow 82.4%

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

      3. +-commutative 82.4%

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

   6. Applied egg-rr 82.4%

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

      1. unpow-1 82.4%

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

      2. associate-/l* 99.5%

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

   8. Simplified 99.5%

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

      1. associate-*l/ 99.6%

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

      2. un-div-inv 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around inf 78.9%

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

   if -1.3499999999999999e30 < x < -8.1999999999999991e-59 or -2.60000000000000013e-152 < x

   1. Initial program 64.3%

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

      1. associate-*l* 64.3%

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

      2. times-frac 93.8%

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

      3. +-commutative 93.8%

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

      4. +-commutative 93.8%

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

      5. associate-+r+ 93.8%

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

      6. +-commutative 93.8%

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

      7. associate-+l+ 93.8%

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

   4. Applied egg-rr 93.8%

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

   5. Taylor expanded in x around 0 59.8%

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

      1. associate-/r* 61.2%

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

      2. +-commutative 61.2%

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

   7. Simplified 61.2%

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

   if -8.1999999999999991e-59 < x < -2.60000000000000013e-152

   1. Initial program 78.0%

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

      1. associate-/l* 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 47.0%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{+30}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + y}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-59} \lor \neg \left(x \leq -2.6 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\frac{y}{x}}}\\ \mathbf{elif}\;x \leq -3.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -3.7e+102)
     (/ 1.0 (/ t_0 (/ y x)))
     (if (<= x -3.95e-224)
       (/ y (* (+ x y) t_0))
       (* (/ x (+ x y)) (/ 1.0 (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -3.7e+102) {
		tmp = 1.0 / (t_0 / (y / x));
	} else if (x <= -3.95e-224) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 (x <= (-3.7d+102)) then
        tmp = 1.0d0 / (t_0 / (y / x))
    else if (x <= (-3.95d-224)) then
        tmp = y / ((x + y) * t_0)
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -3.7e+102) {
		tmp = 1.0 / (t_0 / (y / x));
	} else if (x <= -3.95e-224) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -3.7e+102:
		tmp = 1.0 / (t_0 / (y / x))
	elif x <= -3.95e-224:
		tmp = y / ((x + y) * t_0)
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -3.7e+102)
		tmp = Float64(1.0 / Float64(t_0 / Float64(y / x)));
	elseif (x <= -3.95e-224)
		tmp = Float64(y / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -3.7e+102)
		tmp = 1.0 / (t_0 / (y / x));
	elseif (x <= -3.95e-224)
		tmp = y / ((x + y) * t_0);
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+102], N[(1.0 / N[(t$95$0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.95e-224], N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.70000000000000023e102

   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. Step-by-step derivation

      1. associate-/l* 66.0%

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

      2. associate-+l+ 66.0%

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

   3. Simplified 66.0%

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

      1. associate-*r/ 60.2%

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

      2. associate-+r+ 60.2%

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

      3. associate-/r* 73.5%

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

      4. clear-num 73.5%

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

      5. associate-+r+ 73.5%

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

      6. +-commutative 73.5%

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

      7. associate-+l+ 73.5%

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

      8. *-commutative 73.5%

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

      9. associate-/l* 81.2%

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

      10. pow2 81.2%

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

      11. +-commutative 81.2%

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in y around 0 84.7%

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

   if -3.70000000000000023e102 < x < -3.9499999999999999e-224

   1. Initial program 72.5%

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

      1. associate-*l* 72.5%

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

      2. times-frac 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l+ 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in x around inf 65.2%

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

   if -3.9499999999999999e-224 < x

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. times-frac 93.1%

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

      3. +-commutative 93.1%

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

      4. +-commutative 93.1%

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

      5. associate-+r+ 93.1%

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

      6. +-commutative 93.1%

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

      7. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. +-commutative 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + 1\right)}{\frac{y}{x}}}\\ \mathbf{elif}\;x \leq -3.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{y \cdot \frac{1}{x}}}\\ \mathbf{elif}\;x \leq -3.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -3.7e+102)
     (/ 1.0 (/ t_0 (* y (/ 1.0 x))))
     (if (<= x -3.95e-224)
       (/ y (* (+ x y) t_0))
       (* (/ x (+ x y)) (/ 1.0 (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -3.7e+102) {
		tmp = 1.0 / (t_0 / (y * (1.0 / x)));
	} else if (x <= -3.95e-224) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 (x <= (-3.7d+102)) then
        tmp = 1.0d0 / (t_0 / (y * (1.0d0 / x)))
    else if (x <= (-3.95d-224)) then
        tmp = y / ((x + y) * t_0)
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -3.7e+102) {
		tmp = 1.0 / (t_0 / (y * (1.0 / x)));
	} else if (x <= -3.95e-224) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -3.7e+102:
		tmp = 1.0 / (t_0 / (y * (1.0 / x)))
	elif x <= -3.95e-224:
		tmp = y / ((x + y) * t_0)
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -3.7e+102)
		tmp = Float64(1.0 / Float64(t_0 / Float64(y * Float64(1.0 / x))));
	elseif (x <= -3.95e-224)
		tmp = Float64(y / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -3.7e+102)
		tmp = 1.0 / (t_0 / (y * (1.0 / x)));
	elseif (x <= -3.95e-224)
		tmp = y / ((x + y) * t_0);
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.7e+102], N[(1.0 / N[(t$95$0 / N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.95e-224], N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.70000000000000023e102

   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. Step-by-step derivation

      1. associate-/l* 66.0%

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

      2. associate-+l+ 66.0%

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

   3. Simplified 66.0%

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

      1. associate-*r/ 60.2%

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

      2. associate-+r+ 60.2%

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

      3. associate-/r* 73.5%

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

      4. clear-num 73.5%

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

      5. associate-+r+ 73.5%

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

      6. +-commutative 73.5%

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

      7. associate-+l+ 73.5%

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

      8. *-commutative 73.5%

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

      9. associate-/l* 81.2%

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

      10. pow2 81.2%

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

      11. +-commutative 81.2%

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

   6. Applied egg-rr 81.2%

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

   7. Taylor expanded in x around inf 84.7%

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

   if -3.70000000000000023e102 < x < -3.9499999999999999e-224

   1. Initial program 72.5%

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

      1. associate-*l* 72.5%

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

      2. times-frac 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l+ 97.9%

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

   4. Applied egg-rr 97.9%

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

   5. Taylor expanded in x around inf 65.2%

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

   if -3.9499999999999999e-224 < x

   1. Initial program 63.0%

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

      1. associate-*l* 63.0%

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

      2. times-frac 93.1%

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

      3. +-commutative 93.1%

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

      4. +-commutative 93.1%

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

      5. associate-+r+ 93.1%

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

      6. +-commutative 93.1%

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

      7. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

   5. Taylor expanded in x around 0 59.5%

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

      1. +-commutative 59.5%

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

   7. Simplified 59.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + 1\right)}{y \cdot \frac{1}{x}}}\\ \mathbf{elif}\;x \leq -3.95 \cdot 10^{-224}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4e-157)
   (/ y x)
   (if (<= y 1.0) (* x (/ 1.0 y)) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4e-157) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x * (1.0 / y);
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-157) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x * (1.0d0 / y)
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-157) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x * (1.0 / y);
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4e-157:
		tmp = y / x
	elif y <= 1.0:
		tmp = x * (1.0 / y)
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4e-157)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x * Float64(1.0 / y));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-157)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x * (1.0 / y);
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4e-157], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.99999999999999977e-157

   1. Initial program 60.9%

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

      1. associate-/l* 72.2%

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

      2. associate-+l+ 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in y around 0 54.5%

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

   6. Taylor expanded in x around 0 41.6%

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

   if 3.99999999999999977e-157 < y < 1

   1. Initial program 86.3%

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

      1. associate-/l* 94.9%

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

      2. associate-+l+ 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in x around 0 38.7%

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

      1. +-commutative 38.7%

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

   7. Simplified 38.7%

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

      1. *-un-lft-identity 38.7%

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

      2. times-frac 38.4%

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

   9. Applied egg-rr 38.4%

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

   10. Taylor expanded in y around 0 37.4%

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

   if 1 < y

   1. Initial program 59.7%

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

      1. associate-*l* 59.7%

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

      2. times-frac 86.9%

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

      3. +-commutative 86.9%

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

      4. +-commutative 86.9%

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

      5. associate-+r+ 86.9%

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

      6. +-commutative 86.9%

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

      7. associate-+l+ 86.9%

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

   4. Applied egg-rr 86.9%

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

   5. Taylor expanded in y around inf 73.7%

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

   6. Taylor expanded in x around 0 73.3%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.8e-157)
   (/ y x)
   (if (<= y 2.35e+66) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-157) {
		tmp = y / x;
	} else if (y <= 2.35e+66) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.8d-157) then
        tmp = y / x
    else if (y <= 2.35d+66) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.8e-157) {
		tmp = y / x;
	} else if (y <= 2.35e+66) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.8e-157:
		tmp = y / x
	elif y <= 2.35e+66:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.8e-157)
		tmp = Float64(y / x);
	elseif (y <= 2.35e+66)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.8e-157)
		tmp = y / x;
	elseif (y <= 2.35e+66)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.8e-157], N[(y / x), $MachinePrecision], If[LessEqual[y, 2.35e+66], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.8e-157

   1. Initial program 60.9%

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

      1. associate-/l* 72.2%

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

      2. associate-+l+ 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in y around 0 54.5%

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

   6. Taylor expanded in x around 0 41.6%

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

   if 1.8e-157 < y < 2.3500000000000001e66

   1. Initial program 84.2%

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

      1. associate-/l* 93.4%

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

      2. associate-+l+ 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in x around 0 43.6%

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

      1. +-commutative 43.6%

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

   7. Simplified 43.6%

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

   if 2.3500000000000001e66 < y

   1. Initial program 52.7%

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

      1. associate-*l* 52.7%

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

      2. times-frac 85.8%

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

      3. +-commutative 85.8%

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

      4. +-commutative 85.8%

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

      5. associate-+r+ 85.8%

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

      6. +-commutative 85.8%

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

      7. associate-+l+ 85.8%

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

   4. Applied egg-rr 85.8%

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

   5. Taylor expanded in y around inf 81.8%

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

   6. Taylor expanded in x around 0 81.4%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-157}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{+66}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-152)
   (/ y (* x (+ x 1.0)))
   (if (<= x 7e-10) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 7e-10) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d-152)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 7d-10) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 7e-10) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e-152:
		tmp = y / (x * (x + 1.0))
	elif x <= 7e-10:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-152)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 7e-10)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-152)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 7e-10)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e-152], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7e-10], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.60000000000000013e-152

   1. Initial program 65.4%

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

      1. associate-/l* 75.4%

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

      2. associate-+l+ 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in y around 0 59.4%

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

   if -2.60000000000000013e-152 < x < 6.99999999999999961e-10

   1. Initial program 66.8%

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

      1. associate-/l* 81.2%

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

      2. associate-+l+ 81.2%

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

   3. Simplified 81.2%

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

   5. Taylor expanded in x around 0 78.6%

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

      1. +-commutative 78.6%

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

   7. Simplified 78.6%

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

   if 6.99999999999999961e-10 < x

   1. Initial program 59.6%

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

      1. associate-*l* 59.6%

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

      2. times-frac 84.0%

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

      3. +-commutative 84.0%

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

      4. +-commutative 84.0%

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

      5. associate-+r+ 84.0%

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

      6. +-commutative 84.0%

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

      7. associate-+l+ 84.0%

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

   4. Applied egg-rr 84.0%

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

   5. Taylor expanded in y around inf 34.0%

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

   6. Taylor expanded in x around 0 33.3%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + 1\right)}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-152)
   (/ 1.0 (/ (+ y (+ x 1.0)) (/ y x)))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = 1.0 / ((y + (x + 1.0)) / (y / x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d-152)) then
        tmp = 1.0d0 / ((y + (x + 1.0d0)) / (y / x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = 1.0 / ((y + (x + 1.0)) / (y / x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e-152:
		tmp = 1.0 / ((y + (x + 1.0)) / (y / x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-152)
		tmp = Float64(1.0 / Float64(Float64(y + Float64(x + 1.0)) / Float64(y / x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-152)
		tmp = 1.0 / ((y + (x + 1.0)) / (y / x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e-152], N[(1.0 / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000013e-152

   1. Initial program 65.4%

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

      1. associate-/l* 75.4%

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

      2. associate-+l+ 75.4%

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

   3. Simplified 75.4%

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

      1. associate-*r/ 65.4%

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

      2. associate-+r+ 65.4%

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

      3. associate-/r* 76.3%

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

      4. clear-num 75.6%

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

      5. associate-+r+ 75.6%

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

      6. +-commutative 75.6%

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

      7. associate-+l+ 75.6%

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

      8. *-commutative 75.6%

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

      9. associate-/l* 88.1%

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

      10. pow2 88.1%

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

      11. +-commutative 88.1%

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

   6. Applied egg-rr 88.1%

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

   7. Taylor expanded in y around 0 64.5%

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

   if -2.60000000000000013e-152 < x

   1. Initial program 63.8%

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

      1. associate-*l* 63.9%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in x around 0 58.6%

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

      1. associate-/r* 60.1%

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

      2. +-commutative 60.1%

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

   7. Simplified 60.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + 1\right)}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.8e-152) (/ (/ y (+ x y)) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-152) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.8d-152)) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.8e-152) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.8e-152:
		tmp = (y / (x + y)) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.8e-152)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.8e-152)
		tmp = (y / (x + y)) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -1.8e-152], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.8e-152

   1. Initial program 65.4%

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

      1. associate-*l* 65.4%

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

      2. times-frac 88.8%

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

      3. +-commutative 88.8%

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

      4. +-commutative 88.8%

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

      5. associate-+r+ 88.8%

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

      6. +-commutative 88.8%

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

      7. associate-+l+ 88.8%

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

   4. Applied egg-rr 88.8%

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

      1. clear-num 88.8%

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

      2. associate-/r* 99.7%

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

      3. frac-times 99.2%

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

      4. *-un-lft-identity 99.2%

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

      5. +-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in y around 0 64.6%

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

      1. +-commutative 64.6%

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

   9. Simplified 64.6%

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

   if -1.8e-152 < x

   1. Initial program 63.8%

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

      1. associate-*l* 63.9%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in x around 0 58.6%

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

      1. associate-/r* 60.1%

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

      2. +-commutative 60.1%

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

   7. Simplified 60.1%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e-152) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.6d-152)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e-152) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.6e-152:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.6e-152)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e-152)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.6e-152], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.60000000000000013e-152

   1. Initial program 65.4%

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

      1. associate-/l* 75.4%

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

      2. associate-+l+ 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in y around 0 59.4%

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

      1. associate-/r* 64.2%

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

      2. +-commutative 64.2%

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

   7. Simplified 64.2%

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

   if -2.60000000000000013e-152 < x

   1. Initial program 63.8%

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

      1. associate-*l* 63.9%

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

      2. times-frac 93.4%

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

      3. +-commutative 93.4%

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

      4. +-commutative 93.4%

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

      5. associate-+r+ 93.4%

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

      6. +-commutative 93.4%

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

      7. associate-+l+ 93.4%

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

   4. Applied egg-rr 93.4%

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

   5. Taylor expanded in x around 0 58.6%

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

      1. associate-/r* 60.1%

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

      2. +-commutative 60.1%

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

   7. Simplified 60.1%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e-197) (/ y x) (* x (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-197) {
		tmp = y / x;
	} else {
		tmp = x * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d-197)) then
        tmp = y / x
    else
        tmp = x * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-197) {
		tmp = y / x;
	} else {
		tmp = x * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.2e-197:
		tmp = y / x
	else:
		tmp = x * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-197)
		tmp = Float64(y / x);
	else
		tmp = Float64(x * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e-197)
		tmp = y / x;
	else
		tmp = x * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3.2e-197], N[(y / x), $MachinePrecision], N[(x * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999997e-197

   1. Initial program 66.2%

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

      1. associate-/l* 75.6%

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

      2. associate-+l+ 75.6%

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

   3. Simplified 75.6%

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

   5. Taylor expanded in y around 0 57.4%

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

   6. Taylor expanded in x around 0 32.1%

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

   if -3.1999999999999997e-197 < x

   1. Initial program 63.3%

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

      1. associate-/l* 76.9%

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

      2. associate-+l+ 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in x around 0 58.0%

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

      1. +-commutative 58.0%

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

   7. Simplified 58.0%

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

      1. *-un-lft-identity 58.0%

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

      2. times-frac 59.4%

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

   9. Applied egg-rr 59.4%

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

   10. Taylor expanded in y around 0 38.9%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-197}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 76.4%

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

   2. associate-+l+ 76.4%

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

3. Simplified 76.4%

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

   1. associate-*r/ 64.4%

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

   2. associate-+r+ 64.4%

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

   3. associate-/r* 71.5%

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

   4. clear-num 71.1%

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

   5. associate-+r+ 71.1%

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

   6. +-commutative 71.1%

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

   7. associate-+l+ 71.1%

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

   8. *-commutative 71.1%

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

   9. associate-/l* 84.6%

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

   10. pow2 84.6%

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

   11. +-commutative 84.6%

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

6. Applied egg-rr 84.6%

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

7. Taylor expanded in y around 0 49.1%

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

8. Taylor expanded in y around inf 4.2%

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

9. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]
10. Add Preprocessing

Alternative 19: 26.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ y x))

C

assert(x < y);
double code(double x, double y) {
	return y / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return y / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return y / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(y / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = y / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(y / x), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 76.4%

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

   2. associate-+l+ 76.4%

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

3. Simplified 76.4%

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

5. Taylor expanded in y around 0 45.8%

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

6. Taylor expanded in x around 0 28.1%

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

7. Final simplification 28.1%

   \[\leadsto \frac{y}{x} \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.9× 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 2024082 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))