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

Percentage Accurate: 68.9% → 99.8%

Time: 14.0s

Alternatives: 16

Speedup: 1.0×

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: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. associate-/l* 77.7%

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

   2. associate-+l+ 77.7%

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

3. Simplified 77.7%

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

   1. associate-*r/ 68.2%

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

   2. associate-+r+ 68.2%

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

   3. associate-/r* 73.9%

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

   4. clear-num 73.4%

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

   5. associate-+r+ 73.4%

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

   6. associate-/l* 85.6%

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

   7. pow2 85.6%

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

6. Applied egg-rr 85.6%

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

   1. *-un-lft-identity 85.6%

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

   2. unpow2 85.6%

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

   3. times-frac 98.3%

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

   4. +-commutative 98.3%

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

   5. +-commutative 98.3%

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

8. Applied egg-rr 98.3%

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

   1. clear-num 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-un-lft-identity 99.7%

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

   4. times-frac 99.6%

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

   5. un-div-inv 99.8%

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

   6. +-commutative 99.8%

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

   7. +-commutative 99.8%

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

   8. +-commutative 99.8%

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

10. Applied egg-rr 99.8%

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

    1. /-rgt-identity 99.8%

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

    2. associate-*r/ 99.9%

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

    3. +-commutative 99.9%

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

12. Simplified 99.9%

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

Alternative 2: 77.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq 650000000000:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\frac{x}{y}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -5.6e+74)
     (/ (/ y (+ x y)) t_0)
     (if (<= x 650000000000.0)
       (* x (/ (/ y (* (+ x y) t_0)) (+ x y)))
       (/ 1.0 (/ t_0 (/ x y)))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -5.6e+74) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= 650000000000.0) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = 1.0 / (t_0 / (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-5.6d+74)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= 650000000000.0d0) then
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    else
        tmp = 1.0d0 / (t_0 / (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -5.6e+74) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= 650000000000.0) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = 1.0 / (t_0 / (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -5.6e+74:
		tmp = (y / (x + y)) / t_0
	elif x <= 650000000000.0:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	else:
		tmp = 1.0 / (t_0 / (x / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -5.6e+74)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= 650000000000.0)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	else
		tmp = Float64(1.0 / Float64(t_0 / Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -5.6e+74)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= 650000000000.0)
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	else
		tmp = 1.0 / (t_0 / (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+74], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, 650000000000.0], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(t$95$0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.60000000000000003e74

   1. Initial program 51.2%

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

      1. associate-/l* 65.5%

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

      2. associate-+l+ 65.5%

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

   3. Simplified 65.5%

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

      1. associate-*r/ 51.2%

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

      2. associate-+r+ 51.2%

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

      3. associate-/r* 58.9%

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

      4. clear-num 58.8%

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

      5. associate-+r+ 58.8%

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

      6. associate-/l* 75.7%

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

      7. pow2 75.7%

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

   6. Applied egg-rr 75.7%

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

      1. *-un-lft-identity 75.7%

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

      2. unpow2 75.7%

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

      3. times-frac 96.2%

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

      4. +-commutative 96.2%

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

      5. +-commutative 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. clear-num 99.8%

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

      2. associate-*r* 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. times-frac 99.7%

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

      5. un-div-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. /-rgt-identity 99.8%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around inf 86.4%

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

   if -5.60000000000000003e74 < x < 6.5e11

   1. Initial program 76.4%

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

      1. associate-/l* 87.3%

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

      2. associate-+l+ 87.3%

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

   3. Simplified 87.3%

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

      1. *-un-lft-identity 87.3%

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

      2. associate-+r+ 87.3%

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

      3. associate-*l* 87.3%

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

      4. times-frac 98.5%

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

      5. associate-+r+ 98.5%

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

   6. Applied egg-rr 98.5%

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

      1. associate-*l/ 98.7%

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

      2. *-lft-identity 98.7%

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

   8. Simplified 98.7%

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

   if 6.5e11 < x

   1. Initial program 56.4%

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

      1. associate-/l* 59.2%

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

      2. associate-+l+ 59.2%

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

   3. Simplified 59.2%

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

      1. associate-*r/ 56.4%

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

      2. associate-+r+ 56.4%

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

      3. associate-/r* 68.8%

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

      4. clear-num 68.8%

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

      5. associate-+r+ 68.8%

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

      6. associate-/l* 78.7%

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

      7. pow2 78.7%

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

   6. Applied egg-rr 78.7%

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

      1. *-un-lft-identity 78.7%

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

      2. unpow2 78.7%

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

      3. times-frac 98.2%

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

      4. +-commutative 98.2%

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

      5. +-commutative 98.2%

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

   8. Applied egg-rr 98.2%

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

   9. Taylor expanded in x around 0 20.4%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq 650000000000:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + 1\right)}{\frac{x}{y}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 68.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.46 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{y}{t\_0 \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.46e+22)
     (/ (/ y (+ x y)) t_0)
     (if (<= x -3.3e-164)
       (* x (/ y (* t_0 (* (+ x y) (+ x y)))))
       (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.46e+22) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -3.3e-164) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.46d+22)) then
        tmp = (y / (x + y)) / t_0
    else if (x <= (-3.3d-164)) then
        tmp = x * (y / (t_0 * ((x + y) * (x + y))))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.46e+22) {
		tmp = (y / (x + y)) / t_0;
	} else if (x <= -3.3e-164) {
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.46e+22:
		tmp = (y / (x + y)) / t_0
	elif x <= -3.3e-164:
		tmp = x * (y / (t_0 * ((x + y) * (x + y))))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.46e+22)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	elseif (x <= -3.3e-164)
		tmp = Float64(x * Float64(y / Float64(t_0 * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.46e+22)
		tmp = (y / (x + y)) / t_0;
	elseif (x <= -3.3e-164)
		tmp = x * (y / (t_0 * ((x + y) * (x + y))));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.46e+22], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[x, -3.3e-164], N[(x * N[(y / N[(t$95$0 * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.46e22

   1. Initial program 60.5%

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

      1. associate-/l* 67.7%

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

      2. associate-+l+ 67.7%

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

   3. Simplified 67.7%

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

      1. associate-*r/ 60.5%

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

      2. associate-+r+ 60.5%

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

      3. associate-/r* 68.4%

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

      4. clear-num 68.3%

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

      5. associate-+r+ 68.3%

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

      6. associate-/l* 81.3%

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

      7. pow2 81.3%

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

   6. Applied egg-rr 81.3%

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

      1. *-un-lft-identity 81.3%

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

      2. unpow2 81.3%

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

      3. times-frac 96.9%

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

      4. +-commutative 96.9%

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

      5. +-commutative 96.9%

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

   8. Applied egg-rr 96.9%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.8%

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

      3. *-un-lft-identity 99.8%

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

      4. times-frac 99.7%

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

      5. un-div-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. /-rgt-identity 99.8%

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

       2. associate-*r/ 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 82.9%

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

   if -1.46e22 < x < -3.3e-164

   1. Initial program 88.9%

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

      1. associate-/l* 94.1%

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

      2. associate-+l+ 94.1%

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

   3. Simplified 94.1%

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

   if -3.3e-164 < x

   1. Initial program 64.9%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

      1. associate-*r/ 64.9%

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

      2. associate-+r+ 64.9%

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

      3. associate-/r* 70.6%

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

      4. clear-num 69.8%

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

      5. associate-+r+ 69.8%

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

      6. associate-/l* 83.6%

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

      7. pow2 83.6%

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

   6. Applied egg-rr 83.6%

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

      1. *-un-lft-identity 83.6%

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

      2. unpow2 83.6%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.7%

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

      5. un-div-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around 0 56.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.46 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-164}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{if}\;x \leq -2.2 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-146}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))) (t_1 (/ (/ y (+ x y)) t_0)))
   (if (<= x -2.2e-73)
     t_1
     (if (<= x -3.7e-106)
       (/ 1.0 (/ t_0 (/ x y)))
       (if (<= x -1.85e-146) t_1 (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / (x + y)) / t_0;
	double tmp;
	if (x <= -2.2e-73) {
		tmp = t_1;
	} else if (x <= -3.7e-106) {
		tmp = 1.0 / (t_0 / (x / y));
	} else if (x <= -1.85e-146) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    t_1 = (y / (x + y)) / t_0
    if (x <= (-2.2d-73)) then
        tmp = t_1
    else if (x <= (-3.7d-106)) then
        tmp = 1.0d0 / (t_0 / (x / y))
    else if (x <= (-1.85d-146)) then
        tmp = t_1
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / (x + y)) / t_0;
	double tmp;
	if (x <= -2.2e-73) {
		tmp = t_1;
	} else if (x <= -3.7e-106) {
		tmp = 1.0 / (t_0 / (x / y));
	} else if (x <= -1.85e-146) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	t_1 = (y / (x + y)) / t_0
	tmp = 0
	if x <= -2.2e-73:
		tmp = t_1
	elif x <= -3.7e-106:
		tmp = 1.0 / (t_0 / (x / y))
	elif x <= -1.85e-146:
		tmp = t_1
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	t_1 = Float64(Float64(y / Float64(x + y)) / t_0)
	tmp = 0.0
	if (x <= -2.2e-73)
		tmp = t_1;
	elseif (x <= -3.7e-106)
		tmp = Float64(1.0 / Float64(t_0 / Float64(x / y)));
	elseif (x <= -1.85e-146)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	t_1 = (y / (x + y)) / t_0;
	tmp = 0.0;
	if (x <= -2.2e-73)
		tmp = t_1;
	elseif (x <= -3.7e-106)
		tmp = 1.0 / (t_0 / (x / y));
	elseif (x <= -1.85e-146)
		tmp = t_1;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -2.2e-73], t$95$1, If[LessEqual[x, -3.7e-106], N[(1.0 / N[(t$95$0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-146], t$95$1, N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.2e-73 or -3.69999999999999979e-106 < x < -1.84999999999999993e-146

   1. Initial program 71.4%

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

      1. associate-/l* 78.1%

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

      2. associate-+l+ 78.1%

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

   3. Simplified 78.1%

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

      1. associate-*r/ 71.4%

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

      2. associate-+r+ 71.4%

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

      3. associate-/r* 78.4%

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

      4. clear-num 78.3%

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

      5. associate-+r+ 78.3%

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

      6. associate-/l* 88.6%

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

      7. pow2 88.6%

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

   6. Applied egg-rr 88.6%

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

      1. *-un-lft-identity 88.6%

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

      2. unpow2 88.6%

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

      3. times-frac 97.9%

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

      4. +-commutative 97.9%

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

      5. +-commutative 97.9%

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

   8. Applied egg-rr 97.9%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.6%

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

      5. un-div-inv 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. /-rgt-identity 99.8%

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

       2. associate-*r/ 99.8%

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

       3. +-commutative 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 67.8%

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

   if -2.2e-73 < x < -3.69999999999999979e-106

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-/r* 99.6%

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

      4. clear-num 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/l* 99.4%

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

      7. pow2 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. unpow2 99.4%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around 0 87.1%

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

   if -1.84999999999999993e-146 < x

   1. Initial program 65.4%

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

      1. associate-/l* 76.6%

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

      2. associate-+l+ 76.6%

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

   3. Simplified 76.6%

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

      1. associate-*r/ 65.4%

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

      2. associate-+r+ 65.4%

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

      3. associate-/r* 70.8%

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

      4. clear-num 70.1%

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

      5. associate-+r+ 70.1%

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

      6. associate-/l* 83.7%

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

      7. pow2 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. *-un-lft-identity 83.7%

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

      2. unpow2 83.7%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.7%

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

      5. un-div-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around 0 57.4%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\frac{x + \left(y + 1\right)}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-146}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t\_0}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{\frac{x}{y}}}\\ \mathbf{elif}\;x \leq -2 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))) (t_1 (/ (/ y x) t_0)))
   (if (<= x -1.25e-76)
     t_1
     (if (<= x -5.5e-108)
       (/ 1.0 (/ t_0 (/ x y)))
       (if (<= x -2e-140) t_1 (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -1.25e-76) {
		tmp = t_1;
	} else if (x <= -5.5e-108) {
		tmp = 1.0 / (t_0 / (x / y));
	} else if (x <= -2e-140) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    t_1 = (y / x) / t_0
    if (x <= (-1.25d-76)) then
        tmp = t_1
    else if (x <= (-5.5d-108)) then
        tmp = 1.0d0 / (t_0 / (x / y))
    else if (x <= (-2d-140)) then
        tmp = t_1
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -1.25e-76) {
		tmp = t_1;
	} else if (x <= -5.5e-108) {
		tmp = 1.0 / (t_0 / (x / y));
	} else if (x <= -2e-140) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	t_1 = (y / x) / t_0
	tmp = 0
	if x <= -1.25e-76:
		tmp = t_1
	elif x <= -5.5e-108:
		tmp = 1.0 / (t_0 / (x / y))
	elif x <= -2e-140:
		tmp = t_1
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	t_1 = Float64(Float64(y / x) / t_0)
	tmp = 0.0
	if (x <= -1.25e-76)
		tmp = t_1;
	elseif (x <= -5.5e-108)
		tmp = Float64(1.0 / Float64(t_0 / Float64(x / y)));
	elseif (x <= -2e-140)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	t_1 = (y / x) / t_0;
	tmp = 0.0;
	if (x <= -1.25e-76)
		tmp = t_1;
	elseif (x <= -5.5e-108)
		tmp = 1.0 / (t_0 / (x / y));
	elseif (x <= -2e-140)
		tmp = t_1;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -1.25e-76], t$95$1, If[LessEqual[x, -5.5e-108], N[(1.0 / N[(t$95$0 / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2e-140], t$95$1, N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.2499999999999999e-76 or -5.50000000000000031e-108 < x < -2e-140

   1. Initial program 71.0%

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

      1. associate-/l* 77.9%

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

      2. associate-+l+ 77.9%

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

   3. Simplified 77.9%

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

      1. associate-*r/ 71.0%

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

      2. associate-+r+ 71.0%

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

      3. associate-/r* 78.2%

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

      4. clear-num 78.1%

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

      5. associate-+r+ 78.1%

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

      6. associate-/l* 88.5%

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

      7. pow2 88.5%

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

   6. Applied egg-rr 88.5%

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

   7. Taylor expanded in x around inf 66.9%

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

      1. clear-num 67.9%

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

      2. add-cube-cbrt 67.1%

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

      3. *-un-lft-identity 67.1%

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

      4. times-frac 67.1%

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

      5. pow2 67.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(y + 1\right)} \]

      6. +-commutative 67.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \color{blue}{\left(1 + y\right)}} \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)}} \]
   10. Step-by-step derivation

       1. /-rgt-identity 67.1%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)} \]

       2. associate-*r/ 67.1%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2} \cdot \sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)}} \]

       3. unpow2 67.1%

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

       4. rem-3cbrt-lft 67.9%

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

       5. +-commutative 67.9%

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

   11. Simplified 67.9%

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

   if -1.2499999999999999e-76 < x < -5.50000000000000031e-108

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-/r* 99.6%

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

      4. clear-num 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/l* 99.4%

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

      7. pow2 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-un-lft-identity 99.4%

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

      2. unpow2 99.4%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in x around 0 87.1%

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

   if -2e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

      1. associate-*r/ 65.6%

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

      2. associate-+r+ 65.6%

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

      3. associate-/r* 71.0%

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

      4. clear-num 70.3%

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

      5. associate-+r+ 70.3%

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

      6. associate-/l* 83.8%

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

      7. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. unpow2 83.8%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.6%

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

      5. un-div-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around 0 57.6%

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

Alternative 6: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ t_1 := \frac{\frac{y}{x}}{t\_0}\\ \mathbf{if}\;x \leq -8.6 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))) (t_1 (/ (/ y x) t_0)))
   (if (<= x -8.6e-75)
     t_1
     (if (<= x -5.5e-108)
       (/ x (* y (+ y 1.0)))
       (if (<= x -4.5e-140) t_1 (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -8.6e-75) {
		tmp = t_1;
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    t_1 = (y / x) / t_0
    if (x <= (-8.6d-75)) then
        tmp = t_1
    else if (x <= (-5.5d-108)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.5d-140)) then
        tmp = t_1
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double t_1 = (y / x) / t_0;
	double tmp;
	if (x <= -8.6e-75) {
		tmp = t_1;
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = t_1;
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	t_1 = (y / x) / t_0
	tmp = 0
	if x <= -8.6e-75:
		tmp = t_1
	elif x <= -5.5e-108:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.5e-140:
		tmp = t_1
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	t_1 = Float64(Float64(y / x) / t_0)
	tmp = 0.0
	if (x <= -8.6e-75)
		tmp = t_1;
	elseif (x <= -5.5e-108)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.5e-140)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	t_1 = (y / x) / t_0;
	tmp = 0.0;
	if (x <= -8.6e-75)
		tmp = t_1;
	elseif (x <= -5.5e-108)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.5e-140)
		tmp = t_1;
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x, -8.6e-75], t$95$1, If[LessEqual[x, -5.5e-108], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-140], t$95$1, N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5999999999999998e-75 or -5.50000000000000031e-108 < x < -4.50000000000000004e-140

   1. Initial program 71.0%

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

      1. associate-/l* 77.9%

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

      2. associate-+l+ 77.9%

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

   3. Simplified 77.9%

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

      1. associate-*r/ 71.0%

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

      2. associate-+r+ 71.0%

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

      3. associate-/r* 78.2%

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

      4. clear-num 78.1%

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

      5. associate-+r+ 78.1%

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

      6. associate-/l* 88.5%

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

      7. pow2 88.5%

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

   6. Applied egg-rr 88.5%

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

   7. Taylor expanded in x around inf 66.9%

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

      1. clear-num 67.9%

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

      2. add-cube-cbrt 67.1%

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

      3. *-un-lft-identity 67.1%

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

      4. times-frac 67.1%

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

      5. pow2 67.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(y + 1\right)} \]

      6. +-commutative 67.1%

         \[\leadsto \frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \color{blue}{\left(1 + y\right)}} \]

   9. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}}{1} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)}} \]
   10. Step-by-step derivation

       1. /-rgt-identity 67.1%

          \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2}} \cdot \frac{\sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)} \]

       2. associate-*r/ 67.1%

          \[\leadsto \color{blue}{\frac{{\left(\sqrt[3]{\frac{y}{x}}\right)}^{2} \cdot \sqrt[3]{\frac{y}{x}}}{x + \left(1 + y\right)}} \]

       3. unpow2 67.1%

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

       4. rem-3cbrt-lft 67.9%

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

       5. +-commutative 67.9%

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

   11. Simplified 67.9%

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

   if -8.5999999999999998e-75 < x < -5.50000000000000031e-108

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   if -4.50000000000000004e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

      1. associate-*r/ 65.6%

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

      2. associate-+r+ 65.6%

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

      3. associate-/r* 71.0%

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

      4. clear-num 70.3%

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

      5. associate-+r+ 70.3%

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

      6. associate-/l* 83.8%

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

      7. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. unpow2 83.8%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.6%

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

      5. un-div-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around 0 57.6%

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

Alternative 7: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.95e-73)
     (/ (/ y x) (+ x 1.0))
     (if (<= x -5.5e-108)
       (/ x (* y (+ y 1.0)))
       (if (<= x -4.5e-140) (/ y (* x t_0)) (/ (/ x y) t_0))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.95e-73) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.95d-73)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-5.5d-108)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.5d-140)) then
        tmp = y / (x * t_0)
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.95e-73) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * t_0);
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.95e-73:
		tmp = (y / x) / (x + 1.0)
	elif x <= -5.5e-108:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.5e-140:
		tmp = y / (x * t_0)
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.95e-73)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -5.5e-108)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.5e-140)
		tmp = Float64(y / Float64(x * t_0));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.95e-73)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -5.5e-108)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.5e-140)
		tmp = y / (x * t_0);
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-73], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e-108], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-140], N[(y / N[(x * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.94999999999999991e-73

   1. Initial program 70.0%

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

      1. associate-/l* 76.0%

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

      2. associate-+l+ 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in y around 0 61.1%

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

      1. associate-/r* 68.9%

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

      2. +-commutative 68.9%

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

   7. Simplified 68.9%

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

   if -1.94999999999999991e-73 < x < -5.50000000000000031e-108

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   if -5.50000000000000031e-108 < x < -4.50000000000000004e-140

   1. Initial program 83.5%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 83.5%

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

      2. associate-+r+ 83.5%

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

      3. associate-/r* 83.7%

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

      4. clear-num 83.5%

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

      5. associate-+r+ 83.5%

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

      6. associate-/l* 99.5%

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

      7. pow2 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/r/ 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. frac-times 50.4%

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

      4. *-un-lft-identity 50.4%

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

      5. +-commutative 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. *-lft-identity 50.4%

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

       2. *-commutative 50.4%

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

       3. +-commutative 50.4%

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

   11. Simplified 50.4%

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

   if -4.50000000000000004e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

      1. associate-*r/ 65.6%

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

      2. associate-+r+ 65.6%

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

      3. associate-/r* 71.0%

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

      4. clear-num 70.3%

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

      5. associate-+r+ 70.3%

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

      6. associate-/l* 83.8%

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

      7. pow2 83.8%

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

   6. Applied egg-rr 83.8%

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

      1. *-un-lft-identity 83.8%

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

      2. unpow2 83.8%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. +-commutative 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. clear-num 99.7%

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

      2. associate-*r* 99.7%

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

      3. *-un-lft-identity 99.7%

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

      4. times-frac 99.6%

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

      5. un-div-inv 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. +-commutative 99.9%

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

   10. Applied egg-rr 99.9%

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

       1. /-rgt-identity 99.9%

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

       2. associate-*r/ 99.9%

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

       3. +-commutative 99.9%

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

   12. Simplified 99.9%

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

   13. Taylor expanded in x around 0 57.6%

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

Alternative 8: 61.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.2e-77)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -5.5e-108)
     (/ x (* y (+ y 1.0)))
     (if (<= x -4.5e-140) (/ y (* x (+ x (+ y 1.0)))) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-77) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.2d-77)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-5.5d-108)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.5d-140)) then
        tmp = y / (x * (x + (y + 1.0d0)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.2e-77) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * (x + (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.2e-77:
		tmp = (y / x) / (x + 1.0)
	elif x <= -5.5e-108:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.5e-140:
		tmp = y / (x * (x + (y + 1.0)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.2e-77)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -5.5e-108)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.5e-140)
		tmp = Float64(y / Float64(x * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.2e-77)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -5.5e-108)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.5e-140)
		tmp = y / (x * (x + (y + 1.0)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.2e-77], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e-108], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-140], N[(y / N[(x * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.20000000000000007e-77

   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* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in y around 0 60.3%

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

      1. associate-/r* 68.0%

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

      2. +-commutative 68.0%

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

   7. Simplified 68.0%

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

   if -2.20000000000000007e-77 < x < -5.50000000000000031e-108

   1. Initial program 99.7%

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

      1. associate-/l* 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 84.9%

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

      1. +-commutative 84.9%

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

   7. Simplified 84.9%

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

   if -5.50000000000000031e-108 < x < -4.50000000000000004e-140

   1. Initial program 83.5%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 83.5%

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

      2. associate-+r+ 83.5%

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

      3. associate-/r* 83.7%

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

      4. clear-num 83.5%

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

      5. associate-+r+ 83.5%

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

      6. associate-/l* 99.5%

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

      7. pow2 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 49.8%

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

      1. associate-/r/ 50.3%

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

      2. *-un-lft-identity 50.3%

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

      3. frac-times 50.4%

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

      4. *-un-lft-identity 50.4%

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

      5. +-commutative 50.4%

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

   9. Applied egg-rr 50.4%

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

       1. *-lft-identity 50.4%

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

       2. *-commutative 50.4%

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

       3. +-commutative 50.4%

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

   11. Simplified 50.4%

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

   if -4.50000000000000004e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in x around 0 55.4%

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

      1. +-commutative 55.4%

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

   7. Simplified 55.4%

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

      1. div-inv 55.5%

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

      2. associate-/r* 57.4%

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

   9. Applied egg-rr 57.4%

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

Alternative 9: 61.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-73)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -5.5e-108)
     (/ x (* y (+ y 1.0)))
     (if (<= x -4.5e-140) (* x (/ (/ y x) (+ x y))) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-73) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = x * ((y / x) / (x + y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.8d-73)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-5.5d-108)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.5d-140)) then
        tmp = x * ((y / x) / (x + y))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-73) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = x * ((y / x) / (x + y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e-73:
		tmp = (y / x) / (x + 1.0)
	elif x <= -5.5e-108:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.5e-140:
		tmp = x * ((y / x) / (x + y))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e-73)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -5.5e-108)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.5e-140)
		tmp = Float64(x * Float64(Float64(y / x) / Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e-73)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -5.5e-108)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.5e-140)
		tmp = x * ((y / x) / (x + y));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e-73], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e-108], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-140], N[(x * N[(N[(y / x), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -5.8e-73

   1. Initial program 70.0%

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

      1. associate-/l* 76.0%

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

      2. associate-+l+ 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in y around 0 61.1%

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

      1. associate-/r* 68.9%

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

      2. +-commutative 68.9%

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

   7. Simplified 68.9%

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

   if -5.8e-73 < x < -5.50000000000000031e-108

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   if -5.50000000000000031e-108 < x < -4.50000000000000004e-140

   1. Initial program 83.5%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

      1. *-un-lft-identity 99.6%

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

      2. associate-+r+ 99.6%

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

      3. associate-*l* 100.0%

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

      4. times-frac 99.5%

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

      5. associate-+r+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in y around 0 50.1%

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

       1. +-commutative 50.1%

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

   11. Simplified 50.1%

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

   12. Taylor expanded in x around 0 50.1%

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

   if -4.50000000000000004e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in x around 0 55.4%

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

      1. +-commutative 55.4%

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

   7. Simplified 55.4%

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

      1. div-inv 55.5%

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

      2. associate-/r* 57.4%

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

   9. Applied egg-rr 57.4%

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

Alternative 10: 61.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e-74)
   (/ (/ y x) (+ x 1.0))
   (if (<= x -5.5e-108)
     (/ x (* y (+ y 1.0)))
     (if (<= x -4.5e-140) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.8d-74)) then
        tmp = (y / x) / (x + 1.0d0)
    else if (x <= (-5.5d-108)) then
        tmp = x / (y * (y + 1.0d0))
    else if (x <= (-4.5d-140)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e-74) {
		tmp = (y / x) / (x + 1.0);
	} else if (x <= -5.5e-108) {
		tmp = x / (y * (y + 1.0));
	} else if (x <= -4.5e-140) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e-74:
		tmp = (y / x) / (x + 1.0)
	elif x <= -5.5e-108:
		tmp = x / (y * (y + 1.0))
	elif x <= -4.5e-140:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e-74)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (x <= -5.5e-108)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	elseif (x <= -4.5e-140)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e-74)
		tmp = (y / x) / (x + 1.0);
	elseif (x <= -5.5e-108)
		tmp = x / (y * (y + 1.0));
	elseif (x <= -4.5e-140)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e-74], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.5e-108], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.5e-140], 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 4 regimes

2. if x < -5.8e-74

   1. Initial program 70.0%

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

      1. associate-/l* 76.0%

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

      2. associate-+l+ 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in y around 0 61.1%

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

      1. associate-/r* 68.9%

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

      2. +-commutative 68.9%

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

   7. Simplified 68.9%

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

   if -5.8e-74 < x < -5.50000000000000031e-108

   1. Initial program 99.6%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 87.1%

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

      1. +-commutative 87.1%

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

   7. Simplified 87.1%

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

   if -5.50000000000000031e-108 < x < -4.50000000000000004e-140

   1. Initial program 83.5%

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

      1. associate-/l* 99.6%

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

      2. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around 0 49.7%

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

   if -4.50000000000000004e-140 < x

   1. Initial program 65.6%

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

      1. associate-/l* 76.7%

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

      2. associate-+l+ 76.7%

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

   3. Simplified 76.7%

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

   5. Taylor expanded in x around 0 55.4%

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

      1. +-commutative 55.4%

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

   7. Simplified 55.4%

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

      1. div-inv 55.5%

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

      2. associate-/r* 57.4%

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

   9. Applied egg-rr 57.4%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -4.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-83) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-83) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-83) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-83) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-83:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-83)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-83)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-83], 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 y < 3.0000000000000001e-83

   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* 77.8%

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

      2. associate-+l+ 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in y around 0 52.2%

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

   if 3.0000000000000001e-83 < y

   1. Initial program 72.5%

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

      1. associate-/l* 77.2%

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

      2. associate-+l+ 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. +-commutative 65.0%

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

   7. Simplified 65.0%

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

      1. div-inv 65.0%

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

      2. associate-/r* 66.9%

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

   9. Applied egg-rr 66.9%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.65e-82) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.65d-82) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-82:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-82)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-82)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-82], 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 y < 1.65000000000000011e-82

   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* 77.8%

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

      2. associate-+l+ 77.8%

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

   3. Simplified 77.8%

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

   5. Taylor expanded in y around 0 52.2%

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

   if 1.65000000000000011e-82 < y

   1. Initial program 72.5%

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

      1. associate-/l* 77.2%

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

      2. associate-+l+ 77.2%

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

   3. Simplified 77.2%

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

   5. Taylor expanded in x around 0 65.0%

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

      1. +-commutative 65.0%

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

   7. Simplified 65.0%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 27.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -105:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -105.0) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -105.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-105.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -105.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -105.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -105.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -105.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -105.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -105

   1. Initial program 65.0%

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

      1. associate-/l* 71.3%

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

      2. associate-+l+ 71.3%

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

   3. Simplified 71.3%

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

      1. associate-*r/ 65.0%

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

      2. associate-+r+ 65.0%

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

      3. associate-/r* 72.0%

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

      4. clear-num 71.9%

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

      5. associate-+r+ 71.9%

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

      6. associate-/l* 83.4%

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

      7. pow2 83.4%

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

   6. Applied egg-rr 83.4%

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

   7. Taylor expanded in x around inf 79.4%

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

   8. Taylor expanded in y around inf 5.8%

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

   if -105 < x

   1. Initial program 69.0%

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

      1. associate-/l* 79.3%

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

      2. associate-+l+ 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in x around 0 57.6%

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

      1. +-commutative 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in y around 0 34.6%

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

Alternative 14: 49.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. associate-/l* 77.7%

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

   2. associate-+l+ 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in x around 0 49.4%

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

   1. +-commutative 49.4%

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

7. Simplified 49.4%

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

Alternative 15: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.2%

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

   1. associate-/l* 77.7%

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

   2. associate-+l+ 77.7%

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

3. Simplified 77.7%

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

   1. associate-*r/ 68.2%

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

   2. associate-+r+ 68.2%

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

   3. associate-/r* 73.9%

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

   4. clear-num 73.4%

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

   5. associate-+r+ 73.4%

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

   6. associate-/l* 85.6%

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

   7. pow2 85.6%

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

6. Applied egg-rr 85.6%

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

7. Taylor expanded in x around inf 49.6%

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

8. Taylor expanded in y around inf 3.9%

   \[\leadsto \frac{1}{\color{blue}{x}} \]
9. Add Preprocessing

Alternative 16: 3.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

double code(double x, double y) {
	return -x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x
end function

Java

public static double code(double x, double y) {
	return -x;
}

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y)
	tmp = -x;
end

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 68.2%

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

   1. associate-/l* 77.7%

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

   2. associate-+l+ 77.7%

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

3. Simplified 77.7%

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

5. Taylor expanded in x around 0 49.4%

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

   1. +-commutative 49.4%

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

7. Simplified 49.4%

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

8. Taylor expanded in y around 0 13.0%

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

   1. neg-mul-1 13.0%

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

10. Simplified 13.0%

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

11. Taylor expanded in y around inf 3.8%

    \[\leadsto \color{blue}{-1 \cdot x} \]
12. Step-by-step derivation

    1. neg-mul-1 3.8%

       \[\leadsto \color{blue}{-x} \]

13. Simplified 3.8%

    \[\leadsto \color{blue}{-x} \]
14. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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