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

Percentage Accurate: 69.6% → 99.8%

Time: 14.4s

Alternatives: 19

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: 69.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. *-commutative 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 92.1%

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

   4. +-commutative 92.1%

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

   5. +-commutative 92.1%

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

   6. associate-+r+ 92.1%

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

   7. +-commutative 92.1%

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

   8. associate-+l+ 92.1%

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

4. Applied egg-rr 92.1%

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

   1. add-exp-log 63.9%

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

   2. associate-/r* 56.5%

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

   3. log-div 41.8%

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

   4. +-commutative 41.8%

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

   5. +-commutative 41.8%

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

   6. associate-+l+ 41.8%

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

   7. +-commutative 41.8%

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

   8. log1p-undefine 41.8%

      \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

   9. +-commutative 41.8%

      \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

6. Applied egg-rr 41.8%

   \[\leadsto \frac{y}{y + x} \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
7. Step-by-step derivation

   1. exp-diff 41.8%

      \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

   2. rem-exp-log 59.9%

      \[\leadsto \frac{y}{y + x} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

   3. log1p-undefine 59.9%

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

   4. rem-exp-log 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 69.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.7e-165)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.95e+103)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
     (* (/ y (+ y x)) (* (/ x (+ y x)) (/ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-165) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.95e+103) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (y / (y + x)) * ((x / (y + x)) * (1.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.7d-165) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.95d+103) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else
        tmp = (y / (y + x)) * ((x / (y + x)) * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.7e-165) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.95e+103) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else {
		tmp = (y / (y + x)) * ((x / (y + x)) * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.7e-165:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.95e+103:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	else:
		tmp = (y / (y + x)) * ((x / (y + x)) * (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.7e-165)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.95e+103)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.7e-165)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.95e+103)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	else
		tmp = (y / (y + x)) * ((x / (y + x)) * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.7e-165], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+103], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.6999999999999998e-165

   1. Initial program 68.9%

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

      1. associate-/l* 78.5%

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

      2. associate-+l+ 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in y around 0 57.2%

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

      1. associate-/r* 57.6%

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

      2. +-commutative 57.6%

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

   7. Simplified 57.6%

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

   if 2.6999999999999998e-165 < y < 2.9499999999999999e103

   1. Initial program 78.4%

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

      1. associate-/l* 89.1%

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

      2. associate-+l+ 89.1%

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

   3. Simplified 89.1%

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

   if 2.9499999999999999e103 < y

   1. Initial program 47.5%

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

      1. *-commutative 47.5%

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

      2. associate-*l* 47.5%

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

      3. times-frac 74.3%

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

      4. +-commutative 74.3%

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

      5. +-commutative 74.3%

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

      6. associate-+r+ 74.3%

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

      7. +-commutative 74.3%

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

      8. associate-+l+ 74.3%

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

   4. Applied egg-rr 74.3%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 80.1%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.7 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+103}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(t\_0 \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -7.2e+42)
     (* (/ t_0 (+ (+ y x) 1.0)) (/ y x))
     (if (<= x 4.35e+15)
       (* x (/ (/ y (* (+ y x) (+ y (+ x 1.0)))) (+ y x)))
       (* (/ y (+ y x)) (* t_0 (/ 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -7.2e+42) {
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 4.35e+15) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = (y / (y + x)) * (t_0 * (1.0 / 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 + x)
    if (x <= (-7.2d+42)) then
        tmp = (t_0 / ((y + x) + 1.0d0)) * (y / x)
    else if (x <= 4.35d+15) then
        tmp = x * ((y / ((y + x) * (y + (x + 1.0d0)))) / (y + x))
    else
        tmp = (y / (y + x)) * (t_0 * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -7.2e+42) {
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 4.35e+15) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = (y / (y + x)) * (t_0 * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -7.2e+42:
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x)
	elif x <= 4.35e+15:
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x))
	else:
		tmp = (y / (y + x)) * (t_0 * (1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -7.2e+42)
		tmp = Float64(Float64(t_0 / Float64(Float64(y + x) + 1.0)) * Float64(y / x));
	elseif (x <= 4.35e+15)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) / Float64(y + x)));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(t_0 * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -7.2e+42)
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	elseif (x <= 4.35e+15)
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	else
		tmp = (y / (y + x)) * (t_0 * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.2e+42], N[(N[(t$95$0 / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.35e+15], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.2000000000000002e42

   1. Initial program 61.1%

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

      1. *-commutative 61.1%

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

      2. associate-*l* 61.1%

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

      3. times-frac 81.3%

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

      4. +-commutative 81.3%

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

      5. +-commutative 81.3%

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

      6. associate-+r+ 81.3%

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

      7. +-commutative 81.3%

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

      8. associate-+l+ 81.3%

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

   4. Applied egg-rr 81.3%

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

   5. Taylor expanded in y around 0 75.9%

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

      1. add-exp-log 44.0%

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

      2. associate-/r* 3.1%

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

      3. log-div 0.0%

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

      4. +-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+l+ 0.0%

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

      7. +-commutative 0.0%

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

      8. log1p-undefine 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

      9. +-commutative 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{y}{x} \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 0.0%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

      2. rem-exp-log 8.6%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

      3. log1p-undefine 8.6%

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

      4. rem-exp-log 99.8%

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

   9. Simplified 85.8%

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

   if -7.2000000000000002e42 < x < 4.35e15

   1. Initial program 72.1%

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

      1. associate-/l* 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-un-lft-identity 82.8%

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

      2. associate-+r+ 82.8%

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

      3. associate-*l* 82.8%

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

      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. +-commutative 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-+r+ 99.5%

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

      8. +-commutative 99.5%

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

      9. associate-+l+ 99.5%

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

   6. Applied egg-rr 99.5%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

   8. Simplified 99.5%

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

   if 4.35e15 < x

   1. Initial program 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*l* 64.4%

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

      3. times-frac 84.5%

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

      4. +-commutative 84.5%

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

      5. +-commutative 84.5%

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

      6. associate-+r+ 84.5%

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

      7. +-commutative 84.5%

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

      8. associate-+l+ 84.5%

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

   4. Applied egg-rr 84.5%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 50.2%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 4.35 \cdot 10^{+15}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -3.75 \cdot 10^{+168}:\\ \;\;\;\;\frac{t\_0}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-86}:\\ \;\;\;\;t\_0 \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(t\_0 \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -3.75e+168)
     (* (/ t_0 (+ (+ y x) 1.0)) (/ y x))
     (if (<= x 4.2e-86)
       (* t_0 (/ y (* (+ y x) (+ y (+ x 1.0)))))
       (* (/ y (+ y x)) (* t_0 (/ 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -3.75e+168) {
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 4.2e-86) {
		tmp = t_0 * (y / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = (y / (y + x)) * (t_0 * (1.0 / 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 + x)
    if (x <= (-3.75d+168)) then
        tmp = (t_0 / ((y + x) + 1.0d0)) * (y / x)
    else if (x <= 4.2d-86) then
        tmp = t_0 * (y / ((y + x) * (y + (x + 1.0d0))))
    else
        tmp = (y / (y + x)) * (t_0 * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -3.75e+168) {
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 4.2e-86) {
		tmp = t_0 * (y / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = (y / (y + x)) * (t_0 * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -3.75e+168:
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x)
	elif x <= 4.2e-86:
		tmp = t_0 * (y / ((y + x) * (y + (x + 1.0))))
	else:
		tmp = (y / (y + x)) * (t_0 * (1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -3.75e+168)
		tmp = Float64(Float64(t_0 / Float64(Float64(y + x) + 1.0)) * Float64(y / x));
	elseif (x <= 4.2e-86)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(t_0 * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -3.75e+168)
		tmp = (t_0 / ((y + x) + 1.0)) * (y / x);
	elseif (x <= 4.2e-86)
		tmp = t_0 * (y / ((y + x) * (y + (x + 1.0))));
	else
		tmp = (y / (y + x)) * (t_0 * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.75e+168], N[(N[(t$95$0 / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-86], N[(t$95$0 * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.75e168

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

      3. times-frac 73.9%

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

      4. +-commutative 73.9%

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

      5. +-commutative 73.9%

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

      6. associate-+r+ 73.9%

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

      7. +-commutative 73.9%

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

      8. associate-+l+ 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in y around 0 73.9%

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

      1. add-exp-log 73.9%

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

      2. associate-/r* 2.9%

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

      3. log-div 0.0%

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

      4. +-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+l+ 0.0%

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

      7. +-commutative 0.0%

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

      8. log1p-undefine 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

      9. +-commutative 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{y}{x} \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 0.0%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

      2. rem-exp-log 5.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 100.0%

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

   9. Simplified 92.2%

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

   if -3.75e168 < x < 4.2e-86

   1. Initial program 69.2%

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

      1. associate-*l* 69.2%

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

      2. times-frac 97.9%

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

      3. +-commutative 97.9%

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

      4. +-commutative 97.9%

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

      5. associate-+r+ 97.9%

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

      6. +-commutative 97.9%

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

      7. associate-+l+ 97.9%

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

   4. Applied egg-rr 97.9%

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

   if 4.2e-86 < x

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

      3. times-frac 88.0%

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

      4. +-commutative 88.0%

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

      5. +-commutative 88.0%

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

      6. associate-+r+ 88.0%

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

      7. +-commutative 88.0%

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

      8. associate-+l+ 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 51.9%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.75 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{y + x}\\ t_1 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -5.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{t\_1}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(t\_1 \cdot \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ y x))) (t_1 (/ x (+ y x))))
   (if (<= x -5.9e+168)
     (* (/ t_1 (+ (+ y x) 1.0)) (/ y x))
     (if (<= x 9e-11)
       (* t_0 (/ x (* (+ y x) (+ y (+ x 1.0)))))
       (* t_0 (* t_1 (/ 1.0 y)))))))

C

double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = x / (y + x);
	double tmp;
	if (x <= -5.9e+168) {
		tmp = (t_1 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 9e-11) {
		tmp = t_0 * (x / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = t_0 * (t_1 * (1.0 / 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) :: t_1
    real(8) :: tmp
    t_0 = y / (y + x)
    t_1 = x / (y + x)
    if (x <= (-5.9d+168)) then
        tmp = (t_1 / ((y + x) + 1.0d0)) * (y / x)
    else if (x <= 9d-11) then
        tmp = t_0 * (x / ((y + x) * (y + (x + 1.0d0))))
    else
        tmp = t_0 * (t_1 * (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (y + x);
	double t_1 = x / (y + x);
	double tmp;
	if (x <= -5.9e+168) {
		tmp = (t_1 / ((y + x) + 1.0)) * (y / x);
	} else if (x <= 9e-11) {
		tmp = t_0 * (x / ((y + x) * (y + (x + 1.0))));
	} else {
		tmp = t_0 * (t_1 * (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (y + x)
	t_1 = x / (y + x)
	tmp = 0
	if x <= -5.9e+168:
		tmp = (t_1 / ((y + x) + 1.0)) * (y / x)
	elif x <= 9e-11:
		tmp = t_0 * (x / ((y + x) * (y + (x + 1.0))))
	else:
		tmp = t_0 * (t_1 * (1.0 / y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(y + x))
	t_1 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -5.9e+168)
		tmp = Float64(Float64(t_1 / Float64(Float64(y + x) + 1.0)) * Float64(y / x));
	elseif (x <= 9e-11)
		tmp = Float64(t_0 * Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))));
	else
		tmp = Float64(t_0 * Float64(t_1 * Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (y + x);
	t_1 = x / (y + x);
	tmp = 0.0;
	if (x <= -5.9e+168)
		tmp = (t_1 / ((y + x) + 1.0)) * (y / x);
	elseif (x <= 9e-11)
		tmp = t_0 * (x / ((y + x) * (y + (x + 1.0))));
	else
		tmp = t_0 * (t_1 * (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.9e+168], N[(N[(t$95$1 / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9e-11], N[(t$95$0 * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(t$95$1 * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.89999999999999986e168

   1. Initial program 55.1%

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

      1. *-commutative 55.1%

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

      2. associate-*l* 55.1%

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

      3. times-frac 73.9%

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

      4. +-commutative 73.9%

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

      5. +-commutative 73.9%

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

      6. associate-+r+ 73.9%

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

      7. +-commutative 73.9%

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

      8. associate-+l+ 73.9%

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

   4. Applied egg-rr 73.9%

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

   5. Taylor expanded in y around 0 73.9%

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

      1. add-exp-log 73.9%

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

      2. associate-/r* 2.9%

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

      3. log-div 0.0%

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

      4. +-commutative 0.0%

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

      5. +-commutative 0.0%

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

      6. associate-+l+ 0.0%

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

      7. +-commutative 0.0%

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

      8. log1p-undefine 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

      9. +-commutative 0.0%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

   7. Applied egg-rr 0.0%

      \[\leadsto \frac{y}{x} \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 0.0%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

      2. rem-exp-log 5.4%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

      3. log1p-undefine 5.4%

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

      4. rem-exp-log 100.0%

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

   9. Simplified 92.2%

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

   if -5.89999999999999986e168 < x < 8.9999999999999999e-11

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

      2. associate-*l* 70.8%

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

      3. times-frac 98.2%

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

      4. +-commutative 98.2%

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

      5. +-commutative 98.2%

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

      6. associate-+r+ 98.2%

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

      7. +-commutative 98.2%

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

      8. associate-+l+ 98.2%

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

   4. Applied egg-rr 98.2%

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

   if 8.9999999999999999e-11 < x

   1. Initial program 66.9%

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

      1. *-commutative 66.9%

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

      2. associate-*l* 66.9%

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

      3. times-frac 85.5%

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

      4. +-commutative 85.5%

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

      5. +-commutative 85.5%

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

      6. associate-+r+ 85.5%

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

      7. +-commutative 85.5%

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

      8. associate-+l+ 85.5%

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

   4. Applied egg-rr 85.5%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around inf 48.5%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.9 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-11}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{1}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.4e+64)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -5e-119)
     (/ y (* x (+ x 1.0)))
     (if (<= x 4.2e-86) (/ x (* y (+ y 1.0))) (/ (/ x y) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5e-119) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 4.2e-86) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.4d+64)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-5d-119)) then
        tmp = y / (x * (x + 1.0d0))
    else if (x <= 4.2d-86) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.4e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5e-119) {
		tmp = y / (x * (x + 1.0));
	} else if (x <= 4.2e-86) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.4e+64:
		tmp = (y / x) * (1.0 / x)
	elif x <= -5e-119:
		tmp = y / (x * (x + 1.0))
	elif x <= 4.2e-86:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.4e+64)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -5e-119)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (x <= 4.2e-86)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.4e+64)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -5e-119)
		tmp = y / (x * (x + 1.0));
	elseif (x <= 4.2e-86)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.4e+64], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-119], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.2e-86], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.39999999999999999e64

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*l* 59.2%

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

      3. times-frac 80.4%

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

      4. +-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. associate-+r+ 80.4%

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

      7. +-commutative 80.4%

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

      8. associate-+l+ 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in y around 0 74.8%

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

   6. Taylor expanded in x around inf 81.7%

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

   if -2.39999999999999999e64 < x < -4.99999999999999993e-119

   1. Initial program 82.3%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in y around 0 48.1%

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

   if -4.99999999999999993e-119 < x < 4.2e-86

   1. Initial program 63.6%

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

      1. associate-/l* 77.6%

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

      2. associate-+l+ 77.6%

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

   3. Simplified 77.6%

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

   5. Taylor expanded in x around 0 84.1%

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

      1. +-commutative 84.1%

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

   7. Simplified 84.1%

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

   if 4.2e-86 < x

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. associate-*l* 71.3%

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

      3. times-frac 88.0%

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

      4. +-commutative 88.0%

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

      5. +-commutative 88.0%

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

      6. associate-+r+ 88.0%

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

      7. +-commutative 88.0%

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

      8. associate-+l+ 88.0%

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

   4. Applied egg-rr 88.0%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/r* 88.1%

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

      7. clear-num 87.8%

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

      8. un-div-inv 87.9%

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

      9. associate-+r+ 87.9%

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

      10. +-commutative 87.9%

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

      11. *-commutative 87.9%

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

      12. *-un-lft-identity 87.9%

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

      13. times-frac 99.5%

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

      14. +-commutative 99.5%

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

      15. /-rgt-identity 99.5%

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

      16. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in y around inf 30.3%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 4.2 \cdot 10^{-86}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-144)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 2.7e+146) (/ x (* (+ y x) (+ y (+ x 1.0)))) (/ (/ x y) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.7e+146) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-144) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 2.7d+146) then
        tmp = x / ((y + x) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-144) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 2.7e+146) {
		tmp = x / ((y + x) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-144:
		tmp = (y / x) / (x + 1.0)
	elif y <= 2.7e+146:
		tmp = x / ((y + x) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-144)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 2.7e+146)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-144)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 2.7e+146)
		tmp = x / ((y + x) * (y + (x + 1.0)));
	else
		tmp = (x / y) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-144], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+146], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4999999999999998e-144

   1. Initial program 68.9%

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

      1. associate-/l* 78.4%

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

      2. associate-+l+ 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in y around 0 57.4%

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

      1. associate-/r* 57.8%

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

      2. +-commutative 57.8%

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

   7. Simplified 57.8%

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

   if 4.4999999999999998e-144 < y < 2.69999999999999989e146

   1. Initial program 72.0%

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

      1. *-commutative 72.0%

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

      2. associate-*l* 72.0%

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

      3. times-frac 92.8%

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

      4. +-commutative 92.8%

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

      5. +-commutative 92.8%

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

      6. associate-+r+ 92.8%

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

      7. +-commutative 92.8%

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

      8. associate-+l+ 92.8%

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in y around inf 68.7%

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

   if 2.69999999999999989e146 < y

   1. Initial program 52.6%

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

      1. *-commutative 52.6%

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

      2. associate-*l* 52.6%

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

      3. times-frac 68.7%

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

      4. +-commutative 68.7%

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

      5. +-commutative 68.7%

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

      6. associate-+r+ 68.7%

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

      7. +-commutative 68.7%

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

      8. associate-+l+ 68.7%

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

   4. Applied egg-rr 68.7%

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

      1. associate-/r* 99.9%

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

      2. div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+l+ 99.9%

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

      7. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. un-div-inv 99.9%

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

      3. +-commutative 99.9%

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

      4. +-commutative 99.9%

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

      5. associate-+r+ 99.9%

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

      6. associate-/r* 68.7%

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

      7. clear-num 68.7%

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

      8. un-div-inv 68.7%

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

      9. associate-+r+ 68.7%

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

      10. +-commutative 68.7%

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

      11. *-commutative 68.7%

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

      12. *-un-lft-identity 68.7%

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

      13. times-frac 99.9%

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

      14. +-commutative 99.9%

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

      15. /-rgt-identity 99.9%

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

      16. +-commutative 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 81.0%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-144}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-149)
   (* (/ (/ x (+ y x)) (+ (+ y x) 1.0)) (/ y x))
   (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-149) {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * (y / x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.6d-149)) then
        tmp = ((x / (y + x)) / ((y + x) + 1.0d0)) * (y / x)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-149) {
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * (y / x);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-149:
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * (y / x)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.6e-149)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) + 1.0)) * Float64(y / x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.6e-149)
		tmp = ((x / (y + x)) / ((y + x) + 1.0)) * (y / x);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.6e-149], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6000000000000002e-149

   1. Initial program 70.1%

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

      1. *-commutative 70.1%

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

      2. associate-*l* 70.1%

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

      3. times-frac 89.1%

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

      4. +-commutative 89.1%

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

      5. +-commutative 89.1%

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

      6. associate-+r+ 89.1%

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

      7. +-commutative 89.1%

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

      8. associate-+l+ 89.1%

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

   4. Applied egg-rr 89.1%

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

   5. Taylor expanded in y around 0 70.2%

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

      1. add-exp-log 50.1%

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

      2. associate-/r* 26.3%

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

      3. log-div 13.6%

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

      4. +-commutative 13.6%

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

      5. +-commutative 13.6%

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

      6. associate-+l+ 13.6%

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

      7. +-commutative 13.6%

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

      8. log1p-undefine 13.6%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

      9. +-commutative 13.6%

         \[\leadsto \frac{y}{y + x} \cdot e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

   7. Applied egg-rr 12.8%

      \[\leadsto \frac{y}{x} \cdot \color{blue}{e^{\log \left(\frac{x}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
   8. Step-by-step derivation

      1. exp-diff 13.6%

         \[\leadsto \frac{y}{y + x} \cdot \color{blue}{\frac{e^{\log \left(\frac{x}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

      2. rem-exp-log 30.9%

         \[\leadsto \frac{y}{y + x} \cdot \frac{\color{blue}{\frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

      3. log1p-undefine 30.9%

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

      4. rem-exp-log 99.8%

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

   9. Simplified 76.0%

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

   if -3.6000000000000002e-149 < x

   1. Initial program 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*l* 66.1%

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

      3. times-frac 94.4%

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

      4. +-commutative 94.4%

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

      5. +-commutative 94.4%

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

      6. associate-+r+ 94.4%

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

      7. +-commutative 94.4%

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

      8. associate-+l+ 94.4%

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

   4. Applied egg-rr 94.4%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/r* 94.5%

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

      7. clear-num 94.1%

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

      8. un-div-inv 94.2%

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

      9. associate-+r+ 94.2%

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

      10. +-commutative 94.2%

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

      11. *-commutative 94.2%

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

      12. *-un-lft-identity 94.2%

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

      13. times-frac 99.5%

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

      14. +-commutative 99.5%

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

      15. /-rgt-identity 99.5%

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

      16. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around 0 59.3%

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

       1. +-commutative 59.3%

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

   11. Simplified 59.3%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) + 1} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -5e-119) (/ y x) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5e-119) {
		tmp = y / x;
	} 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 <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-5d-119)) then
        tmp = y / x
    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 <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -5e-119) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -5e-119:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -5e-119)
		tmp = Float64(y / x);
	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 (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -5e-119)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-119], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*l* 62.4%

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

      3. times-frac 83.8%

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

      4. +-commutative 83.8%

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

      5. +-commutative 83.8%

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

      6. associate-+r+ 83.8%

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

      7. +-commutative 83.8%

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

      8. associate-+l+ 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in y around 0 75.3%

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

   6. Taylor expanded in x around inf 77.0%

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

   if -1 < x < -4.99999999999999993e-119

   1. Initial program 84.3%

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

      1. associate-/l* 89.2%

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

      2. associate-+l+ 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around 0 45.3%

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

      1. associate-/r* 45.3%

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

      2. +-commutative 45.3%

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

   7. Simplified 45.3%

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

   8. Taylor expanded in x around 0 40.9%

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

   if -4.99999999999999993e-119 < x

   1. Initial program 67.0%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. +-commutative 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-119}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.5e+64)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -2.1e-125) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.5e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.1e-125) {
		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 <= (-1.5d+64)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-2.1d-125)) 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 <= -1.5e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.1e-125) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.5e+64:
		tmp = (y / x) * (1.0 / x)
	elif x <= -2.1e-125:
		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 <= -1.5e+64)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -2.1e-125)
		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 (x <= -1.5e+64)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -2.1e-125)
		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, -1.5e+64], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.1e-125], 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 3 regimes

2. if x < -1.5000000000000001e64

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*l* 59.2%

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

      3. times-frac 80.4%

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

      4. +-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. associate-+r+ 80.4%

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

      7. +-commutative 80.4%

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

      8. associate-+l+ 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in y around 0 74.8%

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

   6. Taylor expanded in x around inf 81.7%

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

   if -1.5000000000000001e64 < x < -2.1e-125

   1. Initial program 82.3%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in y around 0 48.1%

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

   if -2.1e-125 < x

   1. Initial program 67.0%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 58.3%

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

      1. +-commutative 58.3%

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

   7. Simplified 58.3%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e+64)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -1.1e-120) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.1e-120) {
		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 <= (-1.2d+64)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-1.1d-120)) 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 <= -1.2e+64) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -1.1e-120) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e+64:
		tmp = (y / x) * (1.0 / x)
	elif x <= -1.1e-120:
		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 <= -1.2e+64)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -1.1e-120)
		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 <= -1.2e+64)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -1.1e-120)
		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, -1.2e+64], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.1e-120], 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 3 regimes

2. if x < -1.2e64

   1. Initial program 59.2%

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

      1. *-commutative 59.2%

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

      2. associate-*l* 59.2%

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

      3. times-frac 80.4%

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

      4. +-commutative 80.4%

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

      5. +-commutative 80.4%

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

      6. associate-+r+ 80.4%

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

      7. +-commutative 80.4%

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

      8. associate-+l+ 80.4%

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

   4. Applied egg-rr 80.4%

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

   5. Taylor expanded in y around 0 74.8%

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

   6. Taylor expanded in x around inf 81.7%

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

   if -1.2e64 < x < -1.10000000000000006e-120

   1. Initial program 82.3%

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

      1. associate-/l* 85.6%

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

      2. associate-+l+ 85.6%

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

   3. Simplified 85.6%

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

   5. Taylor expanded in y around 0 48.1%

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

   if -1.10000000000000006e-120 < x

   1. Initial program 67.0%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. associate-/r* 58.3%

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

      2. +-commutative 58.3%

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

   7. Simplified 58.3%

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

      1. associate-*r/ 60.0%

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

      2. div-inv 60.1%

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

   9. Applied egg-rr 60.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -1.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.35e-120) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-120) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.35d-120)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.35e-120) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.35e-120:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.35e-120)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.35e-120)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.35e-120], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3499999999999999e-120

   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* 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 62.3%

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

      1. associate-/r* 67.4%

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

      2. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if -1.3499999999999999e-120 < x

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 67.0%

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

      3. times-frac 94.6%

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

      4. +-commutative 94.6%

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

      5. +-commutative 94.6%

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

      6. associate-+r+ 94.6%

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

      7. +-commutative 94.6%

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

      8. associate-+l+ 94.6%

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

   4. Applied egg-rr 94.6%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/r* 94.6%

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

      7. clear-num 94.3%

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

      8. un-div-inv 94.3%

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

      9. associate-+r+ 94.3%

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

      10. +-commutative 94.3%

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

      11. *-commutative 94.3%

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

      12. *-un-lft-identity 94.3%

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

      13. times-frac 99.5%

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

      14. +-commutative 99.5%

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

      15. /-rgt-identity 99.5%

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

      16. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around 0 60.4%

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

       1. +-commutative 60.4%

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

   11. Simplified 60.4%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-119) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-119) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d-119)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-119) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-119:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-119)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e-119)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-119], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000017e-119

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 69.0%

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

      2. associate-*l* 69.0%

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

      3. times-frac 88.6%

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

      4. +-commutative 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-+r+ 88.6%

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

      7. +-commutative 88.6%

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

      8. associate-+l+ 88.6%

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

   4. Applied egg-rr 88.6%

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

   5. Taylor expanded in y around 0 67.8%

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

      1. +-commutative 67.8%

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

   7. Simplified 67.8%

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

      1. un-div-inv 67.8%

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

      2. +-commutative 67.8%

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

   9. Applied egg-rr 67.8%

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

   if -4.80000000000000017e-119 < x

   1. Initial program 67.0%

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

      1. *-commutative 67.0%

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

      2. associate-*l* 67.0%

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

      3. times-frac 94.6%

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

      4. +-commutative 94.6%

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

      5. +-commutative 94.6%

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

      6. associate-+r+ 94.6%

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

      7. +-commutative 94.6%

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

      8. associate-+l+ 94.6%

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

   4. Applied egg-rr 94.6%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+l+ 99.8%

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

      7. +-commutative 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.8%

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

      2. un-div-inv 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. associate-/r* 94.6%

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

      7. clear-num 94.3%

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

      8. un-div-inv 94.3%

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

      9. associate-+r+ 94.3%

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

      10. +-commutative 94.3%

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

      11. *-commutative 94.3%

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

      12. *-un-lft-identity 94.3%

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

      13. times-frac 99.5%

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

      14. +-commutative 99.5%

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

      15. /-rgt-identity 99.5%

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

      16. +-commutative 99.5%

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

   8. Applied egg-rr 99.5%

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

   9. Taylor expanded in x around 0 60.4%

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

       1. +-commutative 60.4%

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

   11. Simplified 60.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (* (/ y x) (/ 1.0 x)) (if (<= x -7e-151) (/ y x) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7e-151) {
		tmp = y / 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 <= (-1.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-7d-151)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -7e-151) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -7e-151:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -7e-151)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -7e-151)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -7e-151], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 62.4%

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

      1. *-commutative 62.4%

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

      2. associate-*l* 62.4%

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

      3. times-frac 83.8%

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

      4. +-commutative 83.8%

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

      5. +-commutative 83.8%

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

      6. associate-+r+ 83.8%

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

      7. +-commutative 83.8%

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

      8. associate-+l+ 83.8%

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

   4. Applied egg-rr 83.8%

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

   5. Taylor expanded in y around 0 75.3%

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

   6. Taylor expanded in x around inf 77.0%

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

   if -1 < x < -6.99999999999999991e-151

   1. Initial program 83.8%

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

      1. associate-/l* 90.5%

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

      2. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in y around 0 42.0%

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

      1. associate-/r* 42.1%

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

      2. +-commutative 42.1%

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

   7. Simplified 42.1%

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

   8. Taylor expanded in x around 0 38.3%

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

   if -6.99999999999999991e-151 < x

   1. Initial program 66.5%

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

      1. associate-/l* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

   5. 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 40.3%

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

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -7 \cdot 10^{-151}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.6e-122) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.6e-122) {
		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 <= (-3.6d-122)) 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 <= -3.6e-122) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.6e-122:
		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 <= -3.6e-122)
		tmp = Float64(Float64(y / 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 <= -3.6e-122)
		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, -3.6e-122], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.59999999999999994e-122

   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* 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 62.3%

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

      1. associate-/r* 67.4%

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

      2. +-commutative 67.4%

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

   7. Simplified 67.4%

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

   if -3.59999999999999994e-122 < x

   1. Initial program 67.0%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 58.2%

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

      1. associate-/r* 58.3%

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

      2. +-commutative 58.3%

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

   7. Simplified 58.3%

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

      1. associate-*r/ 60.0%

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

      2. div-inv 60.1%

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

   9. Applied egg-rr 60.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.6 \cdot 10^{-122}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 34.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.36e-148) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-148) {
		tmp = y / 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 (y <= 1.36d-148) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.36e-148) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.36e-148:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.36e-148)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.36e-148)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.36e-148], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.36e-148

   1. Initial program 68.9%

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

      1. associate-/l* 78.4%

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

      2. associate-+l+ 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in y around 0 57.4%

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

      1. associate-/r* 57.8%

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

      2. +-commutative 57.8%

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

   7. Simplified 57.8%

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

   8. Taylor expanded in x around 0 39.0%

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

   if 1.36e-148 < y

   1. Initial program 66.1%

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

      1. associate-/l* 78.6%

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

      2. associate-+l+ 78.6%

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

   3. Simplified 78.6%

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

   5. Taylor expanded in x around 0 55.8%

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

      1. +-commutative 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in y around 0 29.2%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 4.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. *-commutative 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 92.1%

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

   4. +-commutative 92.1%

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

   5. +-commutative 92.1%

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

   6. associate-+r+ 92.1%

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

   7. +-commutative 92.1%

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

   8. associate-+l+ 92.1%

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

4. Applied egg-rr 92.1%

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

5. Taylor expanded in y around 0 56.0%

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

6. Taylor expanded in y around inf 3.5%

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

   1. mul-1-neg 3.5%

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

   2. unsub-neg 3.5%

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

   3. *-commutative 3.5%

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

8. Simplified 3.5%

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

9. Taylor expanded in y around inf 4.0%

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

10. Final simplification 4.0%

    \[\leadsto \frac{1}{y} \]
11. Add Preprocessing

Alternative 18: 26.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 78.5%

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

   2. associate-+l+ 78.5%

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

3. Simplified 78.5%

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

5. Taylor expanded in x around 0 46.1%

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

   1. +-commutative 46.1%

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

7. Simplified 46.1%

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

8. Taylor expanded in y around 0 25.7%

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

9. Final simplification 25.7%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Alternative 19: 3.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.8%

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

   1. *-commutative 67.8%

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

   2. associate-*l* 67.8%

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

   3. times-frac 92.1%

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

   4. +-commutative 92.1%

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

   5. +-commutative 92.1%

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

   6. associate-+r+ 92.1%

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

   7. +-commutative 92.1%

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

   8. associate-+l+ 92.1%

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

4. Applied egg-rr 92.1%

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

5. Taylor expanded in y around 0 51.6%

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

   1. +-commutative 51.6%

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

7. Simplified 51.6%

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

8. Taylor expanded in x around 0 3.3%

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

9. Final simplification 3.3%

   \[\leadsto 1 \]
10. 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 2024130 
(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))))