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

Percentage Accurate: 69.1% → 99.8%

Time: 19.4s

Alternatives: 22

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: 69.1% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. times-frac 89.8%

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

   2. /-rgt-identity 89.8%

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

   3. associate-/l/ 89.8%

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

   4. *-lft-identity 89.8%

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

   5. associate-+l+ 89.8%

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

3. Simplified 89.8%

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

   1. *-commutative 89.8%

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

   2. associate-/r* 99.7%

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

   3. associate-*r/ 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 67.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-234}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e+154)
   (/ (/ y (+ x (+ y 1.0))) (+ y x))
   (if (<= x -5.2e-137)
     (* (/ y (+ x 1.0)) (/ x (* (+ y x) (+ y x))))
     (if (<= x -8e-234)
       (* (/ x (+ y x)) (/ y (* (+ y x) (+ x 1.0))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+154) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -5.2e-137) {
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	} else if (x <= -8e-234) {
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d+154)) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else if (x <= (-5.2d-137)) then
        tmp = (y / (x + 1.0d0)) * (x / ((y + x) * (y + x)))
    else if (x <= (-8d-234)) then
        tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e+154) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -5.2e-137) {
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	} else if (x <= -8e-234) {
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e+154:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	elif x <= -5.2e-137:
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)))
	elif x <= -8e-234:
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e+154)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	elseif (x <= -5.2e-137)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	elseif (x <= -8e-234)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e+154)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	elseif (x <= -5.2e-137)
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	elseif (x <= -8e-234)
		tmp = (x / (y + x)) * (y / ((y + x) * (x + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e+154], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-137], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -8e-234], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.4499999999999999e154

   1. Initial program 48.4%

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

      1. times-frac 82.8%

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

      2. /-rgt-identity 82.8%

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

      3. associate-/l/ 82.8%

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

      4. *-lft-identity 82.8%

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

      5. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 88.0%

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

   if -1.4499999999999999e154 < x < -5.1999999999999999e-137

   1. Initial program 70.6%

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

      1. times-frac 95.2%

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

      2. /-rgt-identity 95.2%

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

      3. associate-/l/ 95.2%

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

      4. *-lft-identity 95.2%

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

      5. associate-+l+ 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around 0 89.3%

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

      1. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   if -5.1999999999999999e-137 < x < -7.9999999999999997e-234

   1. Initial program 74.1%

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

      1. times-frac 83.0%

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

      2. /-rgt-identity 83.0%

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

      3. associate-/l/ 83.0%

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

      4. *-lft-identity 83.0%

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

      5. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 87.2%

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

      1. +-commutative 74.9%

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

   8. Simplified 87.2%

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

      1. expm1-log1p-u 87.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + 1} \cdot \frac{x}{x + y}}{x + y}\right)\right)} \]

      2. expm1-udef 51.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + 1} \cdot \frac{x}{x + y}}{x + y}\right)} - 1} \]

      3. associate-/l* 51.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}}\right)} - 1 \]

   10. Applied egg-rr 51.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 93.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}\right)\right)} \]

       2. expm1-log1p 93.0%

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

       3. associate-/r/ 87.2%

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

       4. *-rgt-identity 87.2%

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

       5. associate-*r/ 87.2%

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

       6. associate-*r/ 87.1%

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

       7. associate-/l/ 87.1%

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

       8. associate-*r/ 87.2%

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

       9. *-rgt-identity 87.2%

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

       10. *-commutative 87.2%

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

   12. Simplified 87.2%

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

   if -7.9999999999999997e-234 < 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. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 58.7%

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

      1. +-commutative 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-234}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 3: 67.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-233}:\\ \;\;\;\;t_0 \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + \left(x + 1\right)\right)}{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -1.32e+154)
     (/ (/ y (+ x (+ y 1.0))) (+ y x))
     (if (<= x -5.2e-137)
       (* (/ y (+ x 1.0)) (/ x (* (+ y x) (+ y x))))
       (if (<= x -9e-233)
         (* t_0 (/ y (* (+ y x) (+ x 1.0))))
         (/ 1.0 (/ (+ y (+ x (+ x 1.0))) t_0)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -5.2e-137) {
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	} else if (x <= -9e-233) {
		tmp = t_0 * (y / ((y + x) * (x + 1.0)));
	} else {
		tmp = 1.0 / ((y + (x + (x + 1.0))) / t_0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y + x)
    if (x <= (-1.32d+154)) then
        tmp = (y / (x + (y + 1.0d0))) / (y + x)
    else if (x <= (-5.2d-137)) then
        tmp = (y / (x + 1.0d0)) * (x / ((y + x) * (y + x)))
    else if (x <= (-9d-233)) then
        tmp = t_0 * (y / ((y + x) * (x + 1.0d0)))
    else
        tmp = 1.0d0 / ((y + (x + (x + 1.0d0))) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -1.32e+154) {
		tmp = (y / (x + (y + 1.0))) / (y + x);
	} else if (x <= -5.2e-137) {
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	} else if (x <= -9e-233) {
		tmp = t_0 * (y / ((y + x) * (x + 1.0)));
	} else {
		tmp = 1.0 / ((y + (x + (x + 1.0))) / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -1.32e+154:
		tmp = (y / (x + (y + 1.0))) / (y + x)
	elif x <= -5.2e-137:
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)))
	elif x <= -9e-233:
		tmp = t_0 * (y / ((y + x) * (x + 1.0)))
	else:
		tmp = 1.0 / ((y + (x + (x + 1.0))) / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -1.32e+154)
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) / Float64(y + x));
	elseif (x <= -5.2e-137)
		tmp = Float64(Float64(y / Float64(x + 1.0)) * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	elseif (x <= -9e-233)
		tmp = Float64(t_0 * Float64(y / Float64(Float64(y + x) * Float64(x + 1.0))));
	else
		tmp = Float64(1.0 / Float64(Float64(y + Float64(x + Float64(x + 1.0))) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -1.32e+154)
		tmp = (y / (x + (y + 1.0))) / (y + x);
	elseif (x <= -5.2e-137)
		tmp = (y / (x + 1.0)) * (x / ((y + x) * (y + x)));
	elseif (x <= -9e-233)
		tmp = t_0 * (y / ((y + x) * (x + 1.0)));
	else
		tmp = 1.0 / ((y + (x + (x + 1.0))) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.32e+154], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.2e-137], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -9e-233], N[(t$95$0 * N[(y / N[(N[(y + x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(y + N[(x + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.31999999999999998e154

   1. Initial program 48.4%

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

      1. times-frac 82.8%

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

      2. /-rgt-identity 82.8%

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

      3. associate-/l/ 82.8%

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

      4. *-lft-identity 82.8%

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

      5. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 88.0%

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

   if -1.31999999999999998e154 < x < -5.1999999999999999e-137

   1. Initial program 70.6%

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

      1. times-frac 95.2%

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

      2. /-rgt-identity 95.2%

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

      3. associate-/l/ 95.2%

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

      4. *-lft-identity 95.2%

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

      5. associate-+l+ 95.2%

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

   3. Simplified 95.2%

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

   4. Taylor expanded in y around 0 89.3%

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

      1. +-commutative 89.3%

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

   6. Simplified 89.3%

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

   if -5.1999999999999999e-137 < x < -9.0000000000000004e-233

   1. Initial program 74.1%

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

      1. times-frac 83.0%

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

      2. /-rgt-identity 83.0%

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

      3. associate-/l/ 83.0%

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

      4. *-lft-identity 83.0%

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

      5. associate-+l+ 83.0%

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

   3. Simplified 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around 0 87.2%

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

      1. +-commutative 74.9%

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

   8. Simplified 87.2%

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

      1. expm1-log1p-u 87.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + 1} \cdot \frac{x}{x + y}}{x + y}\right)\right)} \]

      2. expm1-udef 51.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + 1} \cdot \frac{x}{x + y}}{x + y}\right)} - 1} \]

      3. associate-/l* 51.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}}\right)} - 1 \]

   10. Applied egg-rr 51.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 93.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{y}{x + 1}}{\frac{x + y}{\frac{x}{x + y}}}\right)\right)} \]

       2. expm1-log1p 93.0%

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

       3. associate-/r/ 87.2%

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

       4. *-rgt-identity 87.2%

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

       5. associate-*r/ 87.2%

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

       6. associate-*r/ 87.1%

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

       7. associate-/l/ 87.1%

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

       8. associate-*r/ 87.2%

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

       9. *-rgt-identity 87.2%

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

       10. *-commutative 87.2%

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

   12. Simplified 87.2%

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

   if -9.0000000000000004e-233 < 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. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. +-commutative 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. clear-num 99.0%

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

      2. inv-pow 99.0%

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

      3. div-inv 99.0%

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

      4. clear-num 99.0%

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

      5. +-commutative 99.0%

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

   9. Applied egg-rr 99.0%

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

       1. unpow-1 99.0%

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

       2. associate-*r/ 99.0%

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

       3. +-commutative 99.0%

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

   11. Simplified 99.0%

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

   12. Taylor expanded in y around -inf 59.3%

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

       1. mul-1-neg 59.3%

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

       2. unsub-neg 59.3%

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

       3. neg-mul-1 59.3%

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

       4. unsub-neg 59.3%

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

       5. distribute-lft-in 59.3%

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

       6. metadata-eval 59.3%

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

       7. neg-mul-1 59.3%

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

       8. unsub-neg 59.3%

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

   14. Simplified 59.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -5.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{y}{x + 1} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{elif}\;x \leq -9 \cdot 10^{-233}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y + \left(x + \left(x + 1\right)\right)}{\frac{x}{y + x}}}\\ \end{array} \]

Alternative 4: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.8e-21)
   (/ (* (/ x (+ y x)) (/ y (+ x 1.0))) (+ y x))
   (* (/ y (+ x (+ y 1.0))) (/ x (* (+ y x) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-21) {
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	} else {
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.8d-21) then
        tmp = ((x / (y + x)) * (y / (x + 1.0d0))) / (y + x)
    else
        tmp = (y / (x + (y + 1.0d0))) * (x / ((y + x) * (y + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.8e-21) {
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	} else {
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.8e-21:
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x)
	else:
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.8e-21)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(x + Float64(y + 1.0))) * Float64(x / Float64(Float64(y + x) * Float64(y + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.8e-21)
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	else
		tmp = (y / (x + (y + 1.0))) * (x / ((y + x) * (y + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.8e-21], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.8e-21

   1. Initial program 67.8%

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

      1. times-frac 88.7%

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

      2. /-rgt-identity 88.7%

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

      3. associate-/l/ 88.7%

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

      4. *-lft-identity 88.7%

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

      5. associate-+l+ 88.7%

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

   3. Simplified 88.7%

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

      1. *-commutative 88.7%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 81.3%

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

      1. +-commutative 81.3%

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

   8. Simplified 81.3%

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

   if 5.8e-21 < y

   1. Initial program 71.1%

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

      1. times-frac 93.6%

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

      2. /-rgt-identity 93.6%

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

      3. associate-/l/ 93.6%

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

      4. *-lft-identity 93.6%

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

      5. associate-+l+ 93.6%

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

   3. Simplified 93.6%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.8 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 5: 82.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.5)
   (/ (* (/ x (+ y x)) (/ y (+ x 1.0))) (+ y x))
   (/ x (* (+ y x) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5) {
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.5d0) then
        tmp = ((x / (y + x)) * (y / (x + 1.0d0))) / (y + x)
    else
        tmp = x / ((y + x) * (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5) {
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	} else {
		tmp = x / ((y + x) * (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5:
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x)
	else:
		tmp = x / ((y + x) * (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(x / Float64(Float64(y + x) * Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5)
		tmp = ((x / (y + x)) * (y / (x + 1.0))) / (y + x);
	else
		tmp = x / ((y + x) * (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.5

   1. Initial program 68.4%

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

      1. times-frac 88.9%

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

      2. /-rgt-identity 88.9%

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

      3. associate-/l/ 88.9%

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

      4. *-lft-identity 88.9%

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

      5. associate-+l+ 88.9%

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

   3. Simplified 88.9%

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

      1. *-commutative 88.9%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 81.4%

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

      1. +-commutative 81.4%

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

   8. Simplified 81.4%

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

   if 9.5 < y

   1. Initial program 69.2%

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

      1. times-frac 93.2%

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

      2. /-rgt-identity 93.2%

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

      3. associate-/l/ 93.2%

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

      4. *-lft-identity 93.2%

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

      5. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

   4. Taylor expanded in y around inf 83.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. times-frac 89.8%

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

   2. associate-+r+ 89.8%

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

   3. associate-*l/ 84.0%

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

   4. times-frac 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 7: 54.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -1.1e-159)
   (/ (/ y x) x)
   (if (<= y 5.1e-138)
     (/ y x)
     (if (<= y 8.5e-61)
       (- (/ x y) x)
       (if (<= y 9.5) (/ 1.0 (* x (/ x y))) (/ x (* y y)))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -1.1e-159) {
		tmp = (y / x) / x;
	} else if (y <= 5.1e-138) {
		tmp = y / x;
	} else if (y <= 8.5e-61) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		tmp = 1.0 / (x * (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 <= (-1.1d-159)) then
        tmp = (y / x) / x
    else if (y <= 5.1d-138) then
        tmp = y / x
    else if (y <= 8.5d-61) then
        tmp = (x / y) - x
    else if (y <= 9.5d0) then
        tmp = 1.0d0 / (x * (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 <= -1.1e-159) {
		tmp = (y / x) / x;
	} else if (y <= 5.1e-138) {
		tmp = y / x;
	} else if (y <= 8.5e-61) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		tmp = 1.0 / (x * (x / y));
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -1.1e-159:
		tmp = (y / x) / x
	elif y <= 5.1e-138:
		tmp = y / x
	elif y <= 8.5e-61:
		tmp = (x / y) - x
	elif y <= 9.5:
		tmp = 1.0 / (x * (x / y))
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -1.1e-159)
		tmp = Float64(Float64(y / x) / x);
	elseif (y <= 5.1e-138)
		tmp = Float64(y / x);
	elseif (y <= 8.5e-61)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 9.5)
		tmp = Float64(1.0 / Float64(x * 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 <= -1.1e-159)
		tmp = (y / x) / x;
	elseif (y <= 5.1e-138)
		tmp = y / x;
	elseif (y <= 8.5e-61)
		tmp = (x / y) - x;
	elseif (y <= 9.5)
		tmp = 1.0 / (x * (x / y));
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -1.1e-159], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[y, 5.1e-138], N[(y / x), $MachinePrecision], If[LessEqual[y, 8.5e-61], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 9.5], N[(1.0 / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -1.1e-159

   1. Initial program 62.9%

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

      1. times-frac 90.6%

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

      2. /-rgt-identity 90.6%

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

      3. associate-/l/ 90.6%

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

      4. *-lft-identity 90.6%

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

      5. associate-+l+ 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around inf 32.7%

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

      1. unpow2 32.7%

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

   6. Simplified 32.7%

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

      1. associate-/r* 32.7%

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

      2. div-inv 32.6%

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

   8. Applied egg-rr 32.6%

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

      1. un-div-inv 32.7%

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

   10. Applied egg-rr 32.7%

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

   if -1.1e-159 < y < 5.1000000000000002e-138

   1. Initial program 64.0%

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

      1. times-frac 79.9%

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

      2. /-rgt-identity 79.9%

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

      3. associate-/l/ 79.9%

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

      4. *-lft-identity 79.9%

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

      5. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in y around 0 85.2%

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

   5. Taylor expanded in x around 0 72.3%

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

   if 5.1000000000000002e-138 < y < 8.50000000000000016e-61

   1. Initial program 85.7%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 38.0%

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

      1. associate-/r* 38.0%

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

      2. +-commutative 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in y around 0 38.0%

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

      1. neg-mul-1 38.0%

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

      2. unsub-neg 38.0%

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

   9. Simplified 38.0%

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

   if 8.50000000000000016e-61 < y < 9.5

   1. Initial program 92.8%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. 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 x around inf 54.1%

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

      1. unpow2 54.1%

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

   6. Simplified 54.1%

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

      1. associate-/r* 54.2%

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

      2. div-inv 54.2%

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

   8. Applied egg-rr 54.2%

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

      1. clear-num 54.0%

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

      2. frac-times 54.3%

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

      3. metadata-eval 54.3%

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

   10. Applied egg-rr 54.3%

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

   if 9.5 < y

   1. Initial program 69.2%

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

      1. times-frac 93.2%

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

      2. /-rgt-identity 93.2%

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

      3. associate-/l/ 93.2%

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

      4. *-lft-identity 93.2%

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

      5. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-/r* 99.8%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 69.9%

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

      1. unpow2 69.9%

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

   8. Simplified 69.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;\frac{1}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 8: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.3e+154)
   (/ (/ y x) (+ y x))
   (if (<= x -1.35e-229)
     (/ y (* (+ x (+ y 1.0)) (+ y x)))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.3e+154) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.35e-229) {
		tmp = y / ((x + (y + 1.0)) * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.3d+154)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1.35d-229)) then
        tmp = y / ((x + (y + 1.0d0)) * (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.3e+154) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.35e-229) {
		tmp = y / ((x + (y + 1.0)) * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.3e+154:
		tmp = (y / x) / (y + x)
	elif x <= -1.35e-229:
		tmp = y / ((x + (y + 1.0)) * (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.3e+154)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1.35e-229)
		tmp = Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.3e+154)
		tmp = (y / x) / (y + x);
	elseif (x <= -1.35e-229)
		tmp = y / ((x + (y + 1.0)) * (y + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.3e+154], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-229], N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2999999999999998e154

   1. Initial program 48.4%

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

      1. times-frac 82.8%

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

      2. /-rgt-identity 82.8%

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

      3. associate-/l/ 82.8%

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

      4. *-lft-identity 82.8%

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

      5. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 87.0%

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

   if -4.2999999999999998e154 < x < -1.3499999999999999e-229

   1. Initial program 71.6%

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

      1. times-frac 91.9%

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

      2. /-rgt-identity 91.9%

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

      3. associate-/l/ 91.9%

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

      4. *-lft-identity 91.9%

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

      5. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-/r* 99.7%

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

      3. frac-times 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in y around 0 77.0%

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

   if -1.3499999999999999e-229 < 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. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 58.7%

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

      1. +-commutative 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 9: 63.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;x \leq -2.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{t_0}}{y + x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{t_0 \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= x -2.65e+154)
     (/ (/ y t_0) (+ y x))
     (if (<= x -1.35e-229)
       (/ y (* t_0 (+ y x)))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -2.65e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.35e-229) {
		tmp = y / (t_0 * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (x <= (-2.65d+154)) then
        tmp = (y / t_0) / (y + x)
    else if (x <= (-1.35d-229)) then
        tmp = y / (t_0 * (y + x))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (x <= -2.65e+154) {
		tmp = (y / t_0) / (y + x);
	} else if (x <= -1.35e-229) {
		tmp = y / (t_0 * (y + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if x <= -2.65e+154:
		tmp = (y / t_0) / (y + x)
	elif x <= -1.35e-229:
		tmp = y / (t_0 * (y + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (x <= -2.65e+154)
		tmp = Float64(Float64(y / t_0) / Float64(y + x));
	elseif (x <= -1.35e-229)
		tmp = Float64(y / Float64(t_0 * Float64(y + x)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (x <= -2.65e+154)
		tmp = (y / t_0) / (y + x);
	elseif (x <= -1.35e-229)
		tmp = y / (t_0 * (y + x));
	else
		tmp = (x / (y + 1.0)) / (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, -2.65e+154], N[(N[(y / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.35e-229], N[(y / N[(t$95$0 * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.65000000000000012e154

   1. Initial program 48.4%

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

      1. times-frac 82.8%

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

      2. /-rgt-identity 82.8%

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

      3. associate-/l/ 82.8%

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

      4. *-lft-identity 82.8%

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

      5. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

      1. *-commutative 82.8%

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

      2. associate-/r* 99.9%

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

      3. associate-*r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around inf 88.0%

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

   if -2.65000000000000012e154 < x < -1.3499999999999999e-229

   1. Initial program 71.6%

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

      1. times-frac 91.9%

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

      2. /-rgt-identity 91.9%

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

      3. associate-/l/ 91.9%

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

      4. *-lft-identity 91.9%

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

      5. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

      1. *-commutative 91.9%

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

      2. associate-/r* 99.7%

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

      3. frac-times 96.6%

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

   5. Applied egg-rr 96.6%

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

   6. Taylor expanded in y around 0 77.0%

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

   if -1.3499999999999999e-229 < 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. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

      1. *-commutative 89.8%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 58.7%

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

      1. +-commutative 58.7%

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

   8. Simplified 58.7%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.65 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{y}{x + \left(y + 1\right)}}{y + x}\\ \mathbf{elif}\;x \leq -1.35 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 10: 53.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -9.6 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -9.6e-161)
     t_0
     (if (<= y 7e-138)
       (/ y x)
       (if (<= y 1.6e-60) (- (/ x y) x) (if (<= y 9.5) t_0 (/ x (* y y))))))))

C

double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -9.6e-161) {
		tmp = t_0;
	} else if (y <= 7e-138) {
		tmp = y / x;
	} else if (y <= 1.6e-60) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		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 <= (-9.6d-161)) then
        tmp = t_0
    else if (y <= 7d-138) then
        tmp = y / x
    else if (y <= 1.6d-60) then
        tmp = (x / y) - x
    else if (y <= 9.5d0) 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 <= -9.6e-161) {
		tmp = t_0;
	} else if (y <= 7e-138) {
		tmp = y / x;
	} else if (y <= 1.6e-60) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -9.6e-161:
		tmp = t_0
	elif y <= 7e-138:
		tmp = y / x
	elif y <= 1.6e-60:
		tmp = (x / y) - x
	elif y <= 9.5:
		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 <= -9.6e-161)
		tmp = t_0;
	elseif (y <= 7e-138)
		tmp = Float64(y / x);
	elseif (y <= 1.6e-60)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 9.5)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -9.6e-161)
		tmp = t_0;
	elseif (y <= 7e-138)
		tmp = y / x;
	elseif (y <= 1.6e-60)
		tmp = (x / y) - x;
	elseif (y <= 9.5)
		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, -9.6e-161], t$95$0, If[LessEqual[y, 7e-138], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.6e-60], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 9.5], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -9.59999999999999995e-161 or 1.6000000000000001e-60 < y < 9.5

   1. Initial program 66.5%

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

      1. times-frac 91.5%

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

      2. /-rgt-identity 91.5%

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

      3. associate-/l/ 91.5%

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

      4. *-lft-identity 91.5%

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

      5. associate-+l+ 91.5%

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

   3. Simplified 91.5%

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

   4. Taylor expanded in x around inf 35.4%

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

      1. unpow2 35.4%

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

   6. Simplified 35.4%

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

   if -9.59999999999999995e-161 < y < 6.9999999999999997e-138

   1. Initial program 65.0%

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

      1. times-frac 80.4%

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

      2. /-rgt-identity 80.4%

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

      3. associate-/l/ 80.4%

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

      4. *-lft-identity 80.4%

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

      5. associate-+l+ 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in y around 0 86.4%

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

   5. Taylor expanded in x around 0 73.3%

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

   if 6.9999999999999997e-138 < y < 1.6000000000000001e-60

   1. Initial program 85.7%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 38.0%

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

      1. associate-/r* 38.0%

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

      2. +-commutative 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in y around 0 38.0%

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

      1. neg-mul-1 38.0%

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

      2. unsub-neg 38.0%

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

   9. Simplified 38.0%

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

   if 9.5 < y

   1. Initial program 69.2%

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

      1. times-frac 93.2%

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

      2. /-rgt-identity 93.2%

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

      3. associate-/l/ 93.2%

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

      4. *-lft-identity 93.2%

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

      5. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-/r* 99.8%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 69.9%

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

      1. unpow2 69.9%

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

   8. Simplified 69.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 11: 54.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y x) x)))
   (if (<= y -1.35e-159)
     t_0
     (if (<= y 7e-138)
       (/ y x)
       (if (<= y 1.12e-60) (- (/ x y) x) (if (<= y 9.5) t_0 (/ x (* y y))))))))

C

double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -1.35e-159) {
		tmp = t_0;
	} else if (y <= 7e-138) {
		tmp = y / x;
	} else if (y <= 1.12e-60) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		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 <= (-1.35d-159)) then
        tmp = t_0
    else if (y <= 7d-138) then
        tmp = y / x
    else if (y <= 1.12d-60) then
        tmp = (x / y) - x
    else if (y <= 9.5d0) 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 <= -1.35e-159) {
		tmp = t_0;
	} else if (y <= 7e-138) {
		tmp = y / x;
	} else if (y <= 1.12e-60) {
		tmp = (x / y) - x;
	} else if (y <= 9.5) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) / x
	tmp = 0
	if y <= -1.35e-159:
		tmp = t_0
	elif y <= 7e-138:
		tmp = y / x
	elif y <= 1.12e-60:
		tmp = (x / y) - x
	elif y <= 9.5:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / x) / x)
	tmp = 0.0
	if (y <= -1.35e-159)
		tmp = t_0;
	elseif (y <= 7e-138)
		tmp = Float64(y / x);
	elseif (y <= 1.12e-60)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 9.5)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) / x;
	tmp = 0.0;
	if (y <= -1.35e-159)
		tmp = t_0;
	elseif (y <= 7e-138)
		tmp = y / x;
	elseif (y <= 1.12e-60)
		tmp = (x / y) - x;
	elseif (y <= 9.5)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[y, -1.35e-159], t$95$0, If[LessEqual[y, 7e-138], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.12e-60], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 9.5], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.35e-159 or 1.12e-60 < y < 9.5

   1. Initial program 67.1%

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

      1. times-frac 91.9%

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

      2. /-rgt-identity 91.9%

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

      3. associate-/l/ 91.9%

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

      4. *-lft-identity 91.9%

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

      5. associate-+l+ 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in x around inf 35.7%

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

      1. unpow2 35.7%

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

   6. Simplified 35.7%

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

      1. associate-/r* 35.7%

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

      2. div-inv 35.6%

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

   8. Applied egg-rr 35.6%

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

      1. un-div-inv 35.7%

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

   10. Applied egg-rr 35.7%

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

   if -1.35e-159 < y < 6.9999999999999997e-138

   1. Initial program 64.0%

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

      1. times-frac 79.9%

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

      2. /-rgt-identity 79.9%

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

      3. associate-/l/ 79.9%

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

      4. *-lft-identity 79.9%

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

      5. associate-+l+ 79.9%

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

   3. Simplified 79.9%

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

   4. Taylor expanded in y around 0 85.2%

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

   5. Taylor expanded in x around 0 72.3%

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

   if 6.9999999999999997e-138 < y < 1.12e-60

   1. Initial program 85.7%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 38.0%

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

      1. associate-/r* 38.0%

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

      2. +-commutative 38.0%

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

   6. Simplified 38.0%

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

   7. Taylor expanded in y around 0 38.0%

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

      1. neg-mul-1 38.0%

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

      2. unsub-neg 38.0%

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

   9. Simplified 38.0%

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

   if 9.5 < y

   1. Initial program 69.2%

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

      1. times-frac 93.2%

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

      2. /-rgt-identity 93.2%

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

      3. associate-/l/ 93.2%

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

      4. *-lft-identity 93.2%

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

      5. associate-+l+ 93.2%

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

   3. Simplified 93.2%

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

      1. *-commutative 93.2%

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

      2. associate-/r* 99.8%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 69.9%

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

      1. unpow2 69.9%

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

   8. Simplified 69.9%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-159}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{-60}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 9.5:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 12: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -3.65 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -3.65e+65)
     t_0
     (if (<= y 2.6e-138) (/ y x) (if (<= y 0.75) (- (/ x y) x) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -3.65e+65) {
		tmp = t_0;
	} else if (y <= 2.6e-138) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-3.65d+65)) then
        tmp = t_0
    else if (y <= 2.6d-138) then
        tmp = y / x
    else if (y <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -3.65e+65) {
		tmp = t_0;
	} else if (y <= 2.6e-138) {
		tmp = y / x;
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -3.65e+65:
		tmp = t_0
	elif y <= 2.6e-138:
		tmp = y / x
	elif y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -3.65e+65)
		tmp = t_0;
	elseif (y <= 2.6e-138)
		tmp = Float64(y / x);
	elseif (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -3.65e+65)
		tmp = t_0;
	elseif (y <= 2.6e-138)
		tmp = y / x;
	elseif (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.65e+65], t$95$0, If[LessEqual[y, 2.6e-138], N[(y / x), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.64999999999999987e65 or 0.75 < y

   1. Initial program 58.9%

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

      1. times-frac 89.9%

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

      2. /-rgt-identity 89.9%

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

      3. associate-/l/ 89.9%

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

      4. *-lft-identity 89.9%

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

      5. associate-+l+ 89.9%

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

   3. Simplified 89.9%

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

      1. *-commutative 89.9%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around inf 71.5%

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

      1. unpow2 71.5%

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

   8. Simplified 71.5%

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

   if -3.64999999999999987e65 < y < 2.6e-138

   1. Initial program 72.2%

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

      1. times-frac 85.9%

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

      2. /-rgt-identity 85.9%

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

      3. associate-/l/ 85.9%

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

      4. *-lft-identity 85.9%

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

      5. associate-+l+ 85.9%

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

   3. Simplified 85.9%

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

   4. Taylor expanded in y around 0 72.7%

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

   5. Taylor expanded in x around 0 52.5%

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

   if 2.6e-138 < y < 0.75

   1. Initial program 88.4%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 37.4%

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

      1. associate-/r* 37.3%

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

      2. +-commutative 37.3%

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

   6. Simplified 37.3%

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

   7. Taylor expanded in y around 0 37.3%

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

      1. neg-mul-1 37.3%

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

      2. unsub-neg 37.3%

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

   9. Simplified 37.3%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{+65}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 13: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -1.16e-174) (- (/ y x) y) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.16e-174) {
		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 = (y / x) / x
    else if (x <= (-1.16d-174)) 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 = (y / x) / x;
	} else if (x <= -1.16e-174) {
		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 = (y / x) / x
	elif x <= -1.16e-174:
		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(y / x) / x);
	elseif (x <= -1.16e-174)
		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 = (y / x) / x;
	elseif (x <= -1.16e-174)
		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[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.16e-174], 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 56.1%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around inf 72.6%

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

      1. unpow2 72.6%

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

   6. Simplified 72.6%

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

      1. associate-/r* 74.3%

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

      2. div-inv 74.2%

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

   8. Applied egg-rr 74.2%

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

      1. un-div-inv 74.3%

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

   10. Applied egg-rr 74.3%

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

   if -1 < x < -1.16e-174

   1. Initial program 82.7%

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

      1. times-frac 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. associate-/l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 46.1%

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

   5. Taylor expanded in x around 0 46.1%

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

      1. neg-mul-1 46.1%

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

      2. unsub-neg 46.1%

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

   7. Simplified 46.1%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 14: 59.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.2e+157)
   (/ (/ y x) x)
   (if (<= x -1.16e-174) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+157) {
		tmp = (y / x) / x;
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.2d+157)) then
        tmp = (y / x) / x
    else if (x <= (-1.16d-174)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.2e+157) {
		tmp = (y / x) / x;
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.2e+157:
		tmp = (y / x) / x
	elif x <= -1.16e-174:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.2e+157)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.16e-174)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.2e+157)
		tmp = (y / x) / x;
	elseif (x <= -1.16e-174)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.2e+157], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.16e-174], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1999999999999999e157

   1. Initial program 50.5%

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

      1. times-frac 82.0%

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

      2. /-rgt-identity 82.0%

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

      3. associate-/l/ 82.0%

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

      4. *-lft-identity 82.0%

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

      5. associate-+l+ 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x around inf 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

      1. associate-/r* 90.5%

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

      2. div-inv 90.4%

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

   8. Applied egg-rr 90.4%

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

      1. un-div-inv 90.5%

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

   10. Applied egg-rr 90.5%

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

   if -3.1999999999999999e157 < x < -1.16e-174

   1. Initial program 71.8%

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

      1. times-frac 96.0%

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

      2. /-rgt-identity 96.0%

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

      3. associate-/l/ 96.0%

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

      4. *-lft-identity 96.0%

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

      5. associate-+l+ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 56.9%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]

Alternative 15: 59.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.8e+156)
   (/ (/ y x) x)
   (if (<= x -1.16e-174) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+156) {
		tmp = (y / x) / x;
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.8d+156)) then
        tmp = (y / x) / x
    else if (x <= (-1.16d-174)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.8e+156) {
		tmp = (y / x) / x;
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.8e+156:
		tmp = (y / x) / x
	elif x <= -1.16e-174:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.8e+156)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.16e-174)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.8e+156)
		tmp = (y / x) / x;
	elseif (x <= -1.16e-174)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.8e+156], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.16e-174], 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 < -5.80000000000000021e156

   1. Initial program 50.5%

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

      1. times-frac 82.0%

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

      2. /-rgt-identity 82.0%

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

      3. associate-/l/ 82.0%

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

      4. *-lft-identity 82.0%

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

      5. associate-+l+ 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in x around inf 82.0%

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

      1. unpow2 82.0%

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

   6. Simplified 82.0%

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

      1. associate-/r* 90.5%

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

      2. div-inv 90.4%

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

   8. Applied egg-rr 90.4%

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

      1. un-div-inv 90.5%

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

   10. Applied egg-rr 90.5%

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

   if -5.80000000000000021e156 < x < -1.16e-174

   1. Initial program 71.8%

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

      1. times-frac 96.0%

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

      2. /-rgt-identity 96.0%

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

      3. associate-/l/ 96.0%

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

      4. *-lft-identity 96.0%

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

      5. associate-+l+ 96.0%

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

   3. Simplified 96.0%

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

   4. Taylor expanded in y around 0 56.9%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   6. Simplified 60.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 16: 59.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+14)
   (/ (/ y x) (+ y x))
   (if (<= x -1.16e-174) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+14) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.5d+14)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1.16d-174)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+14) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.16e-174) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e+14:
		tmp = (y / x) / (y + x)
	elif x <= -1.16e-174:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+14)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1.16e-174)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+14)
		tmp = (y / x) / (y + x);
	elseif (x <= -1.16e-174)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e+14], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.16e-174], 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 < -5.5e14

   1. Initial program 52.0%

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

      1. 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. /-rgt-identity 86.9%

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

      3. associate-/l/ 86.9%

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

      4. *-lft-identity 86.9%

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

      5. 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. Step-by-step derivation

      1. *-commutative 86.9%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around inf 77.4%

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

   if -5.5e14 < x < -1.16e-174

   1. Initial program 84.6%

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

      1. times-frac 99.8%

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

      2. /-rgt-identity 99.8%

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

      3. associate-/l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. associate-+l+ 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 47.8%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   6. Simplified 60.0%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 17: 59.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.16e-174) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.16e-174) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.16d-174)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.16e-174) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.16e-174:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.16e-174)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.16e-174)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.16e-174], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e-174

   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 92.7%

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

      2. /-rgt-identity 92.7%

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

      3. associate-/l/ 92.7%

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

      4. *-lft-identity 92.7%

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

      5. associate-+l+ 92.7%

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

   3. Simplified 92.7%

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

      1. *-commutative 92.7%

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

      2. associate-/r* 99.7%

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

      3. associate-*r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 64.0%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   6. Simplified 60.0%

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

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 18: 59.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.16e-174) (/ (/ y (+ x 1.0)) x) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.16e-174) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.16d-174)) then
        tmp = (y / (x + 1.0d0)) / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.16e-174) {
		tmp = (y / (x + 1.0)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.16e-174:
		tmp = (y / (x + 1.0)) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.16e-174)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / 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 <= -1.16e-174)
		tmp = (y / (x + 1.0)) / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.16e-174], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.16e-174

   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 92.7%

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

      2. /-rgt-identity 92.7%

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

      3. associate-/l/ 92.7%

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

      4. *-lft-identity 92.7%

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

      5. associate-+l+ 92.7%

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

   3. Simplified 92.7%

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

   4. Taylor expanded in y around 0 62.7%

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

      1. associate-/r* 63.7%

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

      2. +-commutative 63.7%

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

   6. Simplified 63.7%

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

   if -1.16e-174 < x

   1. Initial program 69.7%

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

      1. times-frac 88.0%

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

      2. /-rgt-identity 88.0%

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

      3. associate-/l/ 88.0%

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

      4. *-lft-identity 88.0%

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

      5. associate-+l+ 88.0%

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

   3. Simplified 88.0%

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

   4. Taylor expanded in x around 0 59.2%

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

      1. associate-/r* 60.0%

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

      2. +-commutative 60.0%

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

   6. Simplified 60.0%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.16 \cdot 10^{-174}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]

Alternative 19: 23.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -8.6e-231) (/ y x) (- (/ x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8.6e-231) {
		tmp = y / x;
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.6d-231)) then
        tmp = y / x
    else
        tmp = (x / y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8.6e-231) {
		tmp = y / x;
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8.6e-231:
		tmp = y / x
	else:
		tmp = (x / y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8.6e-231)
		tmp = Float64(y / x);
	else
		tmp = Float64(Float64(x / y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.6e-231)
		tmp = y / x;
	else
		tmp = (x / y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8.6e-231], N[(y / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.59999999999999996e-231

   1. Initial program 66.5%

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

      1. times-frac 89.9%

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

      2. /-rgt-identity 89.9%

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

      3. associate-/l/ 89.9%

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

      4. *-lft-identity 89.9%

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

      5. associate-+l+ 89.9%

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

   3. Simplified 89.9%

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

   4. Taylor expanded in y around 0 58.5%

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

   5. Taylor expanded in x around 0 30.9%

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

   if -8.59999999999999996e-231 < 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. times-frac 89.8%

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

      2. /-rgt-identity 89.8%

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

      3. associate-/l/ 89.8%

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

      4. *-lft-identity 89.8%

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

      5. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   4. Taylor expanded in x around 0 57.5%

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

      1. associate-/r* 58.3%

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

      2. +-commutative 58.3%

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

   6. Simplified 58.3%

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

   7. Taylor expanded in y around 0 20.7%

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

      1. neg-mul-1 20.7%

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

      2. unsub-neg 20.7%

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

   9. Simplified 20.7%

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

4. Final simplification 25.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-231}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \]

Alternative 20: 26.9% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.6%

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

   1. times-frac 89.8%

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

   2. /-rgt-identity 89.8%

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

   3. associate-/l/ 89.8%

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

   4. *-lft-identity 89.8%

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

   5. associate-+l+ 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in y around 0 49.8%

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

5. Taylor expanded in x around 0 26.1%

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

6. Final simplification 26.1%

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

Alternative 21: 3.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- x))

C

double code(double x, double y) {
	return -x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -x
end function

Java

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

Python

def code(x, y):
	return -x

Julia

function code(x, y)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y)
	tmp = -x;
end

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 68.6%

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

   1. times-frac 89.8%

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

   2. /-rgt-identity 89.8%

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

   3. associate-/l/ 89.8%

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

   4. *-lft-identity 89.8%

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

   5. associate-+l+ 89.8%

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

3. Simplified 89.8%

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

4. Taylor expanded in x around 0 50.2%

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

   1. associate-/r* 51.0%

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

   2. +-commutative 51.0%

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

6. Simplified 51.0%

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

7. Taylor expanded in y around 0 15.8%

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

   1. neg-mul-1 15.8%

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

   2. unsub-neg 15.8%

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

9. Simplified 15.8%

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

10. Taylor expanded in y around inf 3.5%

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

    1. neg-mul-1 3.5%

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

12. Simplified 3.5%

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

13. Final simplification 3.5%

    \[\leadsto -x \]

Alternative 22: 3.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (- y))

C

double code(double x, double y) {
	return -y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -y
end function

Java

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

Python

def code(x, y):
	return -y

Julia

function code(x, y)
	return Float64(-y)
end

MATLAB

function tmp = code(x, y)
	tmp = -y;
end

Wolfram

code[x_, y_] := (-y)

Derivation

1. Initial program 68.6%

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

   1. times-frac 89.8%

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

   2. /-rgt-identity 89.8%

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

   3. associate-/l/ 89.8%

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

   4. *-lft-identity 89.8%

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

   5. associate-+l+ 89.8%

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

3. Simplified 89.8%

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

   1. *-commutative 89.8%

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

   2. associate-/r* 99.7%

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

   3. associate-*r/ 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in y around 0 50.8%

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

7. Taylor expanded in x around 0 16.1%

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

   1. neg-mul-1 16.1%

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

   2. unsub-neg 16.1%

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

   3. *-commutative 16.1%

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

9. Simplified 16.1%

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

10. Taylor expanded in x around inf 3.5%

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

    1. neg-mul-1 3.5%

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

12. Simplified 3.5%

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

13. Final simplification 3.5%

    \[\leadsto -y \]

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 2023185 
(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))))