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

Percentage Accurate: 68.6% → 99.7%

Time: 13.4s

Alternatives: 16

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: 68.6% 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: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((x / (x + y)) / (x + y)) / (1.0 + ((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)) / (x + y)) / (1.0d0 + ((1.0d0 / y) + (x / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((x / (x + y)) / (x + y)) / (1.0 + ((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[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(1.0 / y), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. associate-/l* 79.9%

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

   2. associate-+l+ 79.9%

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

3. Simplified 79.9%

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

   1. clear-num 79.8%

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

   2. associate-+r+ 79.8%

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

   3. *-commutative 79.8%

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

   4. distribute-rgt1-in 64.2%

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

   5. cube-mult 64.2%

      \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

   6. un-div-inv 64.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

   7. cube-mult 64.3%

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

   8. distribute-rgt1-in 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 83.3%

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

   11. pow2 83.3%

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

   12. +-commutative 83.3%

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

6. Applied egg-rr 83.3%

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

   1. associate-/r* 85.7%

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

   2. +-commutative 85.7%

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

8. Simplified 85.7%

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

   1. *-un-lft-identity 85.7%

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

   2. unpow2 85.7%

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

   3. times-frac 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

12. Simplified 99.8%

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

13. Taylor expanded in y around inf 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 89.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -7.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-287}:\\ \;\;\;\;t\_0 \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

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 (<= x -7.5e+172)
     (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
     (if (<= x -9.2e-8)
       (* t_0 (/ y (* (+ x y) (+ x y))))
       (if (<= x -1.6e-287)
         (* t_0 (/ y (+ x y)))
         (/ (/ x (+ y 1.0)) (+ x y)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -7.5e+172) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -9.2e-8) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else if (x <= -1.6e-287) {
		tmp = t_0 * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (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 (x <= (-7.5d+172)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-9.2d-8)) then
        tmp = t_0 * (y / ((x + y) * (x + y)))
    else if (x <= (-1.6d-287)) then
        tmp = t_0 * (y / (x + y))
    else
        tmp = (x / (y + 1.0d0)) / (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 (x <= -7.5e+172) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -9.2e-8) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else if (x <= -1.6e-287) {
		tmp = t_0 * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -7.5e+172:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -9.2e-8:
		tmp = t_0 * (y / ((x + y) * (x + y)))
	elif x <= -1.6e-287:
		tmp = t_0 * (y / (x + y))
	else:
		tmp = (x / (y + 1.0)) / (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 (x <= -7.5e+172)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -9.2e-8)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	elseif (x <= -1.6e-287)
		tmp = Float64(t_0 * Float64(y / Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / 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 (x <= -7.5e+172)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -9.2e-8)
		tmp = t_0 * (y / ((x + y) * (x + y)));
	elseif (x <= -1.6e-287)
		tmp = t_0 * (y / (x + y));
	else
		tmp = (x / (y + 1.0)) / (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[x, -7.5e+172], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.2e-8], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-287], N[(t$95$0 * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.4999999999999994e172

   1. Initial program 73.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

      1. clear-num 82.1%

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

      2. associate-+r+ 82.1%

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

      3. *-commutative 82.1%

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

      4. distribute-rgt1-in 0.7%

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

      5. cube-mult 0.7%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 0.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 0.7%

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

      8. distribute-rgt1-in 82.1%

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

      9. *-commutative 82.1%

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

      10. associate-/l* 82.1%

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

      11. pow2 82.1%

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

      12. +-commutative 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. associate-/r* 82.1%

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

      2. +-commutative 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in x around inf 93.6%

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

   if -7.4999999999999994e172 < x < -9.2000000000000003e-8

   1. Initial program 61.1%

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

      1. associate-*l* 61.1%

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

      2. times-frac 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in x around inf 89.8%

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

   if -9.2000000000000003e-8 < x < -1.60000000000000009e-287

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0 69.3%

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

   4. Taylor expanded in y around 0 49.8%

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

      1. *-rgt-identity 49.8%

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

      2. *-commutative 49.8%

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

      3. *-rgt-identity 49.8%

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

      4. *-rgt-identity 49.8%

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

      5. +-commutative 49.8%

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

      6. +-commutative 49.8%

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

      7. times-frac 72.6%

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

      8. +-commutative 72.6%

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

      9. +-commutative 72.6%

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

   6. Applied egg-rr 72.6%

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

   if -1.60000000000000009e-287 < x

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

      2. times-frac 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 49.5%

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

      1. +-commutative 49.5%

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

   7. Simplified 49.5%

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

      1. associate-*l/ 49.5%

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

      2. un-div-inv 49.5%

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

      3. +-commutative 49.5%

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

   9. Applied egg-rr 49.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+172}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -9.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{t\_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -6.2e+174)
     (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
     (if (<= x -1.0)
       (* t_0 (/ y (* (+ x y) (+ x y))))
       (* (/ y (+ x y)) (/ t_0 (+ y 1.0)))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -6.2e+174) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -1.0) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else {
		tmp = (y / (x + y)) * (t_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 = x / (x + y)
    if (x <= (-6.2d+174)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-1.0d0)) then
        tmp = t_0 * (y / ((x + y) * (x + y)))
    else
        tmp = (y / (x + y)) * (t_0 / (y + 1.0d0))
    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 (x <= -6.2e+174) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -1.0) {
		tmp = t_0 * (y / ((x + y) * (x + y)));
	} else {
		tmp = (y / (x + y)) * (t_0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -6.2e+174:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -1.0:
		tmp = t_0 * (y / ((x + y) * (x + y)))
	else:
		tmp = (y / (x + y)) * (t_0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -6.2e+174)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -1.0)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(t_0 / Float64(y + 1.0)));
	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 (x <= -6.2e+174)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -1.0)
		tmp = t_0 * (y / ((x + y) * (x + y)));
	else
		tmp = (y / (x + y)) * (t_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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.2e+174], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.0], N[(t$95$0 * N[(y / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e174

   1. Initial program 73.7%

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

      1. associate-/l* 82.1%

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

      2. associate-+l+ 82.1%

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

   3. Simplified 82.1%

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

      1. clear-num 82.1%

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

      2. associate-+r+ 82.1%

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

      3. *-commutative 82.1%

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

      4. distribute-rgt1-in 0.7%

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

      5. cube-mult 0.7%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 0.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 0.7%

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

      8. distribute-rgt1-in 82.1%

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

      9. *-commutative 82.1%

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

      10. associate-/l* 82.1%

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

      11. pow2 82.1%

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

      12. +-commutative 82.1%

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

   6. Applied egg-rr 82.1%

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

      1. associate-/r* 82.1%

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

      2. +-commutative 82.1%

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

   8. Simplified 82.1%

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

   9. Taylor expanded in x around inf 93.6%

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

   if -6.2e174 < x < -1

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. times-frac 90.7%

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

      3. +-commutative 90.7%

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

      4. +-commutative 90.7%

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

      5. associate-+r+ 90.7%

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

      6. +-commutative 90.7%

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

      7. associate-+l+ 90.7%

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

   4. Applied egg-rr 90.7%

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

   5. Taylor expanded in x around inf 89.5%

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

   if -1 < x

   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. Taylor expanded in x around 0 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. times-frac 85.3%

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

      4. +-commutative 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. associate-*l/ 60.0%

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

      3. associate-*r/ 85.3%

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

      4. *-commutative 85.3%

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

      5. associate-*r/ 85.3%

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

      6. associate-/r* 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -1:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-285}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.8e-13)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (if (<= x -1.65e-285)
     (* (/ x (+ x y)) (/ y (+ x y)))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-13) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -1.65e-285) {
		tmp = (x / (x + y)) * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (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) :: tmp
    if (x <= (-3.8d-13)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-1.65d-285)) then
        tmp = (x / (x + y)) * (y / (x + y))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.8e-13) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -1.65e-285) {
		tmp = (x / (x + y)) * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.8e-13:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -1.65e-285:
		tmp = (x / (x + y)) * (y / (x + y))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.8e-13)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -1.65e-285)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.8e-13)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -1.65e-285)
		tmp = (x / (x + y)) * (y / (x + y));
	else
		tmp = (x / (y + 1.0)) / (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_] := If[LessEqual[x, -3.8e-13], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.65e-285], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.8e-13

   1. Initial program 68.6%

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

      1. associate-/l* 75.8%

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

      2. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

      1. clear-num 75.7%

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

      2. associate-+r+ 75.7%

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

      3. *-commutative 75.7%

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

      4. distribute-rgt1-in 32.4%

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

      5. cube-mult 32.4%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 32.5%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 32.4%

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

      8. distribute-rgt1-in 75.7%

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

      9. *-commutative 75.7%

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

      10. associate-/l* 83.0%

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

      11. pow2 83.0%

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

      12. +-commutative 83.0%

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

   6. Applied egg-rr 83.0%

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

      1. associate-/r* 87.4%

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

      2. +-commutative 87.4%

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

   8. Simplified 87.4%

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

   9. Taylor expanded in x around inf 75.4%

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

   if -3.8e-13 < x < -1.64999999999999993e-285

   1. Initial program 68.6%

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

   3. Taylor expanded in x around 0 68.6%

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

   4. Taylor expanded in y around 0 49.3%

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

      1. *-rgt-identity 49.3%

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

      2. *-commutative 49.3%

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

      3. *-rgt-identity 49.3%

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

      4. *-rgt-identity 49.3%

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

      5. +-commutative 49.3%

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

      6. +-commutative 49.3%

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

      7. times-frac 74.0%

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

      8. +-commutative 74.0%

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

      9. +-commutative 74.0%

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

   6. Applied egg-rr 74.0%

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

   if -1.64999999999999993e-285 < x

   1. Initial program 64.5%

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

      1. associate-*l* 64.5%

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

      2. times-frac 93.2%

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

      3. +-commutative 93.2%

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

      4. +-commutative 93.2%

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

      5. associate-+r+ 93.2%

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

      6. +-commutative 93.2%

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

      7. associate-+l+ 93.2%

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

   4. Applied egg-rr 93.2%

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

   5. Taylor expanded in x around 0 49.8%

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

      1. +-commutative 49.8%

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

   7. Simplified 49.8%

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

      1. associate-*l/ 49.8%

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

      2. un-div-inv 49.9%

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

      3. +-commutative 49.9%

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

   9. Applied egg-rr 49.9%

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

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-285}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.65e-13)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -1e-286)
     (* (/ x (+ x y)) (/ y (+ x y)))
     (/ (/ x (+ y 1.0)) (+ x y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-13) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1e-286) {
		tmp = (x / (x + y)) * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (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) :: tmp
    if (x <= (-2.65d-13)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-1d-286)) then
        tmp = (x / (x + y)) * (y / (x + y))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.65e-13) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -1e-286) {
		tmp = (x / (x + y)) * (y / (x + y));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.65e-13:
		tmp = (y / x) / (x + 1.0)
	elif x <= -1e-286:
		tmp = (x / (x + y)) * (y / (x + y))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.65e-13)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -1e-286)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(x + y)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.65e-13)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -1e-286)
		tmp = (x / (x + y)) * (y / (x + y));
	else
		tmp = (x / (y + 1.0)) / (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_] := If[LessEqual[x, -2.65e-13], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1e-286], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6499999999999998e-13

   1. Initial program 68.6%

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

      1. associate-/l* 75.8%

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

      2. associate-+l+ 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in y around 0 71.3%

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

      1. associate-/r* 74.9%

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

      2. +-commutative 74.9%

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

   7. Simplified 74.9%

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

   if -2.6499999999999998e-13 < x < -1.00000000000000005e-286

   1. Initial program 69.2%

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

   3. Taylor expanded in x around 0 69.1%

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

   4. Taylor expanded in y around 0 48.5%

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

      1. *-rgt-identity 48.5%

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

      2. *-commutative 48.5%

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

      3. *-rgt-identity 48.5%

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

      4. *-rgt-identity 48.5%

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

      5. +-commutative 48.5%

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

      6. +-commutative 48.5%

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

      7. times-frac 72.7%

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

      8. +-commutative 72.7%

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

      9. +-commutative 72.7%

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

   6. Applied egg-rr 72.7%

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

   if -1.00000000000000005e-286 < x

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

      2. times-frac 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 49.5%

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

      1. +-commutative 49.5%

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

   7. Simplified 49.5%

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

      1. associate-*l/ 49.5%

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

      2. un-div-inv 49.5%

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

      3. +-commutative 49.5%

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

   9. Applied egg-rr 49.5%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -1 \cdot 10^{-286}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{t\_0}{x + y}}{1 + \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{t\_0}{y + 1}\\ \end{array} \end{array} \]

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -1.0) {
		tmp = (t_0 / (x + y)) / (1.0 + (x / y));
	} else {
		tmp = (y / (x + y)) * (t_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 = x / (x + y)
    if (x <= (-1.0d0)) then
        tmp = (t_0 / (x + y)) / (1.0d0 + (x / y))
    else
        tmp = (y / (x + y)) * (t_0 / (y + 1.0d0))
    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 (x <= -1.0) {
		tmp = (t_0 / (x + y)) / (1.0 + (x / y));
	} else {
		tmp = (y / (x + y)) * (t_0 / (y + 1.0));
	}
	return tmp;
}

Python

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

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(t_0 / Float64(x + y)) / Float64(1.0 + Float64(x / y)));
	else
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(t_0 / Float64(y + 1.0)));
	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 (x <= -1.0)
		tmp = (t_0 / (x + y)) / (1.0 + (x / y));
	else
		tmp = (y / (x + y)) * (t_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[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.0], N[(N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 66.5%

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

      1. associate-/l* 74.2%

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

      2. associate-+l+ 74.2%

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

   3. Simplified 74.2%

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

      1. clear-num 74.1%

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

      2. associate-+r+ 74.1%

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

      3. *-commutative 74.1%

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

      4. distribute-rgt1-in 29.6%

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

      5. cube-mult 29.6%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 29.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 29.6%

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

      8. distribute-rgt1-in 74.2%

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

      9. *-commutative 74.2%

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

      10. associate-/l* 81.9%

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

      11. pow2 81.9%

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

      12. +-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-/r* 86.6%

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

      2. +-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. unpow2 86.6%

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

      3. times-frac 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around inf 99.8%

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

   14. Taylor expanded in x around inf 99.2%

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

   if -1 < x

   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. Taylor expanded in x around 0 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. times-frac 85.3%

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

      4. +-commutative 85.3%

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

   5. Applied egg-rr 85.3%

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

      1. associate-*r/ 85.3%

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

      2. associate-*l/ 60.0%

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

      3. associate-*r/ 85.3%

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

      4. *-commutative 85.3%

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

      5. associate-*r/ 85.3%

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

      6. associate-/r* 85.9%

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

   7. Simplified 85.9%

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

4. Final simplification 89.0%

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

Alternative 7: 77.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.2 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -4.2e-96) (/ y x) (/ x (* y (+ y 1.0))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.2e-96) {
		tmp = 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 <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-4.2d-96)) then
        tmp = 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 <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.2e-96) {
		tmp = 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 <= -1.0:
		tmp = y / (x * x)
	elif x <= -4.2e-96:
		tmp = 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 <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -4.2e-96)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(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 <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -4.2e-96)
		tmp = 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, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.2e-96], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.5%

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

      1. associate-/l* 74.2%

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

      2. associate-+l+ 74.2%

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

   3. Simplified 74.2%

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

      1. clear-num 74.1%

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

      2. associate-+r+ 74.1%

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

      3. *-commutative 74.1%

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

      4. distribute-rgt1-in 29.6%

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

      5. cube-mult 29.6%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 29.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 29.6%

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

      8. distribute-rgt1-in 74.2%

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

      9. *-commutative 74.2%

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

      10. associate-/l* 81.9%

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

      11. pow2 81.9%

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

      12. +-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-/r* 86.6%

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

      2. +-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. unpow2 86.6%

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

      3. times-frac 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 71.0%

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

       1. +-commutative 71.0%

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

   15. Simplified 71.0%

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

   16. Taylor expanded in x around inf 70.4%

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

   if -1 < x < -4.20000000000000002e-96

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0 77.1%

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

   4. Taylor expanded in y around 0 47.7%

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

   if -4.20000000000000002e-96 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

Alternative 8: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return ((x / (x + y)) / (x + y)) / ((y + (x + 1.0)) / 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)) / (x + y)) / ((y + (x + 1.0d0)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-/l* 79.9%

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

   2. associate-+l+ 79.9%

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

3. Simplified 79.9%

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

   1. clear-num 79.8%

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

   2. associate-+r+ 79.8%

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

   3. *-commutative 79.8%

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

   4. distribute-rgt1-in 64.2%

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

   5. cube-mult 64.2%

      \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

   6. un-div-inv 64.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

   7. cube-mult 64.3%

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

   8. distribute-rgt1-in 79.8%

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

   9. *-commutative 79.8%

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

   10. associate-/l* 83.3%

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

   11. pow2 83.3%

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

   12. +-commutative 83.3%

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

6. Applied egg-rr 83.3%

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

   1. associate-/r* 85.7%

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

   2. +-commutative 85.7%

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

8. Simplified 85.7%

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

   1. *-un-lft-identity 85.7%

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

   2. unpow2 85.7%

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

   3. times-frac 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.8%

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

    2. *-lft-identity 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

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

Alternative 9: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ y (* x x)) (if (<= x -4.5e-97) (/ y x) (/ 1.0 (/ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.5e-97) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	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 <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-4.5d-97)) then
        tmp = y / x
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -4.5e-97) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -4.5e-97:
		tmp = y / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -4.5e-97)
		tmp = Float64(y / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -4.5e-97)
		tmp = y / x;
	else
		tmp = 1.0 / (y / x);
	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, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-97], N[(y / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.5%

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

      1. associate-/l* 74.2%

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

      2. associate-+l+ 74.2%

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

   3. Simplified 74.2%

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

      1. clear-num 74.1%

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

      2. associate-+r+ 74.1%

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

      3. *-commutative 74.1%

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

      4. distribute-rgt1-in 29.6%

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

      5. cube-mult 29.6%

         \[\leadsto x \cdot \frac{1}{\frac{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{{\left(x + y\right)}^{3}}}{y}} \]

      6. un-div-inv 29.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}{y}}} \]

      7. cube-mult 29.6%

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

      8. distribute-rgt1-in 74.2%

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

      9. *-commutative 74.2%

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

      10. associate-/l* 81.9%

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

      11. pow2 81.9%

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

      12. +-commutative 81.9%

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

   6. Applied egg-rr 81.9%

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

      1. associate-/r* 86.6%

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

      2. +-commutative 86.6%

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

   8. Simplified 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. unpow2 86.6%

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

      3. times-frac 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around 0 71.0%

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

       1. +-commutative 71.0%

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

   15. Simplified 71.0%

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

   16. Taylor expanded in x around inf 70.4%

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

   if -1 < x < -4.5000000000000001e-97

   1. Initial program 80.9%

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

   3. Taylor expanded in x around 0 77.1%

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

   4. Taylor expanded in y around 0 47.7%

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

   if -4.5000000000000001e-97 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in y around 0 28.9%

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

      1. clear-num 29.2%

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

      2. inv-pow 29.2%

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

   10. Applied egg-rr 29.2%

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

       1. unpow-1 29.2%

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

   12. Simplified 29.2%

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

Alternative 10: 82.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4e-96) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e-96) {
		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 <= (-4d-96)) 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 <= -4e-96) {
		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 <= -4e-96:
		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 <= -4e-96)
		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 <= -4e-96)
		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, -4e-96], 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 < -3.9999999999999996e-96

   1. Initial program 70.4%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in y around 0 65.8%

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

      1. associate-/r* 68.6%

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

      2. +-commutative 68.6%

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

   7. Simplified 68.6%

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

   if -3.9999999999999996e-96 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 51.9%

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

      1. +-commutative 51.9%

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

      2. div-inv 52.0%

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

      3. associate-/r* 54.0%

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

   7. Applied egg-rr 54.0%

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

Alternative 11: 80.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3.1e-96) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.1e-96) {
		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 <= (-3.1d-96)) 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 <= -3.1e-96) {
		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 <= -3.1e-96:
		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 <= -3.1e-96)
		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 <= -3.1e-96)
		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, -3.1e-96], 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 < -3.0999999999999999e-96

   1. Initial program 70.4%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in y around 0 65.8%

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

   if -3.0999999999999999e-96 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 51.9%

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

      1. +-commutative 51.9%

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

      2. div-inv 52.0%

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

      3. associate-/r* 54.0%

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

   7. Applied egg-rr 54.0%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4e-96) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4e-96) {
		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 <= (-4d-96)) 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 <= -4e-96) {
		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 <= -4e-96:
		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 <= -4e-96)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(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 <= -4e-96)
		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, -4e-96], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.9999999999999996e-96

   1. Initial program 70.4%

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

      1. associate-/l* 79.6%

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

      2. associate-+l+ 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in y around 0 65.8%

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

   if -3.9999999999999996e-96 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 44.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -9.5e-98) (/ 1.0 (/ x y)) (/ 1.0 (/ y x))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-98) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = 1.0 / (y / x);
	}
	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-98)) then
        tmp = 1.0d0 / (x / y)
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -9.5e-98) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -9.5e-98:
		tmp = 1.0 / (x / y)
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -9.5e-98)
		tmp = Float64(1.0 / Float64(x / y));
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -9.5e-98)
		tmp = 1.0 / (x / y);
	else
		tmp = 1.0 / (y / x);
	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-98], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5000000000000001e-98

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0 53.1%

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

   4. Taylor expanded in y around 0 34.2%

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

      1. clear-num 35.8%

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

      2. inv-pow 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. unpow-1 35.8%

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

   8. Simplified 35.8%

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

   if -9.5000000000000001e-98 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in y around 0 28.9%

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

      1. clear-num 29.2%

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

      2. inv-pow 29.2%

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

   10. Applied egg-rr 29.2%

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

       1. unpow-1 29.2%

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

   12. Simplified 29.2%

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

Alternative 14: 44.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.2e-97) (/ 1.0 (/ x y)) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.2e-97) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = 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 (x <= (-3.2d-97)) then
        tmp = 1.0d0 / (x / y)
    else
        tmp = x / 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-97) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.2e-97:
		tmp = 1.0 / (x / y)
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.2e-97)
		tmp = Float64(1.0 / Float64(x / y));
	else
		tmp = Float64(x / 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-97)
		tmp = 1.0 / (x / y);
	else
		tmp = 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[x, -3.2e-97], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1999999999999998e-97

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0 53.1%

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

   4. Taylor expanded in y around 0 34.2%

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

      1. clear-num 35.8%

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

      2. inv-pow 35.8%

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

   6. Applied egg-rr 35.8%

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

      1. unpow-1 35.8%

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

   8. Simplified 35.8%

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

   if -3.1999999999999998e-97 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in y around 0 28.9%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-96}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -3.6e-96) (/ y x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-96) {
		tmp = y / x;
	} else {
		tmp = 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 (x <= (-3.6d-96)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-96) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3.6e-96:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-96)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e-96)
		tmp = y / x;
	else
		tmp = 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[x, -3.6e-96], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.60000000000000008e-96

   1. Initial program 70.4%

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

   3. Taylor expanded in x around 0 53.1%

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

   4. Taylor expanded in y around 0 34.2%

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

   if -3.60000000000000008e-96 < x

   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* 80.1%

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

      2. associate-+l+ 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in x around 0 52.0%

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

      1. +-commutative 52.0%

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

   7. Simplified 52.0%

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

   8. Taylor expanded in y around 0 28.9%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.1% accurate, 5.7× speedup

Math

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

FPCore

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

C

assert(x < y);
double code(double x, double y) {
	return 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 / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Step-by-step derivation

   1. associate-/l* 79.9%

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

   2. associate-+l+ 79.9%

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

3. Simplified 79.9%

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

5. Taylor expanded in x around 0 44.0%

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

   1. +-commutative 44.0%

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

7. Simplified 44.0%

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

8. Taylor expanded in y around 0 21.6%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Developer Target 1: 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 2024116 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))