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

Percentage Accurate: 69.9% → 99.8%

Time: 15.3s

Alternatives: 26

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: 69.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.7%

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

   1. associate-/r* 73.5%

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

   2. +-commutative 73.5%

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

   3. +-commutative 73.5%

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

   4. +-commutative 73.5%

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

   5. associate-/l/ 68.7%

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

   6. times-frac 88.4%

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

   7. *-commutative 88.4%

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

   8. +-commutative 88.4%

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

   9. +-commutative 88.4%

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

   10. +-commutative 88.4%

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

   11. associate-+l+ 88.4%

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

3. Simplified 88.4%

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

   1. associate-/r* 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+152)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (if (<= x -5e-17)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x (+ y 1.0))))
     (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+152) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -5e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d+152)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-5d-17)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0d0)))
    else
        tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+152) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -5e-17) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+152:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -5e-17:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)))
	else:
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+152)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -5e-17)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e+152)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -5e-17)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	else
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+152], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-17], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e152

   1. Initial program 59.5%

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

      1. times-frac 78.8%

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

      2. associate-+l+ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x around inf 85.9%

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

      1. clear-num 85.9%

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

      2. un-div-inv 86.0%

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

      3. associate-+r+ 86.0%

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

      4. +-commutative 86.0%

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

      5. associate-+l+ 86.0%

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

   6. Applied egg-rr 86.0%

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

   if -1.4499999999999999e152 < x < -4.9999999999999999e-17

   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. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   if -4.9999999999999999e-17 < x

   1. Initial program 69.1%

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

      1. associate-/r* 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. associate-/l/ 69.1%

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

      6. times-frac 87.8%

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

      7. *-commutative 87.8%

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

      8. +-commutative 87.8%

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

      9. +-commutative 87.8%

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

      10. +-commutative 87.8%

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

      11. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 91.2%

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

      1. +-commutative 91.2%

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

   10. Simplified 91.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 3: 90.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+152)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (if (<= x -5e-16)
     (* (/ x (+ x (+ y 1.0))) (/ y (* (+ y x) (+ y x))))
     (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+152) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -5e-16) {
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d+152)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-5d-16)) then
        tmp = (x / (x + (y + 1.0d0))) * (y / ((y + x) * (y + x)))
    else
        tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+152) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -5e-16) {
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+152:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -5e-16:
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)))
	else:
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+152)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -5e-16)
		tmp = Float64(Float64(x / Float64(x + Float64(y + 1.0))) * Float64(y / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e+152)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -5e-16)
		tmp = (x / (x + (y + 1.0))) * (y / ((y + x) * (y + x)));
	else
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+152], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5e-16], N[(N[(x / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.4499999999999999e152

   1. Initial program 59.5%

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

      1. times-frac 78.8%

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

      2. associate-+l+ 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x around inf 85.9%

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

      1. clear-num 85.9%

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

      2. un-div-inv 86.0%

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

      3. associate-+r+ 86.0%

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

      4. +-commutative 86.0%

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

      5. associate-+l+ 86.0%

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

   6. Applied egg-rr 86.0%

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

   if -1.4499999999999999e152 < x < -5.0000000000000004e-16

   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-/r* 89.4%

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

      2. +-commutative 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-/l/ 74.7%

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

      6. times-frac 99.6%

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

      7. *-commutative 99.6%

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

      8. +-commutative 99.6%

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

      9. +-commutative 99.6%

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

      10. +-commutative 99.6%

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

      11. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   if -5.0000000000000004e-16 < x

   1. Initial program 69.1%

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

      1. associate-/r* 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. associate-/l/ 69.1%

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

      6. times-frac 87.8%

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

      7. *-commutative 87.8%

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

      8. +-commutative 87.8%

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

      9. +-commutative 87.8%

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

      10. +-commutative 87.8%

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

      11. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 91.2%

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

      1. +-commutative 91.2%

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

   10. Simplified 91.2%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{x + \left(y + 1\right)} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 4: 68.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.6e+132)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (if (<= x -2.9e-154)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x 1.0)))
     (/ (/ x (+ y x)) (+ y (+ x (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+132) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -2.9e-154) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (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 <= (-2.6d+132)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-2.9d-154)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0d0))
    else
        tmp = (x / (y + x)) / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -2.6e+132) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -2.9e-154) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.6e+132:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -2.9e-154:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0))
	else:
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.6e+132)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -2.9e-154)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.6e+132)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -2.9e-154)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	else
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.6e+132], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.9e-154], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.6e132

   1. Initial program 61.3%

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

      1. times-frac 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in x around inf 85.0%

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

      1. clear-num 85.0%

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

      2. un-div-inv 85.1%

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

      3. associate-+r+ 85.1%

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

      4. +-commutative 85.1%

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

      5. associate-+l+ 85.1%

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

   6. Applied egg-rr 85.1%

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

   if -2.6e132 < x < -2.9e-154

   1. Initial program 82.6%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 82.7%

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

      1. +-commutative 82.7%

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

   6. Simplified 82.7%

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

   if -2.9e-154 < x

   1. Initial program 65.9%

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

      1. associate-/r* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-/r* 65.9%

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

      6. associate-*l/ 81.7%

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

      7. *-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. distribute-rgt1-in 74.5%

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

      10. fma-def 81.7%

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

      11. +-commutative 81.7%

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

      12. +-commutative 81.7%

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

      13. cube-unmult 81.7%

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

      14. +-commutative 81.7%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.9%

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

      2. fma-udef 61.1%

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

      3. cube-mult 61.2%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. *-commutative 65.9%

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

      8. frac-times 86.5%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

      3. neg-mul-1 60.9%

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

      4. unsub-neg 60.9%

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

      5. distribute-lft-in 60.9%

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

      6. metadata-eval 60.9%

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

      7. neg-mul-1 60.9%

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

      8. unsub-neg 60.9%

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

   8. Simplified 60.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 5: 88.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e+132)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (if (<= x -4.8e-12)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ x 1.0)))
     (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+132) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -4.8e-12) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d+132)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else if (x <= (-4.8d-12)) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0d0))
    else
        tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+132) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else if (x <= -4.8e-12) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e+132:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	elif x <= -4.8e-12:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0))
	else:
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e+132)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	elseif (x <= -4.8e-12)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e+132)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	elseif (x <= -4.8e-12)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + 1.0));
	else
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e+132], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-12], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999997e132

   1. Initial program 61.3%

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

      1. times-frac 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in x around inf 85.0%

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

      1. clear-num 85.0%

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

      2. un-div-inv 85.1%

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

      3. associate-+r+ 85.1%

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

      4. +-commutative 85.1%

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

      5. associate-+l+ 85.1%

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

   6. Applied egg-rr 85.1%

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

   if -3.1999999999999997e132 < x < -4.79999999999999974e-12

   1. Initial program 74.6%

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

      1. times-frac 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in y around 0 85.0%

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

      1. +-commutative 85.0%

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

   6. Simplified 85.0%

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

   if -4.79999999999999974e-12 < x

   1. Initial program 69.1%

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

      1. associate-/r* 72.6%

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

      2. +-commutative 72.6%

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

      3. +-commutative 72.6%

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

      4. +-commutative 72.6%

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

      5. associate-/l/ 69.1%

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

      6. times-frac 87.8%

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

      7. *-commutative 87.8%

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

      8. +-commutative 87.8%

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

      9. +-commutative 87.8%

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

      10. +-commutative 87.8%

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

      11. associate-+l+ 87.8%

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

   3. Simplified 87.8%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 91.2%

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

      1. +-commutative 91.2%

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

   10. Simplified 91.2%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+132}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 6: 66.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0018:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0018)
   (* (/ 1.0 x) (/ y (+ x (+ y 1.0))))
   (if (<= x -2.9e-154)
     (* y (/ x (* (+ y x) (+ y x))))
     (/ (/ x (+ y x)) (+ y (+ x (+ x 1.0)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.0018) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else if (x <= -2.9e-154) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + (x + (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 <= (-0.0018d0)) then
        tmp = (1.0d0 / x) * (y / (x + (y + 1.0d0)))
    else if (x <= (-2.9d-154)) then
        tmp = y * (x / ((y + x) * (y + x)))
    else
        tmp = (x / (y + x)) / (y + (x + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.0018) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else if (x <= -2.9e-154) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.0018:
		tmp = (1.0 / x) * (y / (x + (y + 1.0)))
	elif x <= -2.9e-154:
		tmp = y * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.0018)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + Float64(y + 1.0))));
	elseif (x <= -2.9e-154)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + Float64(x + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0018)
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	elseif (x <= -2.9e-154)
		tmp = y * (x / ((y + x) * (y + x)));
	else
		tmp = (x / (y + x)) / (y + (x + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.0018], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.9e-154], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0018

   1. Initial program 66.8%

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

      1. times-frac 89.7%

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

      2. associate-+l+ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 79.3%

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

   if -0.0018 < x < -2.9e-154

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 80.5%

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

      1. +-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in x around 0 79.2%

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

   if -2.9e-154 < x

   1. Initial program 65.9%

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

      1. associate-/r* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-/r* 65.9%

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

      6. associate-*l/ 81.7%

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

      7. *-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. distribute-rgt1-in 74.5%

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

      10. fma-def 81.7%

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

      11. +-commutative 81.7%

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

      12. +-commutative 81.7%

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

      13. cube-unmult 81.7%

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

      14. +-commutative 81.7%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.9%

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

      2. fma-udef 61.1%

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

      3. cube-mult 61.2%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. *-commutative 65.9%

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

      8. frac-times 86.5%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in y around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

      3. neg-mul-1 60.9%

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

      4. unsub-neg 60.9%

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

      5. distribute-lft-in 60.9%

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

      6. metadata-eval 60.9%

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

      7. neg-mul-1 60.9%

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

      8. unsub-neg 60.9%

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

   8. Simplified 60.9%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0018:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -2.9 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + \left(x + \left(x + 1\right)\right)}\\ \end{array} \]

Alternative 7: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.6e+35)
   (/ (/ x (+ y x)) (* (+ y x) (/ (+ x 1.0) y)))
   (* (/ (/ y (+ y x)) (+ y x)) (/ x (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.6e+35) {
		tmp = (x / (y + x)) / ((y + x) * ((x + 1.0) / y));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.6d+35) then
        tmp = (x / (y + x)) / ((y + x) * ((x + 1.0d0) / y))
    else
        tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.6e+35) {
		tmp = (x / (y + x)) / ((y + x) * ((x + 1.0) / y));
	} else {
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.6e+35:
		tmp = (x / (y + x)) / ((y + x) * ((x + 1.0) / y))
	else:
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.6e+35)
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(Float64(y + x) * Float64(Float64(x + 1.0) / y)));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + x)) * Float64(x / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.6e+35)
		tmp = (x / (y + x)) / ((y + x) * ((x + 1.0) / y));
	else
		tmp = ((y / (y + x)) / (y + x)) * (x / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.6e+35], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(x + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.6000000000000003e35

   1. Initial program 70.8%

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

      1. times-frac 89.2%

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

      2. associate-+l+ 89.2%

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

   3. Simplified 89.2%

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

   4. Taylor expanded in y around 0 81.3%

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

      1. +-commutative 81.3%

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

   6. Simplified 81.3%

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

      1. *-commutative 81.3%

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

      2. clear-num 81.2%

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

      3. associate-/r* 90.2%

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

      4. frac-times 83.1%

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

      5. *-un-lft-identity 83.1%

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

      6. +-commutative 83.1%

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

      7. +-commutative 83.1%

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

   8. Applied egg-rr 83.1%

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

   if 6.6000000000000003e35 < y

   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. Step-by-step derivation

      1. associate-/r* 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. +-commutative 68.1%

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

      5. associate-/l/ 61.4%

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

      6. times-frac 85.5%

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

      7. *-commutative 85.5%

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

      8. +-commutative 85.5%

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

      9. +-commutative 85.5%

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

      10. +-commutative 85.5%

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

      11. associate-+l+ 85.5%

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

   3. Simplified 85.5%

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

      1. associate-/r* 99.8%

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

      2. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in x around 0 88.6%

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

      1. +-commutative 88.6%

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

   10. Simplified 88.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{\left(y + x\right) \cdot \frac{x + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x} \cdot \frac{x}{y + 1}\\ \end{array} \]

Alternative 8: 65.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.3e-6)
   (* (/ 1.0 x) (/ y (+ x (+ y 1.0))))
   (if (<= x -9e-155)
     (* y (/ x (* (+ y x) (+ y x))))
     (/ (/ x (+ y x)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.3e-6) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else if (x <= -9e-155) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.3d-6)) then
        tmp = (1.0d0 / x) * (y / (x + (y + 1.0d0)))
    else if (x <= (-9d-155)) then
        tmp = y * (x / ((y + x) * (y + x)))
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.3e-6) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else if (x <= -9e-155) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.3e-6:
		tmp = (1.0 / x) * (y / (x + (y + 1.0)))
	elif x <= -9e-155:
		tmp = y * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.3e-6)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + Float64(y + 1.0))));
	elseif (x <= -9e-155)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.3e-6)
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	elseif (x <= -9e-155)
		tmp = y * (x / ((y + x) * (y + x)));
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.3e-6], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-155], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000005e-6

   1. Initial program 66.8%

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

      1. times-frac 89.7%

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

      2. associate-+l+ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 79.3%

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

   if -1.30000000000000005e-6 < x < -9.0000000000000007e-155

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 80.5%

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

      1. +-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in x around 0 79.2%

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

   if -9.0000000000000007e-155 < x

   1. Initial program 65.9%

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

      1. associate-/r* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-/r* 65.9%

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

      6. associate-*l/ 81.7%

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

      7. *-commutative 81.7%

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

      8. *-commutative 81.7%

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

      9. distribute-rgt1-in 74.5%

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

      10. fma-def 81.7%

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

      11. +-commutative 81.7%

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

      12. +-commutative 81.7%

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

      13. cube-unmult 81.7%

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

      14. +-commutative 81.7%

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

   3. Simplified 81.7%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 65.9%

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

      2. fma-udef 61.1%

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

      3. cube-mult 61.2%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. *-commutative 65.9%

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

      8. frac-times 86.5%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.2%

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

      12. *-un-lft-identity 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around 0 60.1%

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

      1. +-commutative 60.1%

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

   8. Simplified 60.1%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 9: 65.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{t_0}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t_0}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -1.95e-5)
     (* (/ 1.0 x) (/ y t_0))
     (if (<= x -1.7e-154)
       (* y (/ x (* (+ y x) (+ y x))))
       (/ (/ x t_0) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.95e-5) {
		tmp = (1.0 / x) * (y / t_0);
	} else if (x <= -1.7e-154) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-1.95d-5)) then
        tmp = (1.0d0 / x) * (y / t_0)
    else if (x <= (-1.7d-154)) then
        tmp = y * (x / ((y + x) * (y + x)))
    else
        tmp = (x / t_0) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -1.95e-5) {
		tmp = (1.0 / x) * (y / t_0);
	} else if (x <= -1.7e-154) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / t_0) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -1.95e-5:
		tmp = (1.0 / x) * (y / t_0)
	elif x <= -1.7e-154:
		tmp = y * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / t_0) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -1.95e-5)
		tmp = Float64(Float64(1.0 / x) * Float64(y / t_0));
	elseif (x <= -1.7e-154)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / t_0) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -1.95e-5)
		tmp = (1.0 / x) * (y / t_0);
	elseif (x <= -1.7e-154)
		tmp = y * (x / ((y + x) * (y + x)));
	else
		tmp = (x / t_0) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-5], N[(N[(1.0 / x), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.7e-154], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e-5

   1. Initial program 66.8%

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

      1. times-frac 89.7%

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

      2. associate-+l+ 89.7%

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

   3. Simplified 89.7%

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

   4. Taylor expanded in x around inf 79.3%

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

   if -1.95e-5 < x < -1.6999999999999999e-154

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 80.5%

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

      1. +-commutative 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in x around 0 79.2%

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

   if -1.6999999999999999e-154 < x

   1. Initial program 65.9%

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

      1. associate-/r* 69.9%

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

      2. +-commutative 69.9%

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

      3. +-commutative 69.9%

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

      4. +-commutative 69.9%

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

      5. associate-/l/ 65.9%

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

      6. times-frac 86.5%

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

      7. *-commutative 86.5%

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

      8. +-commutative 86.5%

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

      9. +-commutative 86.5%

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

      10. +-commutative 86.5%

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

      11. associate-+l+ 86.5%

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

   3. Simplified 86.5%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in y around inf 60.5%

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

      1. associate-*l/ 60.6%

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

      2. *-un-lft-identity 60.6%

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

   10. Applied egg-rr 60.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-154}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + \left(y + 1\right)}}{y + x}\\ \end{array} \]

Alternative 10: 53.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-175}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -7.8e-175)
     t_0
     (if (<= y 5.4e-148)
       (/ y x)
       (if (<= y 1.9e-17) (/ x y) (if (<= y 2.25e+36) t_0 (/ (/ x y) y)))))))

C

double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.8e-175) {
		tmp = t_0;
	} else if (y <= 5.4e-148) {
		tmp = y / x;
	} else if (y <= 1.9e-17) {
		tmp = x / y;
	} else if (y <= 2.25e+36) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x * x)
    if (y <= (-7.8d-175)) then
        tmp = t_0
    else if (y <= 5.4d-148) then
        tmp = y / x
    else if (y <= 1.9d-17) then
        tmp = x / y
    else if (y <= 2.25d+36) then
        tmp = t_0
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -7.8e-175) {
		tmp = t_0;
	} else if (y <= 5.4e-148) {
		tmp = y / x;
	} else if (y <= 1.9e-17) {
		tmp = x / y;
	} else if (y <= 2.25e+36) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -7.8e-175:
		tmp = t_0
	elif y <= 5.4e-148:
		tmp = y / x
	elif y <= 1.9e-17:
		tmp = x / y
	elif y <= 2.25e+36:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -7.8e-175)
		tmp = t_0;
	elseif (y <= 5.4e-148)
		tmp = Float64(y / x);
	elseif (y <= 1.9e-17)
		tmp = Float64(x / y);
	elseif (y <= 2.25e+36)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -7.8e-175)
		tmp = t_0;
	elseif (y <= 5.4e-148)
		tmp = y / x;
	elseif (y <= 1.9e-17)
		tmp = x / y;
	elseif (y <= 2.25e+36)
		tmp = t_0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.8e-175], t$95$0, If[LessEqual[y, 5.4e-148], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.9e-17], N[(x / y), $MachinePrecision], If[LessEqual[y, 2.25e+36], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.79999999999999997e-175 or 1.9000000000000001e-17 < y < 2.24999999999999998e36

   1. Initial program 69.8%

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

      1. associate-/r* 76.5%

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

      2. +-commutative 76.5%

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

      3. +-commutative 76.5%

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

      4. +-commutative 76.5%

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

      5. associate-/r* 69.8%

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

      6. associate-*l/ 85.1%

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

      7. *-commutative 85.1%

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

      8. *-commutative 85.1%

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

      9. distribute-rgt1-in 61.7%

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

      10. fma-def 85.1%

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

      11. +-commutative 85.1%

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

      12. +-commutative 85.1%

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

      13. cube-unmult 85.1%

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

      14. +-commutative 85.1%

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

   3. Simplified 85.1%

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

   4. Taylor expanded in x around inf 37.6%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 37.6%

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

   6. Simplified 37.6%

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

   if -7.79999999999999997e-175 < y < 5.39999999999999976e-148

   1. Initial program 68.3%

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

      1. times-frac 78.5%

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

      2. associate-+l+ 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in y around 0 87.1%

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

   5. Taylor expanded in x around 0 74.6%

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

   if 5.39999999999999976e-148 < y < 1.9000000000000001e-17

   1. Initial program 80.7%

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

      1. times-frac 99.3%

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

      2. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 43.9%

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

   if 2.24999999999999998e36 < y

   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. Step-by-step derivation

      1. associate-/r* 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. +-commutative 68.1%

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

      5. associate-/r* 61.4%

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

      6. associate-*l/ 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. distribute-rgt1-in 74.1%

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

      10. fma-def 76.3%

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

      11. +-commutative 76.3%

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

      12. +-commutative 76.3%

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

      13. cube-unmult 76.3%

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

      14. +-commutative 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

      1. associate-/r* 86.6%

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

      2. div-inv 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate-*r/ 86.6%

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

      2. *-rgt-identity 86.6%

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

   10. Simplified 86.6%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-175}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+36}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 53.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-175}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -8e-175)
   (* y (/ 1.0 (* x x)))
   (if (<= y 5.2e-148)
     (/ y x)
     (if (<= y 1.72e-18)
       (/ x y)
       (if (<= y 6.6e+35) (/ y (* x x)) (/ (/ x y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -8e-175) {
		tmp = y * (1.0 / (x * x));
	} else if (y <= 5.2e-148) {
		tmp = y / x;
	} else if (y <= 1.72e-18) {
		tmp = x / y;
	} else if (y <= 6.6e+35) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / 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 <= (-8d-175)) then
        tmp = y * (1.0d0 / (x * x))
    else if (y <= 5.2d-148) then
        tmp = y / x
    else if (y <= 1.72d-18) then
        tmp = x / y
    else if (y <= 6.6d+35) then
        tmp = y / (x * x)
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -8e-175) {
		tmp = y * (1.0 / (x * x));
	} else if (y <= 5.2e-148) {
		tmp = y / x;
	} else if (y <= 1.72e-18) {
		tmp = x / y;
	} else if (y <= 6.6e+35) {
		tmp = y / (x * x);
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -8e-175:
		tmp = y * (1.0 / (x * x))
	elif y <= 5.2e-148:
		tmp = y / x
	elif y <= 1.72e-18:
		tmp = x / y
	elif y <= 6.6e+35:
		tmp = y / (x * x)
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -8e-175)
		tmp = Float64(y * Float64(1.0 / Float64(x * x)));
	elseif (y <= 5.2e-148)
		tmp = Float64(y / x);
	elseif (y <= 1.72e-18)
		tmp = Float64(x / y);
	elseif (y <= 6.6e+35)
		tmp = Float64(y / Float64(x * x));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -8e-175)
		tmp = y * (1.0 / (x * x));
	elseif (y <= 5.2e-148)
		tmp = y / x;
	elseif (y <= 1.72e-18)
		tmp = x / y;
	elseif (y <= 6.6e+35)
		tmp = y / (x * x);
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -8e-175], N[(y * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e-148], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.72e-18], N[(x / y), $MachinePrecision], If[LessEqual[y, 6.6e+35], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -8e-175

   1. Initial program 67.4%

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

      1. associate-/r* 74.2%

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

      2. +-commutative 74.2%

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

      3. +-commutative 74.2%

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

      4. +-commutative 74.2%

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

      5. associate-/r* 67.4%

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

      6. associate-*l/ 84.3%

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

      7. *-commutative 84.3%

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

      8. *-commutative 84.3%

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

      9. distribute-rgt1-in 58.7%

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

      10. fma-def 84.3%

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

      11. +-commutative 84.3%

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

      12. +-commutative 84.3%

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

      13. cube-unmult 84.3%

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

      14. +-commutative 84.3%

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

   3. Simplified 84.3%

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

   4. Taylor expanded in x around inf 31.8%

      \[\leadsto y \cdot \color{blue}{\frac{1}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 31.8%

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

   6. Simplified 31.8%

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

   if -8e-175 < y < 5.20000000000000015e-148

   1. Initial program 68.3%

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

      1. times-frac 78.5%

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

      2. associate-+l+ 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in y around 0 87.1%

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

   5. Taylor expanded in x around 0 74.6%

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

   if 5.20000000000000015e-148 < y < 1.72e-18

   1. Initial program 79.9%

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

      1. times-frac 99.3%

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

      2. associate-+l+ 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in y around 0 99.3%

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

      1. +-commutative 99.3%

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

   6. Simplified 99.3%

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

   7. Taylor expanded in x around 0 45.5%

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

   if 1.72e-18 < y < 6.6000000000000003e35

   1. Initial program 85.5%

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

      1. associate-/r* 91.0%

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

      2. +-commutative 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-/r* 85.5%

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

      6. associate-*l/ 90.5%

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

      7. *-commutative 90.5%

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

      8. *-commutative 90.5%

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

      9. distribute-rgt1-in 81.4%

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

      10. fma-def 90.6%

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

      11. +-commutative 90.6%

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

      12. +-commutative 90.6%

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

      13. cube-unmult 90.3%

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

      14. +-commutative 90.3%

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

   3. Simplified 90.3%

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

   4. Taylor expanded in x around inf 74.4%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 74.4%

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

   6. Simplified 74.4%

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

   if 6.6000000000000003e35 < y

   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. Step-by-step derivation

      1. associate-/r* 68.1%

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

      2. +-commutative 68.1%

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

      3. +-commutative 68.1%

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

      4. +-commutative 68.1%

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

      5. associate-/r* 61.4%

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

      6. associate-*l/ 76.3%

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

      7. *-commutative 76.3%

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

      8. *-commutative 76.3%

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

      9. distribute-rgt1-in 74.1%

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

      10. fma-def 76.3%

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

      11. +-commutative 76.3%

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

      12. +-commutative 76.3%

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

      13. cube-unmult 76.3%

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

      14. +-commutative 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in y around inf 81.8%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 81.8%

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

   6. Simplified 81.8%

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

      1. associate-/r* 86.6%

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

      2. div-inv 86.6%

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

   8. Applied egg-rr 86.6%

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

      1. associate-*r/ 86.6%

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

      2. *-rgt-identity 86.6%

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

   10. Simplified 86.6%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-175}:\\ \;\;\;\;y \cdot \frac{1}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12: 54.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -8.2e-154)
     (- (/ y x) y)
     (if (<= x 2.2e-127) (/ x y) (* (/ x y) (/ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -8.2e-154) {
		tmp = (y / x) - y;
	} else if (x <= 2.2e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-8.2d-154)) then
        tmp = (y / x) - y
    else if (x <= 2.2d-127) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -8.2e-154) {
		tmp = (y / x) - y;
	} else if (x <= 2.2e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -8.2e-154:
		tmp = (y / x) - y
	elif x <= 2.2e-127:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -8.2e-154)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 2.2e-127)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -8.2e-154)
		tmp = (y / x) - y;
	elseif (x <= 2.2e-127)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8.2e-154], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 2.2e-127], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 66.3%

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

      1. times-frac 89.6%

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

      2. associate-+l+ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 78.9%

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

   5. Taylor expanded in x around inf 76.4%

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

   if -1 < x < -8.2e-154

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 36.1%

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

   5. Taylor expanded in x around 0 32.1%

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

      1. neg-mul-1 32.1%

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

      2. unsub-neg 32.1%

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

   7. Simplified 32.1%

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

   if -8.2e-154 < x < 2.2000000000000001e-127

   1. Initial program 61.1%

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

      1. times-frac 80.5%

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

      2. associate-+l+ 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. +-commutative 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in x around 0 77.2%

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

   if 2.2000000000000001e-127 < x

   1. Initial program 70.1%

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

      1. associate-/r* 77.5%

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

      2. +-commutative 77.5%

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

      3. +-commutative 77.5%

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

      4. +-commutative 77.5%

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

      5. associate-/r* 70.1%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 75.4%

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

      10. fma-def 83.2%

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

      11. +-commutative 83.2%

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

      12. +-commutative 83.2%

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

      13. cube-unmult 83.2%

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

      14. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in y around inf 32.4%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.4%

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

   6. Simplified 32.4%

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

      1. associate-/r* 35.2%

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

      2. div-inv 35.2%

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

   8. Applied egg-rr 35.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]

Alternative 13: 63.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x + x \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -2.7e+154)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.42e-112) (/ y (+ x (* x (+ y x)))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -2.7e+154) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.42e-112) {
		tmp = y / (x + (x * (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 <= (-2.7d+154)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.42d-112)) then
        tmp = y / (x + (x * (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 <= -2.7e+154) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.42e-112) {
		tmp = y / (x + (x * (y + x)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -2.7e+154:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.42e-112:
		tmp = y / (x + (x * (y + x)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -2.7e+154)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.42e-112)
		tmp = Float64(y / Float64(x + Float64(x * 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 <= -2.7e+154)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.42e-112)
		tmp = y / (x + (x * (y + x)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -2.7e+154], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.42e-112], N[(y / N[(x + N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.70000000000000006e154

   1. Initial program 61.3%

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

      1. times-frac 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in x around inf 81.1%

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

   5. Taylor expanded in x around inf 88.0%

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

   if -2.70000000000000006e154 < x < -1.42e-112

   1. Initial program 79.1%

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

      1. times-frac 97.7%

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

      2. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around inf 61.7%

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

      1. frac-times 69.0%

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

      2. *-un-lft-identity 69.0%

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

      3. associate-+r+ 69.0%

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

      4. distribute-rgt-in 68.9%

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

      5. +-commutative 68.9%

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

      6. *-un-lft-identity 68.9%

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

   6. Applied egg-rr 68.9%

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

   if -1.42e-112 < x

   1. Initial program 66.9%

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

      1. associate-/r* 70.8%

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

      2. +-commutative 70.8%

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

      3. +-commutative 70.8%

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

      4. +-commutative 70.8%

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

      5. associate-/l/ 66.9%

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

      6. times-frac 86.9%

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

      7. *-commutative 86.9%

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

      8. +-commutative 86.9%

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

      9. +-commutative 86.9%

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

      10. +-commutative 86.9%

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

      11. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 58.7%

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

      1. associate-/r* 60.1%

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

      2. +-commutative 60.1%

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

   10. Simplified 60.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.42 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x + x \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 14: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x + x \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.32e+154)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -4.8e-110)
     (/ y (+ x (* x (+ y x))))
     (/ (/ x (+ y x)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.8e-110) {
		tmp = y / (x + (x * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.32d+154)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-4.8d-110)) then
        tmp = y / (x + (x * (y + x)))
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.8e-110) {
		tmp = y / (x + (x * (y + x)));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.32e+154:
		tmp = (1.0 / x) * (y / x)
	elif x <= -4.8e-110:
		tmp = y / (x + (x * (y + x)))
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -4.8e-110)
		tmp = Float64(y / Float64(x + Float64(x * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.32e+154)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -4.8e-110)
		tmp = y / (x + (x * (y + x)));
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.32e+154], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-110], N[(y / N[(x + N[(x * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 61.3%

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

      1. times-frac 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in x around inf 81.1%

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

   5. Taylor expanded in x around inf 88.0%

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

   if -1.31999999999999998e154 < x < -4.80000000000000013e-110

   1. Initial program 79.1%

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

      1. times-frac 97.7%

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

      2. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around inf 61.7%

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

      1. frac-times 69.0%

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

      2. *-un-lft-identity 69.0%

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

      3. associate-+r+ 69.0%

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

      4. distribute-rgt-in 68.9%

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

      5. +-commutative 68.9%

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

      6. *-un-lft-identity 68.9%

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

   6. Applied egg-rr 68.9%

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

   if -4.80000000000000013e-110 < x

   1. Initial program 66.9%

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

      1. associate-/r* 70.8%

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

      2. +-commutative 70.8%

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

      3. +-commutative 70.8%

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

      4. +-commutative 70.8%

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

      5. associate-/r* 66.9%

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

      6. associate-*l/ 82.2%

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

      7. *-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 75.3%

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

      10. fma-def 82.2%

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

      11. +-commutative 82.2%

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

      12. +-commutative 82.2%

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

      13. cube-unmult 82.2%

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

      14. +-commutative 82.2%

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

   3. Simplified 82.2%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.9%

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

      2. fma-udef 62.3%

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

      3. cube-mult 62.3%

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

      4. distribute-rgt1-in 66.9%

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

      5. associate-+r+ 66.9%

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

      6. *-commutative 66.9%

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

      7. *-commutative 66.9%

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

      8. frac-times 86.9%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in x around 0 60.3%

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

      1. +-commutative 60.3%

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

   8. Simplified 60.3%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x + x \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 15: 62.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-110)
   (* (/ 1.0 x) (/ y (+ x (+ y 1.0))))
   (/ (/ x (+ y x)) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-110) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / (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 <= (-4.8d-110)) then
        tmp = (1.0d0 / x) * (y / (x + (y + 1.0d0)))
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-110) {
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-110:
		tmp = (1.0 / x) * (y / (x + (y + 1.0)))
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e-110)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e-110)
		tmp = (1.0 / x) * (y / (x + (y + 1.0)));
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e-110], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.80000000000000013e-110

   1. Initial program 72.3%

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

      1. times-frac 91.4%

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

      2. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in x around inf 71.8%

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

   if -4.80000000000000013e-110 < x

   1. Initial program 66.9%

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

      1. associate-/r* 70.8%

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

      2. +-commutative 70.8%

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

      3. +-commutative 70.8%

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

      4. +-commutative 70.8%

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

      5. associate-/r* 66.9%

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

      6. associate-*l/ 82.2%

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

      7. *-commutative 82.2%

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

      8. *-commutative 82.2%

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

      9. distribute-rgt1-in 75.3%

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

      10. fma-def 82.2%

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

      11. +-commutative 82.2%

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

      12. +-commutative 82.2%

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

      13. cube-unmult 82.2%

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

      14. +-commutative 82.2%

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

   3. Simplified 82.2%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.9%

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

      2. fma-udef 62.3%

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

      3. cube-mult 62.3%

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

      4. distribute-rgt1-in 66.9%

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

      5. associate-+r+ 66.9%

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

      6. *-commutative 66.9%

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

      7. *-commutative 66.9%

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

      8. frac-times 86.9%

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

      9. associate-/r* 99.7%

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

      10. clear-num 99.6%

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

      11. frac-times 99.3%

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

      12. *-un-lft-identity 99.3%

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

   5. Applied egg-rr 99.3%

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

   6. Taylor expanded in x around 0 60.3%

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

      1. +-commutative 60.3%

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

   8. Simplified 60.3%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]

Alternative 16: 53.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -9.5e-154)
     (- (/ y x) y)
     (if (<= x 1.75e-127) (/ x y) (/ x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -9.5e-154) {
		tmp = (y / x) - y;
	} else if (x <= 1.75e-127) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-9.5d-154)) then
        tmp = (y / x) - y
    else if (x <= 1.75d-127) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -9.5e-154) {
		tmp = (y / x) - y;
	} else if (x <= 1.75e-127) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -9.5e-154:
		tmp = (y / x) - y
	elif x <= 1.75e-127:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -9.5e-154)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 1.75e-127)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -9.5e-154)
		tmp = (y / x) - y;
	elseif (x <= 1.75e-127)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-154], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 1.75e-127], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 66.3%

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

      1. associate-/r* 74.7%

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

      2. +-commutative 74.7%

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

      3. +-commutative 74.7%

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

      4. +-commutative 74.7%

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

      5. associate-/r* 66.3%

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

      6. associate-*l/ 81.4%

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

      7. *-commutative 81.4%

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

      8. *-commutative 81.4%

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

      9. distribute-rgt1-in 45.4%

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

      10. fma-def 81.4%

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

      11. +-commutative 81.4%

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

      12. +-commutative 81.4%

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

      13. cube-unmult 81.5%

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

      14. +-commutative 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in x around inf 73.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 73.2%

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

   6. Simplified 73.2%

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

   if -1 < x < -9.50000000000000057e-154

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 36.1%

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

   5. Taylor expanded in x around 0 32.1%

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

      1. neg-mul-1 32.1%

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

      2. unsub-neg 32.1%

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

   7. Simplified 32.1%

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

   if -9.50000000000000057e-154 < x < 1.74999999999999995e-127

   1. Initial program 61.1%

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

      1. times-frac 80.5%

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

      2. associate-+l+ 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. +-commutative 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in x around 0 77.2%

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

   if 1.74999999999999995e-127 < x

   1. Initial program 70.1%

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

      1. associate-/r* 77.5%

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

      2. +-commutative 77.5%

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

      3. +-commutative 77.5%

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

      4. +-commutative 77.5%

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

      5. associate-/r* 70.1%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 75.4%

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

      10. fma-def 83.2%

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

      11. +-commutative 83.2%

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

      12. +-commutative 83.2%

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

      13. cube-unmult 83.2%

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

      14. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in y around inf 32.4%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.4%

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

   6. Simplified 32.4%

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

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 17: 54.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -9.5e-154)
     (- (/ y x) y)
     (if (<= x 5.5e-127) (/ x y) (/ (/ x y) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -9.5e-154) {
		tmp = (y / x) - y;
	} else if (x <= 5.5e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-9.5d-154)) then
        tmp = (y / x) - y
    else if (x <= 5.5d-127) then
        tmp = x / y
    else
        tmp = (x / y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -9.5e-154) {
		tmp = (y / x) - y;
	} else if (x <= 5.5e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -9.5e-154:
		tmp = (y / x) - y
	elif x <= 5.5e-127:
		tmp = x / y
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -9.5e-154)
		tmp = Float64(Float64(y / x) - y);
	elseif (x <= 5.5e-127)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -9.5e-154)
		tmp = (y / x) - y;
	elseif (x <= 5.5e-127)
		tmp = x / y;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9.5e-154], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[LessEqual[x, 5.5e-127], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 66.3%

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

      1. times-frac 89.6%

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

      2. associate-+l+ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 78.9%

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

   5. Taylor expanded in x around inf 76.4%

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

   if -1 < x < -9.50000000000000057e-154

   1. Initial program 98.5%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 36.1%

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

   5. Taylor expanded in x around 0 32.1%

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

      1. neg-mul-1 32.1%

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

      2. unsub-neg 32.1%

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

   7. Simplified 32.1%

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

   if -9.50000000000000057e-154 < x < 5.50000000000000036e-127

   1. Initial program 61.1%

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

      1. times-frac 80.5%

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

      2. associate-+l+ 80.5%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. +-commutative 76.2%

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

   6. Simplified 76.2%

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

   7. Taylor expanded in x around 0 77.2%

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

   if 5.50000000000000036e-127 < x

   1. Initial program 70.1%

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

      1. associate-/r* 77.5%

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

      2. +-commutative 77.5%

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

      3. +-commutative 77.5%

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

      4. +-commutative 77.5%

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

      5. associate-/r* 70.1%

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

      6. associate-*l/ 83.2%

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

      7. *-commutative 83.2%

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

      8. *-commutative 83.2%

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

      9. distribute-rgt1-in 75.4%

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

      10. fma-def 83.2%

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

      11. +-commutative 83.2%

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

      12. +-commutative 83.2%

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

      13. cube-unmult 83.2%

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

      14. +-commutative 83.2%

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

   3. Simplified 83.2%

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

   4. Taylor expanded in y around inf 32.4%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 32.4%

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

   6. Simplified 32.4%

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

      1. associate-/r* 35.2%

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

      2. div-inv 35.2%

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

   8. Applied egg-rr 35.2%

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

      1. associate-*r/ 35.2%

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

      2. *-rgt-identity 35.2%

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

   10. Simplified 35.2%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -9.5 \cdot 10^{-154}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 18: 60.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.6e-112) (- (/ y x) y) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.6e-112) {
		tmp = (y / 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 <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.6d-112)) then
        tmp = (y / 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 <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.6e-112) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.6e-112:
		tmp = (y / x) - y
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.6e-112)
		tmp = Float64(Float64(y / 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 <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.6e-112)
		tmp = (y / x) - y;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.6e-112], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.3%

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

      1. times-frac 89.6%

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

      2. associate-+l+ 89.6%

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

   3. Simplified 89.6%

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

   4. Taylor expanded in x around inf 78.9%

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

   5. Taylor expanded in x around inf 76.4%

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

   if -1 < x < -1.59999999999999997e-112

   1. Initial program 99.8%

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

      1. times-frac 99.7%

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

      2. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y around 0 38.4%

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

   5. Taylor expanded in x around 0 32.7%

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

      1. neg-mul-1 32.7%

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

      2. unsub-neg 32.7%

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

   7. Simplified 32.7%

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

   if -1.59999999999999997e-112 < x

   1. Initial program 66.9%

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

      1. times-frac 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   4. Taylor expanded in x around 0 58.7%

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

      1. +-commutative 58.7%

         \[\leadsto \frac{x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

   6. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(y + 1\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-112}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 19: 60.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.14 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -1.14e-110) (- (/ y x) y) (/ x (+ y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.14e-110) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.14d-110)) then
        tmp = (y / x) - y
    else
        tmp = x / (y + (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.14e-110) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (1.0 / x) * (y / x)
	elif x <= -1.14e-110:
		tmp = (y / x) - y
	else:
		tmp = x / (y + (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -1.14e-110)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / Float64(y + Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -1.14e-110)
		tmp = (y / x) - y;
	else
		tmp = x / (y + (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.14e-110], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 89.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 78.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x}} \]

   5. Taylor expanded in x around inf 76.4%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x} \]

   if -1 < x < -1.14e-110

   1. Initial program 99.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 38.4%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 32.7%

      \[\leadsto \color{blue}{\frac{y}{x} + -1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 32.7%

         \[\leadsto \frac{y}{x} + \color{blue}{\left(-y\right)} \]

      2. unsub-neg 32.7%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   7. Simplified 32.7%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -1.14e-110 < x

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. div-inv 99.6%

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot 1}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. *-rgt-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x + y} \cdot \frac{x}{x + \left(y + 1\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   8. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto \frac{\frac{y}{x + y}}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{x + \left(y + 1\right)}\right)} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{\frac{y}{x + y}}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{x + \left(y + 1\right)}\right)} \]

   10. Taylor expanded in x around 0 58.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   11. Step-by-step derivation

       1. distribute-rgt-in 58.7%

          \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

       2. *-lft-identity 58.7%

          \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   12. Simplified 58.7%

       \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -1.14 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 20: 61.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.08e+15)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -4.1e-110) (/ y (* x (+ x 1.0))) (/ x (+ y (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.08e+15) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.1e-110) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.08d+15)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-4.1d-110)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y + (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.08e+15) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.1e-110) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y + (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.08e+15:
		tmp = (1.0 / x) * (y / x)
	elif x <= -4.1e-110:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y + (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.08e+15)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -4.1e-110)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y + Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.08e+15)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -4.1e-110)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y + (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.08e+15], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.1e-110], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08e15

   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. Step-by-step derivation

      1. times-frac 89.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 89.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 81.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x}} \]

   5. Taylor expanded in x around inf 78.5%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x} \]

   if -1.08e15 < x < -4.09999999999999983e-110

   1. Initial program 94.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 46.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if -4.09999999999999983e-110 < x

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. div-inv 99.6%

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot 1}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. *-rgt-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x + y} \cdot \frac{x}{x + \left(y + 1\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]
   8. Step-by-step derivation

      1. div-inv 99.7%

         \[\leadsto \frac{\frac{y}{x + y}}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{x + \left(y + 1\right)}\right)} \]

   9. Applied egg-rr 99.7%

      \[\leadsto \frac{\frac{y}{x + y}}{x + y} \cdot \color{blue}{\left(x \cdot \frac{1}{x + \left(y + 1\right)}\right)} \]

   10. Taylor expanded in x around 0 58.7%

       \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   11. Step-by-step derivation

       1. distribute-rgt-in 58.7%

          \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

       2. *-lft-identity 58.7%

          \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   12. Simplified 58.7%

       \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.1 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \end{array} \]

Alternative 21: 61.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.08e+15)
   (* (/ 1.0 x) (/ y x))
   (if (<= x -4.8e-110) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.08e+15) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.8e-110) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.08d+15)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-4.8d-110)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.08e+15) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -4.8e-110) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.08e+15:
		tmp = (1.0 / x) * (y / x)
	elif x <= -4.8e-110:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.08e+15)
		tmp = Float64(Float64(1.0 / x) * Float64(y / x));
	elseif (x <= -4.8e-110)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.08e+15)
		tmp = (1.0 / x) * (y / x);
	elseif (x <= -4.8e-110)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.08e+15], N[(N[(1.0 / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.8e-110], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.08e15

   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. Step-by-step derivation

      1. times-frac 89.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 89.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 81.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x}} \]

   5. Taylor expanded in x around inf 78.5%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x} \]

   if -1.08e15 < x < -4.80000000000000013e-110

   1. Initial program 94.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 99.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 46.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if -4.80000000000000013e-110 < x

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. div-inv 99.6%

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot 1}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. *-rgt-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x + y} \cdot \frac{x}{x + \left(y + 1\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   8. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 60.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 60.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   10. Simplified 60.1%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.08 \cdot 10^{+15}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 22: 44.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-150) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-150) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-150) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-150) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-150:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-150)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-150)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-150], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.4e-150

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 59.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 36.1%

      \[\leadsto \frac{y}{\color{blue}{x}} \]

   if 2.4e-150 < y < 1

   1. Initial program 82.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 97.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 97.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 95.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + x}} \]
   5. Step-by-step derivation

      1. +-commutative 95.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + 1}} \]

   6. Simplified 95.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x + 1}} \]

   7. Taylor expanded in x around 0 40.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 64.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 71.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 71.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 71.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 71.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/r* 64.7%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      6. associate-*l/ 78.5%

         \[\leadsto \color{blue}{\frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)} \cdot y} \]

      7. *-commutative 78.5%

         \[\leadsto \color{blue}{y \cdot \frac{x}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\left(y + x\right) + 1\right)}} \]

      8. *-commutative 78.5%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      9. distribute-rgt1-in 75.2%

         \[\leadsto y \cdot \frac{x}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      10. fma-def 78.5%

          \[\leadsto y \cdot \frac{x}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      11. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      12. +-commutative 78.5%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      13. cube-unmult 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      14. +-commutative 78.4%

          \[\leadsto y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 78.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 71.1%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 71.1%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 71.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-150}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 23: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e-110) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e-110) {
		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 <= (-4.8d-110)) 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 <= -4.8e-110) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e-110:
		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 <= -4.8e-110)
		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 <= -4.8e-110)
		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, -4.8e-110], 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 < -4.80000000000000013e-110

   1. Initial program 72.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 79.1%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 79.1%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 79.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 79.1%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 72.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 91.4%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 91.4%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 91.4%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 91.4%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 91.4%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. div-inv 99.8%

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.8%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot 1}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. *-rgt-identity 99.8%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x + y} \cdot \frac{x}{x + \left(y + 1\right)} \]

   7. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   8. Taylor expanded in y around 0 68.5%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   9. Step-by-step derivation

      1. +-commutative 68.5%

         \[\leadsto \frac{y}{\color{blue}{\left(x + 1\right)} \cdot x} \]

      2. *-commutative 68.5%

         \[\leadsto \frac{y}{\color{blue}{x \cdot \left(x + 1\right)}} \]

      3. associate-/r* 71.2%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + 1}} \]

      4. +-commutative 71.2%

         \[\leadsto \frac{\frac{y}{x}}{\color{blue}{1 + x}} \]

   10. Simplified 71.2%

       \[\leadsto \color{blue}{\frac{\frac{y}{x}}{1 + x}} \]

   if -4.80000000000000013e-110 < x

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 70.8%

         \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

      2. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

      3. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

      4. +-commutative 70.8%

         \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

      5. associate-/l/ 66.9%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. times-frac 86.9%

         \[\leadsto \color{blue}{\frac{x}{\left(y + x\right) + 1} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}} \]

      7. *-commutative 86.9%

         \[\leadsto \color{blue}{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1}} \]

      8. +-commutative 86.9%

         \[\leadsto \frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)} \cdot \frac{x}{\left(y + x\right) + 1} \]

      9. +-commutative 86.9%

         \[\leadsto \frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}} \cdot \frac{x}{\left(y + x\right) + 1} \]

      10. +-commutative 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{\left(x + y\right)} + 1} \]

      11. associate-+l+ 86.9%

          \[\leadsto \frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.9%

      \[\leadsto \color{blue}{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{x}{x + \left(y + 1\right)}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. div-inv 99.6%

         \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]

   5. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\left(\frac{y}{x + y} \cdot \frac{1}{x + y}\right)} \cdot \frac{x}{x + \left(y + 1\right)} \]
   6. Step-by-step derivation

      1. associate-*r/ 99.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x + y} \cdot 1}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

      2. *-rgt-identity 99.7%

         \[\leadsto \frac{\color{blue}{\frac{y}{x + y}}}{x + y} \cdot \frac{x}{x + \left(y + 1\right)} \]

   7. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x + y}}{x + y}} \cdot \frac{x}{x + \left(y + 1\right)} \]

   8. Taylor expanded in x around 0 58.7%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   9. Step-by-step derivation

      1. associate-/r* 60.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 60.1%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   10. Simplified 60.1%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 24: 33.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 5.4e-149) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-149) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.4d-149) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.4e-149) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.4e-149:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.4e-149)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.4e-149)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.4e-149], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.40000000000000028e-149

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 86.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 86.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 86.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 59.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 36.1%

      \[\leadsto \frac{y}{\color{blue}{x}} \]

   if 5.40000000000000028e-149 < y

   1. Initial program 70.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 91.0%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. associate-+l+ 91.0%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 91.0%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 80.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + x}} \]
   5. Step-by-step derivation

      1. +-commutative 80.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + 1}} \]

   6. Simplified 80.6%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x + 1}} \]

   7. Taylor expanded in x around 0 34.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-149}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 25: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 68.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 88.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.4%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around inf 50.6%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]

5. Taylor expanded in y around inf 4.2%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]

Alternative 26: 25.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ x y))

C

double code(double x, double y) {
	return x / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

public static double code(double x, double y) {
	return x / y;
}

Python

def code(x, y):
	return x / y

Julia

function code(x, y)
	return Float64(x / y)
end

MATLAB

function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 68.7%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.4%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. associate-+l+ 88.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.4%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around 0 79.3%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + x}} \]
5. Step-by-step derivation

   1. +-commutative 79.3%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + 1}} \]

6. Simplified 79.3%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x + 1}} \]

7. Taylor expanded in x around 0 26.8%

   \[\leadsto \color{blue}{\frac{x}{y}} \]

8. Final simplification 26.8%

   \[\leadsto \frac{x}{y} \]

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 2023268 
(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))))