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

Percentage Accurate: 69.3% → 99.8%

Time: 18.1s

Alternatives: 17

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: 69.3% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

   1. *-commutative 71.0%

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

   2. associate-*l* 71.0%

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

   3. times-frac 94.4%

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

   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. div-inv 94.3%

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

   2. +-commutative 94.3%

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

   3. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

   1. associate-*r/ 94.4%

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

   2. *-rgt-identity 94.4%

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

   3. associate-/r* 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

8. Simplified 99.8%

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

   1. *-commutative 99.8%

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

   2. associate-+r+ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-+r+ 99.8%

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

   5. clear-num 99.8%

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

   6. +-commutative 99.8%

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

   7. times-frac 99.0%

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

   8. associate-/l* 99.0%

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

   9. associate-*l/ 99.0%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 69.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 10^{+92}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-164)
   (/ (/ y (+ x 1.0)) (+ x y))
   (if (<= y 1e+92)
     (* x (/ y (* (+ x (+ y 1.0)) (* (+ x y) (+ x y)))))
     (/ (/ x (+ x y)) (+ (+ y 1.0) (* x 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-164) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 1e+92) {
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.8d-164) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else if (y <= 1d+92) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((x + y) * (x + y))))
    else
        tmp = (x / (x + y)) / ((y + 1.0d0) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-164) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 1e+92) {
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.8e-164:
		tmp = (y / (x + 1.0)) / (x + y)
	elif y <= 1e+92:
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))))
	else:
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-164)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	elseif (y <= 1e+92)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(x + y) * Float64(x + y)))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(y + 1.0) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-164)
		tmp = (y / (x + 1.0)) / (x + y);
	elseif (y <= 1e+92)
		tmp = x * (y / ((x + (y + 1.0)) * ((x + y) * (x + y))));
	else
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.8e-164], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+92], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.8000000000000001e-164

   1. Initial program 71.8%

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

      1. *-commutative 71.8%

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

      2. associate-*l* 71.8%

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

      3. times-frac 94.3%

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

      4. +-commutative 94.3%

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

      5. +-commutative 94.3%

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

      6. associate-+r+ 94.3%

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

      7. +-commutative 94.3%

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

      8. associate-+l+ 94.3%

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

   4. Applied egg-rr 94.3%

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

      1. div-inv 94.2%

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

      2. +-commutative 94.2%

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

      3. +-commutative 94.2%

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

   6. Applied egg-rr 94.2%

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

      1. associate-*r/ 94.3%

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

      2. *-rgt-identity 94.3%

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

      3. associate-/r* 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. times-frac 98.9%

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

      8. associate-/l* 98.9%

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

      9. associate-*l/ 99.0%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in y around 0 51.5%

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

       1. +-commutative 51.5%

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

   13. Simplified 51.5%

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

   if 2.8000000000000001e-164 < y < 1e92

   1. Initial program 84.2%

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

      1. associate-/l* 87.6%

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

      2. associate-+l+ 87.6%

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

   3. Simplified 87.6%

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

   if 1e92 < y

   1. Initial program 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*l* 55.9%

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

      3. times-frac 93.1%

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

      4. +-commutative 93.1%

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

      5. +-commutative 93.1%

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

      6. associate-+r+ 93.1%

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

      7. +-commutative 93.1%

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

      8. associate-+l+ 93.1%

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

   4. Applied egg-rr 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 98.5%

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

      5. +-commutative 98.5%

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

      6. +-commutative 98.5%

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

      7. +-commutative 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in y around inf 78.3%

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

      1. associate-+r+ 78.3%

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

      2. +-commutative 78.3%

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

      3. *-commutative 78.3%

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

   9. Simplified 78.3%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 10^{+92}:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.22e+162)
   (* (/ (/ x (+ x y)) (+ x (+ y 1.0))) (/ y x))
   (* (/ y (+ x y)) (/ x (* (+ x y) (+ y (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.22e+162) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 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 <= (-1.22d+162)) then
        tmp = ((x / (x + y)) / (x + (y + 1.0d0))) * (y / x)
    else
        tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.22e+162) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.22e+162:
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x)
	else:
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.22e+162)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + Float64(y + 1.0))) * Float64(y / x));
	else
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.22e+162)
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	else
		tmp = (y / (x + y)) * (x / ((x + y) * (y + (x + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.22e+162], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e162

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. times-frac 85.1%

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

      4. +-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. associate-+r+ 85.1%

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

      7. +-commutative 85.1%

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

      8. associate-+l+ 85.1%

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

   4. Applied egg-rr 85.1%

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

      1. div-inv 85.1%

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

      2. +-commutative 85.1%

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

      3. +-commutative 85.1%

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

   6. Applied egg-rr 85.1%

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

      1. associate-*r/ 85.1%

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

      2. *-rgt-identity 85.1%

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

      3. associate-/r* 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

   8. Simplified 99.9%

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

   9. Taylor expanded in y around 0 97.4%

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

   if -1.22e162 < x

   1. Initial program 72.8%

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

      1. *-commutative 72.8%

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

      2. associate-*l* 72.8%

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

      3. times-frac 96.0%

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

      4. +-commutative 96.0%

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

      5. +-commutative 96.0%

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

      6. associate-+r+ 96.0%

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

      7. +-commutative 96.0%

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

      8. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.12e-142)
   (/ (/ y (+ x 1.0)) (+ x y))
   (if (<= y 6.8e+165) (/ x (* (+ x y) (+ y (+ x 1.0)))) (/ (/ x y) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.12e-142) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 6.8e+165) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (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 <= 1.12d-142) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else if (y <= 6.8d+165) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.12e-142) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else if (y <= 6.8e+165) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.12e-142:
		tmp = (y / (x + 1.0)) / (x + y)
	elif y <= 6.8e+165:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.12e-142)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	elseif (y <= 6.8e+165)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.12e-142)
		tmp = (y / (x + 1.0)) / (x + y);
	elseif (y <= 6.8e+165)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.12e-142], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+165], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.1199999999999999e-142

   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. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 72.1%

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

      2. associate-*l* 72.1%

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

      3. times-frac 94.3%

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

      4. +-commutative 94.3%

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

      5. +-commutative 94.3%

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

      6. associate-+r+ 94.3%

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

      7. +-commutative 94.3%

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

      8. associate-+l+ 94.3%

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

   4. Applied egg-rr 94.3%

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

      1. div-inv 94.3%

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

      2. +-commutative 94.3%

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

      3. +-commutative 94.3%

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

   6. Applied egg-rr 94.3%

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

      1. associate-*r/ 94.3%

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

      2. *-rgt-identity 94.3%

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

      3. associate-/r* 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. times-frac 99.0%

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

      8. associate-/l* 98.9%

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

      9. associate-*l/ 99.0%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in y around 0 52.1%

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

       1. +-commutative 52.1%

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

   13. Simplified 52.1%

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

   if 1.1199999999999999e-142 < y < 6.80000000000000022e165

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l* 65.7%

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

      3. times-frac 96.1%

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

      4. +-commutative 96.1%

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

      5. +-commutative 96.1%

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

      6. associate-+r+ 96.1%

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

      7. +-commutative 96.1%

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

      8. associate-+l+ 96.1%

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

   4. Applied egg-rr 96.1%

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

   5. Taylor expanded in y around inf 72.9%

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

   if 6.80000000000000022e165 < y

   1. Initial program 76.9%

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

      1. *-commutative 76.9%

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

      2. associate-*l* 76.9%

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

      3. times-frac 90.7%

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

      4. +-commutative 90.7%

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

      5. +-commutative 90.7%

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

      6. associate-+r+ 90.7%

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

      7. +-commutative 90.7%

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

      8. associate-+l+ 90.7%

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

   4. Applied egg-rr 90.7%

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

      1. div-inv 90.7%

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

      2. +-commutative 90.7%

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

      3. +-commutative 90.7%

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

   6. Applied egg-rr 90.7%

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

      1. associate-*r/ 90.7%

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

      2. *-rgt-identity 90.7%

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

      3. associate-/r* 99.9%

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

      4. associate-+r+ 99.9%

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

      5. +-commutative 99.9%

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

      6. associate-+r+ 99.9%

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

   8. Simplified 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-+r+ 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-+r+ 99.9%

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

      5. clear-num 99.9%

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

      6. +-commutative 99.9%

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

      7. times-frac 99.9%

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

      8. associate-/l* 99.9%

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

      9. associate-*l/ 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in y around inf 99.9%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.12 \cdot 10^{-142}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+165}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1e-147)
   (* (/ (/ x (+ x y)) (+ x (+ y 1.0))) (/ y x))
   (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1e-147) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (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 <= (-1d-147)) then
        tmp = ((x / (x + y)) / (x + (y + 1.0d0))) * (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 <= -1e-147) {
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1e-147:
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1e-147)
		tmp = Float64(Float64(Float64(x / Float64(x + y)) / Float64(x + Float64(y + 1.0))) * Float64(y / x));
	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 <= -1e-147)
		tmp = ((x / (x + y)) / (x + (y + 1.0))) * (y / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1e-147], N[(N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999997e-148

   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 92.4%

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

      4. +-commutative 92.4%

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

      5. +-commutative 92.4%

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

      6. associate-+r+ 92.4%

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

      7. +-commutative 92.4%

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

      8. associate-+l+ 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. div-inv 92.3%

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

      2. +-commutative 92.3%

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

      3. +-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. associate-*r/ 92.4%

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

      2. *-rgt-identity 92.4%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 82.9%

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

   if -9.9999999999999997e-148 < x

   1. Initial program 73.0%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 62.6%

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

      1. associate-/r* 63.8%

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

      2. +-commutative 63.8%

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

   7. Simplified 63.8%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{t\_0}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= x -4.2e-148)
     (* (/ t_0 (+ x (+ y 1.0))) (/ y x))
     (/ t_0 (+ (+ y 1.0) (* x 2.0))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -4.2e-148) {
		tmp = (t_0 / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = t_0 / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (x <= (-4.2d-148)) then
        tmp = (t_0 / (x + (y + 1.0d0))) * (y / x)
    else
        tmp = t_0 / ((y + 1.0d0) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (x <= -4.2e-148) {
		tmp = (t_0 / (x + (y + 1.0))) * (y / x);
	} else {
		tmp = t_0 / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if x <= -4.2e-148:
		tmp = (t_0 / (x + (y + 1.0))) * (y / x)
	else:
		tmp = t_0 / ((y + 1.0) + (x * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (x <= -4.2e-148)
		tmp = Float64(Float64(t_0 / Float64(x + Float64(y + 1.0))) * Float64(y / x));
	else
		tmp = Float64(t_0 / Float64(Float64(y + 1.0) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (x <= -4.2e-148)
		tmp = (t_0 / (x + (y + 1.0))) * (y / x);
	else
		tmp = t_0 / ((y + 1.0) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.2e-148], N[(N[(t$95$0 / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + 1.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2e-148

   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 92.4%

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

      4. +-commutative 92.4%

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

      5. +-commutative 92.4%

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

      6. associate-+r+ 92.4%

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

      7. +-commutative 92.4%

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

      8. associate-+l+ 92.4%

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

   4. Applied egg-rr 92.4%

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

      1. div-inv 92.3%

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

      2. +-commutative 92.3%

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

      3. +-commutative 92.3%

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

   6. Applied egg-rr 92.3%

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

      1. associate-*r/ 92.4%

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

      2. *-rgt-identity 92.4%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in y around 0 82.9%

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

   if -4.2e-148 < x

   1. Initial program 73.0%

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

      1. *-commutative 73.0%

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

      2. associate-*l* 73.0%

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

      3. times-frac 95.7%

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

      4. +-commutative 95.7%

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

      5. +-commutative 95.7%

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

      6. associate-+r+ 95.7%

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

      7. +-commutative 95.7%

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

      8. associate-+l+ 95.7%

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

   4. Applied egg-rr 95.7%

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

      1. *-commutative 95.7%

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

      2. associate-/r* 99.8%

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

      3. clear-num 99.8%

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

      4. frac-times 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. +-commutative 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in y around inf 65.1%

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

      1. associate-+r+ 65.1%

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

      2. +-commutative 65.1%

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

      3. *-commutative 65.1%

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

   9. Simplified 65.1%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{x + \left(y + 1\right)} \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 47.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -550.0)
   (/ (/ y x) x)
   (if (<= x -3.4e-150) (* y (/ 1.0 (+ x y))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / x;
	} else if (x <= -3.4e-150) {
		tmp = y * (1.0 / (x + y));
	} 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 <= (-550.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-3.4d-150)) then
        tmp = y * (1.0d0 / (x + y))
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / x;
	} else if (x <= -3.4e-150) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -550.0:
		tmp = (y / x) / x
	elif x <= -3.4e-150:
		tmp = y * (1.0 / (x + y))
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -3.4e-150)
		tmp = Float64(y * Float64(1.0 / Float64(x + y)));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -550.0)
		tmp = (y / x) / x;
	elseif (x <= -3.4e-150)
		tmp = y * (1.0 / (x + y));
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -550.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -3.4e-150], N[(y * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -550

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l* 61.4%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. un-div-inv 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in y around 0 73.9%

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

   if -550 < x < -3.39999999999999999e-150

   1. Initial program 83.5%

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

      1. *-un-lft-identity 83.5%

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

      2. associate-*l* 83.5%

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

      3. times-frac 89.8%

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

      4. +-commutative 89.8%

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

      5. *-commutative 89.8%

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

      6. +-commutative 89.8%

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

      7. associate-+r+ 89.8%

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

      8. +-commutative 89.8%

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

      9. associate-+l+ 89.8%

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

   4. Applied egg-rr 89.8%

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

   5. Taylor expanded in y around 0 37.3%

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

      1. +-commutative 37.3%

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

   7. Simplified 37.3%

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

   8. Taylor expanded in x around 0 35.5%

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

   if -3.39999999999999999e-150 < x

   1. Initial program 73.0%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 62.6%

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

   6. Taylor expanded in y around 0 38.9%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -3.4 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -550.0)
   (/ (/ y x) x)
   (if (<= x -2.1e-125) (* y (/ 1.0 (+ x y))) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / x;
	} else if (x <= -2.1e-125) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-550.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-2.1d-125)) then
        tmp = y * (1.0d0 / (x + y))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / x;
	} else if (x <= -2.1e-125) {
		tmp = y * (1.0 / (x + y));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -550.0:
		tmp = (y / x) / x
	elif x <= -2.1e-125:
		tmp = y * (1.0 / (x + y))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.1e-125)
		tmp = Float64(y * Float64(1.0 / Float64(x + y)));
	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 <= -550.0)
		tmp = (y / x) / x;
	elseif (x <= -2.1e-125)
		tmp = y * (1.0 / (x + y));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -550.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.1e-125], N[(y * N[(1.0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -550

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l* 61.4%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. un-div-inv 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in y around 0 73.9%

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

   if -550 < x < -2.1e-125

   1. Initial program 84.1%

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

      1. *-un-lft-identity 84.1%

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

      2. associate-*l* 84.2%

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

      3. times-frac 91.7%

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

      4. +-commutative 91.7%

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

      5. *-commutative 91.7%

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

      6. +-commutative 91.7%

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

      7. associate-+r+ 91.7%

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

      8. +-commutative 91.7%

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

      9. associate-+l+ 91.7%

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in y around 0 40.3%

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

      1. +-commutative 40.3%

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

   7. Simplified 40.3%

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

   8. Taylor expanded in x around 0 38.1%

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

   if -2.1e-125 < x

   1. Initial program 73.3%

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

      1. associate-/l* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 63.1%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;y \cdot \frac{1}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.5e+162)
   (/ (/ y x) 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 <= -2.5e+162) {
		tmp = (y / x) / 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 <= (-2.5d+162)) then
        tmp = (y / x) / 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 <= -2.5e+162) {
		tmp = (y / x) / 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 <= -2.5e+162:
		tmp = (y / x) / 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 <= -2.5e+162)
		tmp = Float64(Float64(y / x) / 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 <= -2.5e+162)
		tmp = (y / x) / 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, -2.5e+162], N[(N[(y / x), $MachinePrecision] / x), $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 < -2.4999999999999998e162

   1. Initial program 60.3%

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

      1. *-commutative 60.3%

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

      2. associate-*l* 60.3%

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

      3. times-frac 85.1%

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

      4. +-commutative 85.1%

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

      5. +-commutative 85.1%

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

      6. associate-+r+ 85.1%

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

      7. +-commutative 85.1%

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

      8. associate-+l+ 85.1%

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

   4. Applied egg-rr 85.1%

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

   5. Taylor expanded in x around inf 92.5%

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

      1. un-div-inv 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y around 0 92.4%

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

   if -2.4999999999999998e162 < x < -2.1e-125

   1. Initial program 71.7%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in y around 0 47.3%

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

      1. +-commutative 47.3%

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

   7. Simplified 47.3%

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

   if -2.1e-125 < x

   1. Initial program 73.3%

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

      1. associate-/l* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 63.1%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+162}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.7e+163)
   (/ (/ y x) x)
   (if (<= x -1.6e-125) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.7e+163) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-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 <= (-4.7d+163)) then
        tmp = (y / x) / x
    else if (x <= (-1.6d-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 <= -4.7e+163) {
		tmp = (y / x) / x;
	} else if (x <= -1.6e-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 <= -4.7e+163:
		tmp = (y / x) / x
	elif x <= -1.6e-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 <= -4.7e+163)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.6e-125)
		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 <= -4.7e+163)
		tmp = (y / x) / x;
	elseif (x <= -1.6e-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, -4.7e+163], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.6e-125], 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 < -4.70000000000000019e163

   1. Initial program 61.9%

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

      1. *-commutative 61.9%

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

      2. associate-*l* 61.9%

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

      3. times-frac 84.7%

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

      4. +-commutative 84.7%

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

      5. +-commutative 84.7%

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

      6. associate-+r+ 84.7%

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

      7. +-commutative 84.7%

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

      8. associate-+l+ 84.7%

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

   4. Applied egg-rr 84.7%

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

   5. Taylor expanded in x around inf 94.9%

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

      1. un-div-inv 94.9%

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

   7. Applied egg-rr 94.9%

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

   8. Taylor expanded in y around 0 94.8%

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

   if -4.70000000000000019e163 < x < -1.5999999999999999e-125

   1. Initial program 70.4%

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

      1. associate-/l* 79.5%

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

      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 y around 0 48.2%

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

      1. +-commutative 48.2%

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

   7. Simplified 48.2%

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

   if -1.5999999999999999e-125 < x

   1. Initial program 73.3%

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

      1. associate-/l* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 63.1%

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

      1. associate-/r* 64.3%

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

      2. +-commutative 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-125}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

   1. *-commutative 71.0%

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

   2. associate-*l* 71.0%

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

   3. times-frac 94.4%

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

   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. div-inv 94.3%

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

   2. +-commutative 94.3%

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

   3. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

   1. associate-*r/ 94.4%

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

   2. *-rgt-identity 94.4%

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

   3. associate-/r* 99.8%

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

   4. associate-+r+ 99.8%

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

   5. +-commutative 99.8%

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

   6. associate-+r+ 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 12: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2e-125) (/ (/ y (+ x 1.0)) (+ x y)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2e-125) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d-125)) then
        tmp = (y / (x + 1.0d0)) / (x + y)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2e-125) {
		tmp = (y / (x + 1.0)) / (x + y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2e-125:
		tmp = (y / (x + 1.0)) / (x + y)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2e-125)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(x + y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e-125)
		tmp = (y / (x + 1.0)) / (x + y);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2e-125], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.00000000000000002e-125

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*l* 67.1%

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

      3. times-frac 92.0%

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

      4. +-commutative 92.0%

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

      5. +-commutative 92.0%

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

      6. associate-+r+ 92.0%

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

      7. +-commutative 92.0%

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

      8. associate-+l+ 92.0%

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

   4. Applied egg-rr 92.0%

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

      1. div-inv 92.0%

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

      2. +-commutative 92.0%

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

      3. +-commutative 92.0%

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

   6. Applied egg-rr 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-+r+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. times-frac 97.8%

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

      8. associate-/l* 97.8%

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

      9. associate-*l/ 97.8%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in y around 0 65.8%

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

       1. +-commutative 65.8%

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

   13. Simplified 65.8%

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

   if -2.00000000000000002e-125 < x

   1. Initial program 73.3%

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

      1. associate-/l* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 63.1%

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

      1. associate-/r* 64.3%

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

      2. +-commutative 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	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 <= (-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 <= -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 <= -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 <= -2.1e-125)
		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 <= -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, -2.1e-125], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e-125

   1. Initial program 67.1%

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

      1. *-commutative 67.1%

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

      2. associate-*l* 67.1%

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

      3. times-frac 92.0%

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

      4. +-commutative 92.0%

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

      5. +-commutative 92.0%

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

      6. associate-+r+ 92.0%

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

      7. +-commutative 92.0%

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

      8. associate-+l+ 92.0%

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

   4. Applied egg-rr 92.0%

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

      1. div-inv 92.0%

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

      2. +-commutative 92.0%

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

      3. +-commutative 92.0%

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

   6. Applied egg-rr 92.0%

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

      1. associate-*r/ 92.0%

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

      2. *-rgt-identity 92.0%

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

      3. associate-/r* 99.8%

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

      4. associate-+r+ 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

   8. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-+r+ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-+r+ 99.8%

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

      5. clear-num 99.8%

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

      6. +-commutative 99.8%

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

      7. times-frac 97.8%

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

      8. associate-/l* 97.8%

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

      9. associate-*l/ 97.8%

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

   10. Applied egg-rr 99.9%

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

   11. Taylor expanded in y around 0 62.4%

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

       1. associate-/r* 65.3%

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

       2. +-commutative 65.3%

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

   13. Simplified 65.3%

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

   if -2.1e-125 < x

   1. Initial program 73.3%

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

      1. associate-/l* 82.4%

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

      2. associate-+l+ 82.4%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in x around 0 63.1%

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

      1. associate-/r* 64.3%

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

      2. +-commutative 64.3%

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

   7. Simplified 64.3%

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

4. Final simplification 64.7%

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

Alternative 14: 45.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -550.0) (/ (/ y x) x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / 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 <= (-550.0d0)) then
        tmp = (y / x) / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -550.0) {
		tmp = (y / x) / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -550.0:
		tmp = (y / x) / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -550.0)
		tmp = Float64(Float64(y / x) / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -550.0)
		tmp = (y / x) / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -550.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -550

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l* 61.4%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. un-div-inv 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in y around 0 73.9%

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

   if -550 < x

   1. Initial program 74.7%

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

      1. associate-/l* 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in x around 0 62.8%

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

   6. Taylor expanded in y around 0 35.3%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -550:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -780:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -780.0) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -780.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-780.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -780.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -780.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -780.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -780.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -780.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -780

   1. Initial program 61.4%

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

      1. *-commutative 61.4%

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

      2. associate-*l* 61.4%

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

      3. times-frac 89.3%

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

      4. +-commutative 89.3%

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

      5. +-commutative 89.3%

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

      6. associate-+r+ 89.3%

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

      7. +-commutative 89.3%

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

      8. associate-+l+ 89.3%

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

   4. Applied egg-rr 89.3%

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

   5. Taylor expanded in x around inf 74.3%

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

      1. un-div-inv 74.4%

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

   7. Applied egg-rr 74.4%

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

   8. Taylor expanded in y around inf 5.8%

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

   if -780 < x

   1. Initial program 74.7%

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

      1. associate-/l* 83.7%

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

      2. associate-+l+ 83.7%

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

   3. Simplified 83.7%

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

   5. Taylor expanded in x around 0 62.8%

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

   6. Taylor expanded in y around 0 35.3%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -780:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 4.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.0%

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

   1. *-commutative 71.0%

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

   2. associate-*l* 71.0%

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

   3. times-frac 94.4%

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

   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. Taylor expanded in x around inf 38.2%

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

   1. un-div-inv 38.2%

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

7. Applied egg-rr 38.2%

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

8. Taylor expanded in y around inf 4.3%

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

9. Final simplification 4.3%

   \[\leadsto \frac{1}{x} \]
10. Add Preprocessing

Alternative 17: 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 71.0%

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

   1. *-un-lft-identity 71.0%

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

   2. associate-*l* 71.0%

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

   3. times-frac 77.3%

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

   4. +-commutative 77.3%

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

   5. *-commutative 77.3%

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

   6. +-commutative 77.3%

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

   7. associate-+r+ 77.3%

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

   8. +-commutative 77.3%

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

   9. associate-+l+ 77.3%

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

4. Applied egg-rr 77.3%

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

5. Taylor expanded in y around 0 47.8%

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

   1. +-commutative 47.8%

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

7. Simplified 47.8%

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

8. Taylor expanded in x around 0 3.6%

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

9. Final simplification 3.6%

   \[\leadsto 1 \]
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 2024044 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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