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

Percentage Accurate: 68.5% → 99.8%

Time: 15.0s

Alternatives: 23

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-*r/ 83.5%

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

   2. *-commutative 83.5%

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

   3. distribute-rgt1-in 63.2%

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

   4. fma-def 83.5%

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

   5. cube-unmult 83.5%

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

3. Simplified 83.5%

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

   1. associate-*r/ 67.9%

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

   2. fma-udef 52.9%

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

   3. cube-mult 52.9%

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

   4. distribute-rgt1-in 67.8%

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

   5. associate-+r+ 67.8%

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

   6. *-commutative 67.8%

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

   7. frac-times 87.5%

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

   8. *-commutative 87.5%

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

   9. associate-/r* 99.7%

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

   10. associate-*r/ 99.8%

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

   11. +-commutative 99.8%

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

   12. associate-+l+ 99.8%

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

   13. +-commutative 99.8%

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

   14. +-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{t_0}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 5e-125)
     (/ (/ t_0 (/ (+ 1.0 x) y)) (+ y x))
     (if (<= y 3.6e+141)
       (* (/ x (* (+ y x) (+ y x))) (/ y (+ x (+ y 1.0))))
       (/ (* t_0 (/ y (+ y 1.0))) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 5e-125) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else if (y <= 3.6e+141) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (t_0 * (y / (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 + x)
    if (y <= 5d-125) then
        tmp = (t_0 / ((1.0d0 + x) / y)) / (y + x)
    else if (y <= 3.6d+141) then
        tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0d0)))
    else
        tmp = (t_0 * (y / (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 + x);
	double tmp;
	if (y <= 5e-125) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else if (y <= 3.6e+141) {
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	} else {
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 5e-125:
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x)
	elif y <= 3.6e+141:
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)))
	else:
		tmp = (t_0 * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 5e-125)
		tmp = Float64(Float64(t_0 / Float64(Float64(1.0 + x) / y)) / Float64(y + x));
	elseif (y <= 3.6e+141)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(t_0 * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 5e-125)
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	elseif (y <= 3.6e+141)
		tmp = (x / ((y + x) * (y + x))) * (y / (x + (y + 1.0)));
	else
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 5e-125], N[(N[(t$95$0 / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+141], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.99999999999999967e-125

   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. associate-*r/ 83.1%

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

      2. *-commutative 83.1%

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

      3. distribute-rgt1-in 59.0%

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

      4. fma-def 83.1%

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

      5. cube-unmult 83.1%

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

   3. Simplified 83.1%

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

      1. associate-*r/ 68.4%

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

      2. fma-udef 48.8%

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

      3. cube-mult 48.7%

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

      4. distribute-rgt1-in 68.4%

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

      5. associate-+r+ 68.4%

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

      6. *-commutative 68.4%

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

      7. frac-times 85.8%

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

      8. *-commutative 85.8%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   9. Simplified 99.7%

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

       1. clear-num 99.8%

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

       2. div-inv 99.7%

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

       3. *-un-lft-identity 99.7%

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

       4. times-frac 99.8%

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

       5. /-rgt-identity 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.7%

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

       2. associate-*r/ 99.8%

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

       3. *-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in y around 0 79.4%

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

   if 4.99999999999999967e-125 < y < 3.6000000000000001e141

   1. Initial program 75.8%

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

      1. times-frac 97.2%

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

      2. /-rgt-identity 97.2%

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

      3. associate-/l/ 97.2%

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

      4. *-lft-identity 97.2%

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

      5. associate-+l+ 97.2%

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

   3. Simplified 97.2%

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

   if 3.6000000000000001e141 < y

   1. Initial program 42.5%

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

      1. associate-*r/ 73.0%

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

      2. *-commutative 73.0%

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

      3. distribute-rgt1-in 68.8%

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

      4. fma-def 73.0%

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

      5. cube-unmult 73.0%

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

   3. Simplified 73.0%

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

      1. associate-*r/ 42.5%

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

      2. fma-udef 42.5%

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

      3. cube-mult 42.5%

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

      4. distribute-rgt1-in 42.5%

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

      5. associate-+r+ 42.5%

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

      6. *-commutative 42.5%

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

      7. frac-times 73.0%

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

      8. *-commutative 73.0%

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

      9. associate-/r* 100.0%

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

      10. associate-*r/ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate-+l+ 100.0%

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

      13. +-commutative 100.0%

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

      14. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{\frac{x}{y + x}}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 3: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;y \leq 1.55 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{t_0}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= y 1.55e-83)
     (/ (/ t_0 (/ (+ 1.0 x) y)) (+ y x))
     (if (<= y 3.6e+141)
       (/ x (* (+ y x) (/ (* (+ y (+ 1.0 x)) (+ y x)) y)))
       (/ (* t_0 (/ y (+ y 1.0))) (+ y x))))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (y <= 1.55e-83) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else if (y <= 3.6e+141) {
		tmp = x / ((y + x) * (((y + (1.0 + x)) * (y + x)) / y));
	} else {
		tmp = (t_0 * (y / (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 + x)
    if (y <= 1.55d-83) then
        tmp = (t_0 / ((1.0d0 + x) / y)) / (y + x)
    else if (y <= 3.6d+141) then
        tmp = x / ((y + x) * (((y + (1.0d0 + x)) * (y + x)) / y))
    else
        tmp = (t_0 * (y / (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 + x);
	double tmp;
	if (y <= 1.55e-83) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else if (y <= 3.6e+141) {
		tmp = x / ((y + x) * (((y + (1.0 + x)) * (y + x)) / y));
	} else {
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if y <= 1.55e-83:
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x)
	elif y <= 3.6e+141:
		tmp = x / ((y + x) * (((y + (1.0 + x)) * (y + x)) / y))
	else:
		tmp = (t_0 * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (y <= 1.55e-83)
		tmp = Float64(Float64(t_0 / Float64(Float64(1.0 + x) / y)) / Float64(y + x));
	elseif (y <= 3.6e+141)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(Float64(y + Float64(1.0 + x)) * Float64(y + x)) / y)));
	else
		tmp = Float64(Float64(t_0 * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (y <= 1.55e-83)
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	elseif (y <= 3.6e+141)
		tmp = x / ((y + x) * (((y + (1.0 + x)) * (y + x)) / y));
	else
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.55e-83], N[(N[(t$95$0 / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.6e+141], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.54999999999999996e-83

   1. Initial program 69.6%

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

      1. associate-*r/ 83.5%

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

      2. *-commutative 83.5%

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

      3. distribute-rgt1-in 60.2%

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

      4. fma-def 83.5%

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

      5. cube-unmult 83.5%

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

   3. Simplified 83.5%

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

      1. associate-*r/ 69.7%

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

      2. fma-udef 50.5%

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

      3. cube-mult 50.5%

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

      4. distribute-rgt1-in 69.6%

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

      5. associate-+r+ 69.6%

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

      6. *-commutative 69.6%

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

      7. frac-times 86.6%

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

      8. *-commutative 86.6%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   9. Simplified 99.7%

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

       1. clear-num 99.8%

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

       2. div-inv 99.7%

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

       3. *-un-lft-identity 99.7%

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

       4. times-frac 99.8%

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

       5. /-rgt-identity 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. associate-*r/ 99.7%

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

       2. associate-*r/ 99.8%

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

       3. *-rgt-identity 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in y around 0 80.5%

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

   if 1.54999999999999996e-83 < y < 3.6000000000000001e141

   1. Initial program 73.1%

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

      1. associate-*r/ 88.0%

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

      2. *-commutative 88.0%

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

      3. distribute-rgt1-in 70.7%

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

      4. fma-def 88.0%

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

      5. cube-unmult 88.1%

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

   3. Simplified 88.1%

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

      1. associate-*r/ 73.2%

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

      2. fma-udef 65.2%

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

      3. cube-mult 65.2%

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

      4. distribute-rgt1-in 73.1%

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

      5. associate-+r+ 73.1%

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

      6. *-commutative 73.1%

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

      7. frac-times 96.7%

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

      8. *-commutative 96.7%

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

      9. clear-num 96.7%

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

      10. associate-/r* 99.6%

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

      11. frac-times 98.3%

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

      12. *-un-lft-identity 98.3%

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

      13. +-commutative 98.3%

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

      14. +-commutative 98.3%

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

      15. associate-+l+ 98.3%

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

      16. +-commutative 98.3%

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

   5. Applied egg-rr 98.3%

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

      1. associate-/l/ 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*r/ 95.1%

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

      4. +-commutative 95.1%

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

   7. Simplified 95.1%

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

   if 3.6000000000000001e141 < y

   1. Initial program 42.5%

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

      1. associate-*r/ 73.0%

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

      2. *-commutative 73.0%

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

      3. distribute-rgt1-in 68.8%

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

      4. fma-def 73.0%

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

      5. cube-unmult 73.0%

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

   3. Simplified 73.0%

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

      1. associate-*r/ 42.5%

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

      2. fma-udef 42.5%

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

      3. cube-mult 42.5%

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

      4. distribute-rgt1-in 42.5%

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

      5. associate-+r+ 42.5%

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

      6. *-commutative 42.5%

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

      7. frac-times 73.0%

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

      8. *-commutative 73.0%

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

      9. associate-/r* 100.0%

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

      10. associate-*r/ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate-+l+ 100.0%

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

      13. +-commutative 100.0%

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

      14. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around 0 81.2%

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

      1. +-commutative 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\frac{x}{y + x}}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{\left(y + \left(1 + x\right)\right) \cdot \left(y + x\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 4: 66.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-178)
   (/ (/ y (+ y x)) (+ 1.0 x))
   (if (<= y 8.2e-9)
     (* (/ x (* (+ y x) (+ y x))) (/ y (+ 1.0 x)))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-178) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (y <= 8.2e-9) {
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + 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 (y <= 8d-178) then
        tmp = (y / (y + x)) / (1.0d0 + x)
    else if (y <= 8.2d-9) then
        tmp = (x / ((y + x) * (y + x))) * (y / (1.0d0 + 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 (y <= 8e-178) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (y <= 8.2e-9) {
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + x));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-178:
		tmp = (y / (y + x)) / (1.0 + x)
	elif y <= 8.2e-9:
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + x))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-178)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(1.0 + x));
	elseif (y <= 8.2e-9)
		tmp = Float64(Float64(x / Float64(Float64(y + x) * Float64(y + x))) * Float64(y / Float64(1.0 + 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 (y <= 8e-178)
		tmp = (y / (y + x)) / (1.0 + x);
	elseif (y <= 8.2e-9)
		tmp = (x / ((y + x) * (y + x))) * (y / (1.0 + x));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-178], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-9], N[(N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + 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 y < 7.9999999999999996e-178

   1. Initial program 66.2%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. distribute-rgt1-in 57.4%

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

      4. fma-def 82.0%

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

      5. cube-unmult 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 66.2%

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

      2. fma-udef 46.5%

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

      3. cube-mult 46.4%

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

      4. distribute-rgt1-in 66.2%

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

      5. associate-+r+ 66.2%

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

      6. *-commutative 66.2%

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

      7. frac-times 84.8%

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

      8. *-commutative 84.8%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 54.2%

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

      1. expm1-log1p-u 54.2%

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

      2. expm1-udef 32.2%

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

      3. associate-/l/ 32.2%

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

      4. +-commutative 32.2%

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

      5. +-commutative 32.2%

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

   8. Applied egg-rr 32.2%

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

      1. expm1-def 56.1%

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

      2. expm1-log1p 56.1%

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

      3. associate-/r* 54.2%

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

   10. Simplified 54.2%

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

   if 7.9999999999999996e-178 < y < 8.2000000000000006e-9

   1. Initial program 91.1%

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

      1. times-frac 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 99.8%

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

      1. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 8.2000000000000006e-9 < y

   1. Initial program 60.5%

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

      1. associate-*r/ 82.0%

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

      2. *-commutative 82.0%

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

      3. distribute-rgt1-in 71.1%

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

      4. fma-def 82.1%

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

      5. cube-unmult 82.1%

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

   3. Simplified 82.1%

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

      1. associate-*r/ 60.6%

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

      2. fma-udef 58.6%

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

      3. cube-mult 58.6%

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

      4. distribute-rgt1-in 60.5%

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

      5. associate-+r+ 60.5%

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

      6. *-commutative 60.5%

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

      7. frac-times 87.7%

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

      8. *-commutative 87.7%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 74.4%

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

      1. +-commutative 74.4%

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

   8. Simplified 74.4%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(y + x\right)} \cdot \frac{y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 5: 81.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6:\\ \;\;\;\;\frac{\frac{1}{\frac{y + \left(1 + x\right)}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6)
   (/ (/ 1.0 (/ (+ y (+ 1.0 x)) y)) (+ y x))
   (/ (* (/ x (+ y x)) (/ y (+ y 1.0))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6) {
		tmp = (1.0 / ((y + (1.0 + x)) / y)) / (y + x);
	} else {
		tmp = ((x / (y + x)) * (y / (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 <= (-5.6d0)) then
        tmp = (1.0d0 / ((y + (1.0d0 + x)) / y)) / (y + x)
    else
        tmp = ((x / (y + x)) * (y / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6], N[(N[(1.0 / N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999996

   1. Initial program 62.7%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 33.9%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.6%

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

   3. Simplified 82.6%

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

      1. associate-*r/ 62.7%

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

      2. fma-udef 33.6%

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

      3. cube-mult 33.6%

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

      4. distribute-rgt1-in 62.7%

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

      5. associate-+r+ 62.7%

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

      6. *-commutative 62.7%

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

      7. frac-times 87.2%

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

      8. *-commutative 87.2%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 76.6%

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

   if -5.5999999999999996 < x

   1. Initial program 69.5%

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

      1. associate-*r/ 83.8%

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

      2. *-commutative 83.8%

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

      3. distribute-rgt1-in 72.8%

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

      4. fma-def 83.8%

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

      5. cube-unmult 83.8%

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

   3. Simplified 83.8%

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

      1. associate-*r/ 69.5%

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

      2. fma-udef 59.2%

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

      3. cube-mult 59.1%

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

      4. distribute-rgt1-in 69.5%

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

      5. associate-+r+ 69.5%

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

      6. *-commutative 69.5%

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

      7. frac-times 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in x around 0 86.4%

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

      1. +-commutative 86.4%

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

   8. Simplified 86.4%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6:\\ \;\;\;\;\frac{\frac{1}{\frac{y + \left(1 + x\right)}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 6: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + x}\\ \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;\frac{\frac{t_0}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0 \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y x))))
   (if (<= x -0.35)
     (/ (/ t_0 (/ (+ 1.0 x) y)) (+ y x))
     (/ (* t_0 (/ y (+ y 1.0))) (+ y x)))))

C

double code(double x, double y) {
	double t_0 = x / (y + x);
	double tmp;
	if (x <= -0.35) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else {
		tmp = (t_0 * (y / (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 + x)
    if (x <= (-0.35d0)) then
        tmp = (t_0 / ((1.0d0 + x) / y)) / (y + x)
    else
        tmp = (t_0 * (y / (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 + x);
	double tmp;
	if (x <= -0.35) {
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	} else {
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + x)
	tmp = 0
	if x <= -0.35:
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x)
	else:
		tmp = (t_0 * (y / (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + x))
	tmp = 0.0
	if (x <= -0.35)
		tmp = Float64(Float64(t_0 / Float64(Float64(1.0 + x) / y)) / Float64(y + x));
	else
		tmp = Float64(Float64(t_0 * Float64(y / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + x);
	tmp = 0.0;
	if (x <= -0.35)
		tmp = (t_0 / ((1.0 + x) / y)) / (y + x);
	else
		tmp = (t_0 * (y / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.35], N[(N[(t$95$0 / N[(N[(1.0 + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.34999999999999998

   1. Initial program 63.3%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-rgt1-in 34.9%

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

      4. fma-def 82.8%

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

      5. cube-unmult 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 63.3%

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

      2. fma-udef 34.6%

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

      3. cube-mult 34.6%

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

      4. distribute-rgt1-in 63.3%

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

      5. associate-+r+ 63.3%

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

      6. *-commutative 63.3%

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

      7. frac-times 87.4%

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

      8. *-commutative 87.4%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. clear-num 99.7%

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

      2. clear-num 99.7%

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

      3. frac-times 99.7%

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

      4. metadata-eval 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. associate-/r* 99.7%

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

   9. Simplified 99.7%

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

       1. clear-num 99.7%

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

       2. div-inv 99.7%

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

       3. *-un-lft-identity 99.7%

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

       4. times-frac 99.7%

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

       5. /-rgt-identity 99.7%

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

   11. Applied egg-rr 99.7%

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

       1. associate-*r/ 99.7%

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

       2. associate-*r/ 99.7%

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

       3. *-rgt-identity 99.7%

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

   13. Simplified 99.7%

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

   14. Taylor expanded in y around 0 75.6%

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

   if -0.34999999999999998 < x

   1. Initial program 69.4%

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

      1. associate-*r/ 83.7%

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

      2. *-commutative 83.7%

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

      3. distribute-rgt1-in 72.7%

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

      4. fma-def 83.7%

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

      5. cube-unmult 83.7%

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

   3. Simplified 83.7%

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

      1. associate-*r/ 69.4%

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

      2. fma-udef 59.0%

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

      3. cube-mult 58.9%

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

      4. distribute-rgt1-in 69.4%

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

      5. associate-+r+ 69.4%

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

      6. *-commutative 69.4%

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

      7. frac-times 87.5%

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

      8. *-commutative 87.5%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 86.5%

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

      1. +-commutative 86.5%

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

   8. Simplified 86.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.35:\\ \;\;\;\;\frac{\frac{\frac{x}{y + x}}{\frac{1 + x}{y}}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{y + 1}}{y + x}\\ \end{array} \]

Alternative 7: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -1.2e-97)
     (- (/ y x) y)
     (if (or (<= x -1.85e-137) (not (<= x -6.6e-171)))
       (/ x (+ y (* y y)))
       (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.2e-97) {
		tmp = (y / x) - y;
	} else if ((x <= -1.85e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = 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 <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-1.2d-97)) then
        tmp = (y / x) - y
    else if ((x <= (-1.85d-137)) .or. (.not. (x <= (-6.6d-171)))) then
        tmp = x / (y + (y * y))
    else
        tmp = y / x
    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 <= -1.2e-97) {
		tmp = (y / x) - y;
	} else if ((x <= -1.85e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -1.2e-97:
		tmp = (y / x) - y
	elif (x <= -1.85e-137) or not (x <= -6.6e-171):
		tmp = x / (y + (y * y))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.2e-97)
		tmp = Float64(Float64(y / x) - y);
	elseif ((x <= -1.85e-137) || !(x <= -6.6e-171))
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -1.2e-97)
		tmp = (y / x) - y;
	elseif ((x <= -1.85e-137) || ~((x <= -6.6e-171)))
		tmp = x / (y + (y * y));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.2e-97], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], If[Or[LessEqual[x, -1.85e-137], N[Not[LessEqual[x, -6.6e-171]], $MachinePrecision]], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1

   1. Initial program 63.3%

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

      1. times-frac 87.4%

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

      2. /-rgt-identity 87.4%

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

      3. associate-/l/ 87.4%

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

      4. *-lft-identity 87.4%

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

      5. associate-+l+ 87.4%

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

   3. Simplified 87.4%

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

   4. Taylor expanded in x around inf 78.4%

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

   5. Taylor expanded in x around inf 71.1%

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

      1. associate-*l/ 71.1%

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

      2. *-un-lft-identity 71.1%

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

   7. Applied egg-rr 71.1%

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

   if -1 < x < -1.2e-97

   1. Initial program 92.3%

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

      1. times-frac 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 y around 0 49.7%

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

   5. Taylor expanded in x around 0 49.7%

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

      1. neg-mul-1 49.7%

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

      2. unsub-neg 49.7%

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

   7. Simplified 49.7%

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

   if -1.2e-97 < x < -1.85e-137 or -6.6000000000000004e-171 < x

   1. Initial program 67.7%

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

      1. times-frac 87.1%

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

      2. /-rgt-identity 87.1%

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

      3. associate-/l/ 87.1%

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

      4. *-lft-identity 87.1%

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

      5. associate-+l+ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 61.6%

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

      1. distribute-rgt-in 61.6%

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

      2. *-lft-identity 61.6%

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

   6. Simplified 61.6%

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

   if -1.85e-137 < x < -6.6000000000000004e-171

   1. Initial program 67.0%

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

      1. times-frac 76.6%

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

      2. /-rgt-identity 76.6%

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

      3. associate-/l/ 76.6%

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

      4. *-lft-identity 76.6%

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

      5. associate-+l+ 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 46.0%

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

   5. Taylor expanded in x around 0 46.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 8: 59.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-136} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.8e+156)
   (/ (/ y x) x)
   (if (<= x -1.2e-97)
     (/ y (* x (+ 1.0 x)))
     (if (or (<= x -8e-136) (not (<= x -6.6e-171)))
       (/ x (+ y (* y y)))
       (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+156) {
		tmp = (y / x) / x;
	} else if (x <= -1.2e-97) {
		tmp = y / (x * (1.0 + x));
	} else if ((x <= -8e-136) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.8d+156)) then
        tmp = (y / x) / x
    else if (x <= (-1.2d-97)) then
        tmp = y / (x * (1.0d0 + x))
    else if ((x <= (-8d-136)) .or. (.not. (x <= (-6.6d-171)))) then
        tmp = x / (y + (y * y))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.8e+156) {
		tmp = (y / x) / x;
	} else if (x <= -1.2e-97) {
		tmp = y / (x * (1.0 + x));
	} else if ((x <= -8e-136) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.8e+156:
		tmp = (y / x) / x
	elif x <= -1.2e-97:
		tmp = y / (x * (1.0 + x))
	elif (x <= -8e-136) or not (x <= -6.6e-171):
		tmp = x / (y + (y * y))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.8e+156)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.2e-97)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	elseif ((x <= -8e-136) || !(x <= -6.6e-171))
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.8e+156)
		tmp = (y / x) / x;
	elseif (x <= -1.2e-97)
		tmp = y / (x * (1.0 + x));
	elseif ((x <= -8e-136) || ~((x <= -6.6e-171)))
		tmp = x / (y + (y * y));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.8e+156], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.2e-97], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -8e-136], N[Not[LessEqual[x, -6.6e-171]], $MachinePrecision]], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.8000000000000002e156

   1. Initial program 58.3%

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

      1. times-frac 87.7%

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

      2. /-rgt-identity 87.7%

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

      3. associate-/l/ 87.7%

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

      4. *-lft-identity 87.7%

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

      5. associate-+l+ 87.7%

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

   3. Simplified 87.7%

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

   4. Taylor expanded in x around inf 87.7%

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

   5. Taylor expanded in x around inf 96.6%

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

      1. associate-*l/ 96.6%

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

      2. *-un-lft-identity 96.6%

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

   7. Applied egg-rr 96.6%

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

   if -4.8000000000000002e156 < x < -1.2e-97

   1. Initial program 73.8%

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

      1. times-frac 90.5%

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

      2. /-rgt-identity 90.5%

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

      3. associate-/l/ 90.5%

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

      4. *-lft-identity 90.5%

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

      5. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 59.8%

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

   if -1.2e-97 < x < -8.00000000000000001e-136 or -6.6000000000000004e-171 < x

   1. Initial program 67.5%

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

      1. times-frac 87.1%

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

      2. /-rgt-identity 87.1%

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

      3. associate-/l/ 87.1%

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

      4. *-lft-identity 87.1%

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

      5. associate-+l+ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 61.4%

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

      1. distribute-rgt-in 61.4%

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

      2. *-lft-identity 61.4%

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

   6. Simplified 61.4%

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

   if -8.00000000000000001e-136 < x < -6.6000000000000004e-171

   1. Initial program 70.3%

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

      1. times-frac 79.0%

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

      2. /-rgt-identity 79.0%

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

      3. associate-/l/ 79.0%

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

      4. *-lft-identity 79.0%

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

      5. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in y around 0 41.5%

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

   5. Taylor expanded in x around 0 41.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-136} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 9: 59.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.7e+157)
   (/ (/ y x) (+ y x))
   (if (<= x -1.2e-97)
     (/ y (* x (+ 1.0 x)))
     (if (or (<= x -4.6e-137) (not (<= x -6.6e-171)))
       (/ x (+ y (* y y)))
       (/ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+157) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.2e-97) {
		tmp = y / (x * (1.0 + x));
	} else if ((x <= -4.6e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.7d+157)) then
        tmp = (y / x) / (y + x)
    else if (x <= (-1.2d-97)) then
        tmp = y / (x * (1.0d0 + x))
    else if ((x <= (-4.6d-137)) .or. (.not. (x <= (-6.6d-171)))) then
        tmp = x / (y + (y * y))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.7e+157) {
		tmp = (y / x) / (y + x);
	} else if (x <= -1.2e-97) {
		tmp = y / (x * (1.0 + x));
	} else if ((x <= -4.6e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.7e+157:
		tmp = (y / x) / (y + x)
	elif x <= -1.2e-97:
		tmp = y / (x * (1.0 + x))
	elif (x <= -4.6e-137) or not (x <= -6.6e-171):
		tmp = x / (y + (y * y))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.7e+157)
		tmp = Float64(Float64(y / x) / Float64(y + x));
	elseif (x <= -1.2e-97)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	elseif ((x <= -4.6e-137) || !(x <= -6.6e-171))
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.7e+157)
		tmp = (y / x) / (y + x);
	elseif (x <= -1.2e-97)
		tmp = y / (x * (1.0 + x));
	elseif ((x <= -4.6e-137) || ~((x <= -6.6e-171)))
		tmp = x / (y + (y * y));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.7e+157], N[(N[(y / x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.2e-97], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.6e-137], N[Not[LessEqual[x, -6.6e-171]], $MachinePrecision]], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -3.6999999999999999e157

   1. Initial program 58.3%

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

      1. associate-*r/ 87.7%

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

      2. *-commutative 87.7%

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

      3. distribute-rgt1-in 0.0%

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

      4. fma-def 87.7%

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

      5. cube-unmult 87.7%

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

   3. Simplified 87.7%

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

      1. associate-*r/ 58.3%

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

      2. fma-udef 0.0%

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

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 58.3%

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

      5. associate-+r+ 58.3%

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

      6. *-commutative 58.3%

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

      7. frac-times 87.7%

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

      8. *-commutative 87.7%

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

      9. associate-/r* 99.9%

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

      10. associate-*r/ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate-+l+ 100.0%

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

      13. +-commutative 100.0%

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

      14. +-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in x around inf 96.7%

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

   if -3.6999999999999999e157 < x < -1.2e-97

   1. Initial program 73.8%

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

      1. times-frac 90.5%

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

      2. /-rgt-identity 90.5%

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

      3. associate-/l/ 90.5%

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

      4. *-lft-identity 90.5%

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

      5. associate-+l+ 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around 0 59.8%

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

   if -1.2e-97 < x < -4.60000000000000016e-137 or -6.6000000000000004e-171 < x

   1. Initial program 67.5%

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

      1. times-frac 87.1%

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

      2. /-rgt-identity 87.1%

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

      3. associate-/l/ 87.1%

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

      4. *-lft-identity 87.1%

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

      5. associate-+l+ 87.1%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in x around 0 61.4%

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

      1. distribute-rgt-in 61.4%

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

      2. *-lft-identity 61.4%

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

   6. Simplified 61.4%

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

   if -4.60000000000000016e-137 < x < -6.6000000000000004e-171

   1. Initial program 70.3%

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

      1. times-frac 79.0%

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

      2. /-rgt-identity 79.0%

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

      3. associate-/l/ 79.0%

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

      4. *-lft-identity 79.0%

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

      5. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in y around 0 41.5%

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

   5. Taylor expanded in x around 0 41.5%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.7 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{x}}{y + x}\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 10: 60.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{1 + x}}}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -9.2e-98)
   (/ 1.0 (/ x (/ y (+ 1.0 x))))
   (if (<= x -3.5e-134)
     (/ x (+ y (* y y)))
     (if (<= x -6.6e-171) (/ y x) (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -9.2e-98) {
		tmp = 1.0 / (x / (y / (1.0 + x)));
	} else if (x <= -3.5e-134) {
		tmp = x / (y + (y * y));
	} else if (x <= -6.6e-171) {
		tmp = 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 <= (-9.2d-98)) then
        tmp = 1.0d0 / (x / (y / (1.0d0 + x)))
    else if (x <= (-3.5d-134)) then
        tmp = x / (y + (y * y))
    else if (x <= (-6.6d-171)) then
        tmp = 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 <= -9.2e-98) {
		tmp = 1.0 / (x / (y / (1.0 + x)));
	} else if (x <= -3.5e-134) {
		tmp = x / (y + (y * y));
	} else if (x <= -6.6e-171) {
		tmp = y / x;
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -9.2e-98:
		tmp = 1.0 / (x / (y / (1.0 + x)))
	elif x <= -3.5e-134:
		tmp = x / (y + (y * y))
	elif x <= -6.6e-171:
		tmp = y / x
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -9.2e-98)
		tmp = Float64(1.0 / Float64(x / Float64(y / Float64(1.0 + x))));
	elseif (x <= -3.5e-134)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	elseif (x <= -6.6e-171)
		tmp = 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 <= -9.2e-98)
		tmp = 1.0 / (x / (y / (1.0 + x)));
	elseif (x <= -3.5e-134)
		tmp = x / (y + (y * y));
	elseif (x <= -6.6e-171)
		tmp = y / x;
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -9.2e-98], N[(1.0 / N[(x / N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.5e-134], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-171], N[(y / x), $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 < -9.20000000000000002e-98

   1. Initial program 68.2%

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

      1. times-frac 89.5%

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

      2. /-rgt-identity 89.5%

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

      3. associate-/l/ 89.5%

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

      4. *-lft-identity 89.5%

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

      5. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in y around 0 69.9%

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

      1. clear-num 69.9%

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

      2. inv-pow 69.9%

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

      3. *-commutative 69.9%

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

   6. Applied egg-rr 69.9%

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

      1. unpow-1 69.9%

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

      2. associate-/l* 69.7%

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

   8. Simplified 69.7%

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

   if -9.20000000000000002e-98 < x < -3.4999999999999998e-134

   1. Initial program 99.6%

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

      1. 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 88.3%

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

      1. distribute-rgt-in 88.3%

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

      2. *-lft-identity 88.3%

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

   6. Simplified 88.3%

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

   if -3.4999999999999998e-134 < x < -6.6000000000000004e-171

   1. Initial program 70.3%

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

      1. times-frac 79.0%

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

      2. /-rgt-identity 79.0%

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

      3. associate-/l/ 79.0%

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

      4. *-lft-identity 79.0%

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

      5. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in y around 0 41.5%

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

   5. Taylor expanded in x around 0 41.5%

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

   if -6.6000000000000004e-171 < x

   1. Initial program 65.9%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 71.8%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 56.0%

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

      3. cube-mult 56.0%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. frac-times 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. +-commutative 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{1 + x}}}\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 11: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-97)
   (/ (/ y (+ 1.0 x)) (+ y x))
   (if (<= x -1.85e-137)
     (/ x (+ y (* y y)))
     (if (<= x -3.2e-171) (/ y x) (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-97) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (x <= -1.85e-137) {
		tmp = x / (y + (y * y));
	} else if (x <= -3.2e-171) {
		tmp = 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 <= (-1.05d-97)) then
        tmp = (y / (1.0d0 + x)) / (y + x)
    else if (x <= (-1.85d-137)) then
        tmp = x / (y + (y * y))
    else if (x <= (-3.2d-171)) then
        tmp = 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 <= -1.05e-97) {
		tmp = (y / (1.0 + x)) / (y + x);
	} else if (x <= -1.85e-137) {
		tmp = x / (y + (y * y));
	} else if (x <= -3.2e-171) {
		tmp = y / x;
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-97:
		tmp = (y / (1.0 + x)) / (y + x)
	elif x <= -1.85e-137:
		tmp = x / (y + (y * y))
	elif x <= -3.2e-171:
		tmp = y / x
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-97)
		tmp = Float64(Float64(y / Float64(1.0 + x)) / Float64(y + x));
	elseif (x <= -1.85e-137)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	elseif (x <= -3.2e-171)
		tmp = 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 <= -1.05e-97)
		tmp = (y / (1.0 + x)) / (y + x);
	elseif (x <= -1.85e-137)
		tmp = x / (y + (y * y));
	elseif (x <= -3.2e-171)
		tmp = y / x;
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-97], N[(N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.85e-137], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.2e-171], N[(y / x), $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 < -1.0500000000000001e-97

   1. Initial program 68.2%

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

      1. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

      3. distribute-rgt1-in 42.0%

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

      4. fma-def 84.4%

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

      5. cube-unmult 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 68.2%

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

      2. fma-udef 41.8%

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

      3. cube-mult 41.7%

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

      4. distribute-rgt1-in 68.2%

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

      5. associate-+r+ 68.2%

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

      6. *-commutative 68.2%

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

      7. frac-times 89.5%

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

      8. *-commutative 89.5%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 71.0%

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

   if -1.0500000000000001e-97 < x < -1.85e-137

   1. Initial program 99.7%

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

      1. times-frac 99.7%

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

      2. /-rgt-identity 99.7%

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

      3. associate-/l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. associate-+l+ 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around 0 89.6%

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

      1. distribute-rgt-in 89.6%

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

      2. *-lft-identity 89.6%

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

   6. Simplified 89.6%

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

   if -1.85e-137 < x < -3.2000000000000001e-171

   1. Initial program 67.0%

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

      1. times-frac 76.6%

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

      2. /-rgt-identity 76.6%

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

      3. associate-/l/ 76.6%

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

      4. *-lft-identity 76.6%

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

      5. associate-+l+ 76.6%

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

   3. Simplified 76.6%

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

   4. Taylor expanded in y around 0 46.0%

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

   5. Taylor expanded in x around 0 46.0%

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

   if -3.2000000000000001e-171 < x

   1. Initial program 65.9%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 71.8%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 56.0%

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

      3. cube-mult 56.0%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. frac-times 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. +-commutative 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{1 + x}}{y + x}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -3.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 12: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.2e-97)
   (/ (/ y (+ y x)) (+ 1.0 x))
   (if (<= x -4.6e-137)
     (/ x (+ y (* y y)))
     (if (<= x -6.6e-171) (/ y x) (/ (/ x (+ y 1.0)) (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.2e-97) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (x <= -4.6e-137) {
		tmp = x / (y + (y * y));
	} else if (x <= -6.6e-171) {
		tmp = 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 <= (-1.2d-97)) then
        tmp = (y / (y + x)) / (1.0d0 + x)
    else if (x <= (-4.6d-137)) then
        tmp = x / (y + (y * y))
    else if (x <= (-6.6d-171)) then
        tmp = 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 <= -1.2e-97) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (x <= -4.6e-137) {
		tmp = x / (y + (y * y));
	} else if (x <= -6.6e-171) {
		tmp = y / x;
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.2e-97:
		tmp = (y / (y + x)) / (1.0 + x)
	elif x <= -4.6e-137:
		tmp = x / (y + (y * y))
	elif x <= -6.6e-171:
		tmp = y / x
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.2e-97)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(1.0 + x));
	elseif (x <= -4.6e-137)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	elseif (x <= -6.6e-171)
		tmp = 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 <= -1.2e-97)
		tmp = (y / (y + x)) / (1.0 + x);
	elseif (x <= -4.6e-137)
		tmp = x / (y + (y * y));
	elseif (x <= -6.6e-171)
		tmp = y / x;
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.2e-97], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -4.6e-137], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-171], N[(y / x), $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 < -1.2e-97

   1. Initial program 68.2%

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

      1. associate-*r/ 84.4%

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

      2. *-commutative 84.4%

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

      3. distribute-rgt1-in 42.0%

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

      4. fma-def 84.4%

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

      5. cube-unmult 84.5%

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

   3. Simplified 84.5%

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

      1. associate-*r/ 68.2%

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

      2. fma-udef 41.8%

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

      3. cube-mult 41.7%

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

      4. distribute-rgt1-in 68.2%

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

      5. associate-+r+ 68.2%

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

      6. *-commutative 68.2%

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

      7. frac-times 89.5%

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

      8. *-commutative 89.5%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 71.0%

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

      1. expm1-log1p-u 71.0%

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

      2. expm1-udef 55.5%

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

      3. associate-/l/ 55.5%

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

      4. +-commutative 55.5%

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

      5. +-commutative 55.5%

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

   8. Applied egg-rr 55.5%

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

      1. expm1-def 74.1%

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

      2. expm1-log1p 74.1%

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

      3. associate-/r* 71.0%

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

   10. Simplified 71.0%

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

   if -1.2e-97 < x < -4.60000000000000016e-137

   1. Initial program 99.6%

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

      1. 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 88.3%

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

      1. distribute-rgt-in 88.3%

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

      2. *-lft-identity 88.3%

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

   6. Simplified 88.3%

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

   if -4.60000000000000016e-137 < x < -6.6000000000000004e-171

   1. Initial program 70.3%

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

      1. times-frac 79.0%

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

      2. /-rgt-identity 79.0%

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

      3. associate-/l/ 79.0%

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

      4. *-lft-identity 79.0%

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

      5. associate-+l+ 79.0%

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

   3. Simplified 79.0%

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

   4. Taylor expanded in y around 0 41.5%

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

   5. Taylor expanded in x around 0 41.5%

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

   if -6.6000000000000004e-171 < x

   1. Initial program 65.9%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 71.8%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 56.0%

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

      3. cube-mult 56.0%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. frac-times 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. +-commutative 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-97}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-137}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 13: 65.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.075:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.075)
   (/ (/ y (+ y x)) (+ 1.0 x))
   (if (<= x -6.6e-171)
     (* y (/ x (* (+ y x) (+ y x))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.075) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (x <= -6.6e-171) {
		tmp = y * (x / ((y + x) * (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 <= (-0.075d0)) then
        tmp = (y / (y + x)) / (1.0d0 + x)
    else if (x <= (-6.6d-171)) then
        tmp = y * (x / ((y + x) * (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 <= -0.075) {
		tmp = (y / (y + x)) / (1.0 + x);
	} else if (x <= -6.6e-171) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.075:
		tmp = (y / (y + x)) / (1.0 + x)
	elif x <= -6.6e-171:
		tmp = y * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.075)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(1.0 + x));
	elseif (x <= -6.6e-171)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * 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 <= -0.075)
		tmp = (y / (y + x)) / (1.0 + x);
	elseif (x <= -6.6e-171)
		tmp = y * (x / ((y + x) * (y + x)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.075], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-171], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $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 < -0.0749999999999999972

   1. Initial program 63.3%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

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

      3. distribute-rgt1-in 34.9%

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

      4. fma-def 82.8%

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

      5. cube-unmult 82.8%

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

   3. Simplified 82.8%

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

      1. associate-*r/ 63.3%

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

      2. fma-udef 34.6%

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

      3. cube-mult 34.6%

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

      4. distribute-rgt1-in 63.3%

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

      5. associate-+r+ 63.3%

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

      6. *-commutative 63.3%

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

      7. frac-times 87.4%

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

      8. *-commutative 87.4%

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

      9. associate-/r* 99.8%

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

      10. associate-*r/ 99.8%

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

      11. +-commutative 99.8%

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

      12. associate-+l+ 99.8%

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

      13. +-commutative 99.8%

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

      14. +-commutative 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in y around 0 75.3%

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

      1. expm1-log1p-u 75.3%

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

      2. expm1-udef 65.2%

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

      3. associate-/l/ 65.2%

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

      4. +-commutative 65.2%

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

      5. +-commutative 65.2%

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

   8. Applied egg-rr 65.2%

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

      1. expm1-def 79.1%

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

      2. expm1-log1p 79.1%

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

      3. associate-/r* 75.3%

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

   10. Simplified 75.3%

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

   if -0.0749999999999999972 < x < -6.6000000000000004e-171

   1. Initial program 87.1%

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

      1. times-frac 92.9%

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

      2. /-rgt-identity 92.9%

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

      3. associate-/l/ 92.9%

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

      4. *-lft-identity 92.9%

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

      5. associate-+l+ 92.9%

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

   3. Simplified 92.9%

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

   4. Taylor expanded in y around 0 72.7%

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

      1. +-commutative 72.7%

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

   6. Simplified 72.7%

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

   7. Taylor expanded in x around 0 72.4%

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

   if -6.6000000000000004e-171 < x

   1. Initial program 65.9%

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

      1. associate-*r/ 82.5%

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

      2. *-commutative 82.5%

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

      3. distribute-rgt1-in 71.8%

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

      4. fma-def 82.5%

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

      5. cube-unmult 82.5%

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

   3. Simplified 82.5%

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

      1. associate-*r/ 66.0%

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

      2. fma-udef 56.0%

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

      3. cube-mult 56.0%

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

      4. distribute-rgt1-in 65.9%

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

      5. associate-+r+ 65.9%

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

      6. *-commutative 65.9%

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

      7. frac-times 86.4%

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

      8. *-commutative 86.4%

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

      9. associate-/r* 99.7%

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

      10. associate-*r/ 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

      14. +-commutative 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 59.6%

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

      1. +-commutative 59.6%

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

   8. Simplified 59.6%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.075:\\ \;\;\;\;\frac{\frac{y}{y + x}}{1 + x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 14: 65.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{\frac{1}{\frac{y + \left(1 + x\right)}{y}}}{y + x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.11)
   (/ (/ 1.0 (/ (+ y (+ 1.0 x)) y)) (+ y x))
   (if (<= x -6.6e-171)
     (* y (/ x (* (+ y x) (+ y x))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.11) {
		tmp = (1.0 / ((y + (1.0 + x)) / y)) / (y + x);
	} else if (x <= -6.6e-171) {
		tmp = y * (x / ((y + x) * (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 <= (-0.11d0)) then
        tmp = (1.0d0 / ((y + (1.0d0 + x)) / y)) / (y + x)
    else if (x <= (-6.6d-171)) then
        tmp = y * (x / ((y + x) * (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 <= -0.11) {
		tmp = (1.0 / ((y + (1.0 + x)) / y)) / (y + x);
	} else if (x <= -6.6e-171) {
		tmp = y * (x / ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.11:
		tmp = (1.0 / ((y + (1.0 + x)) / y)) / (y + x)
	elif x <= -6.6e-171:
		tmp = y * (x / ((y + x) * (y + x)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.11)
		tmp = Float64(Float64(1.0 / Float64(Float64(y + Float64(1.0 + x)) / y)) / Float64(y + x));
	elseif (x <= -6.6e-171)
		tmp = Float64(y * Float64(x / Float64(Float64(y + x) * 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 <= -0.11)
		tmp = (1.0 / ((y + (1.0 + x)) / y)) / (y + x);
	elseif (x <= -6.6e-171)
		tmp = y * (x / ((y + x) * (y + x)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.11], N[(N[(1.0 / N[(N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.6e-171], N[(y * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $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 < -0.110000000000000001

   1. Initial program 63.3%

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

      1. associate-*r/ 82.8%

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

      2. *-commutative 82.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 34.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 82.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 82.8%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 63.3%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 34.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 34.6%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 63.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 63.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 63.3%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 87.4%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 87.4%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.8%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.8%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.8%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.8%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]
   6. Step-by-step derivation

      1. clear-num 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y}}} \cdot \frac{x}{y + x}}{y + x} \]

      2. clear-num 99.7%

         \[\leadsto \frac{\frac{1}{\frac{y + \left(1 + x\right)}{y}} \cdot \color{blue}{\frac{1}{\frac{y + x}{x}}}}{y + x} \]

      3. frac-times 99.7%

         \[\leadsto \frac{\color{blue}{\frac{1 \cdot 1}{\frac{y + \left(1 + x\right)}{y} \cdot \frac{y + x}{x}}}}{y + x} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{y + \left(1 + x\right)}{y} \cdot \frac{y + x}{x}}}{y + x} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{y + \left(1 + x\right)}{y} \cdot \frac{y + x}{x}}}}{y + x} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \frac{\frac{1}{\color{blue}{\frac{y + x}{x} \cdot \frac{y + \left(1 + x\right)}{y}}}}{y + x} \]

      2. associate-/r* 99.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{y + x}{x}}}{\frac{y + \left(1 + x\right)}{y}}}}{y + x} \]

   9. Simplified 99.7%

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{y + x}{x}}}{\frac{y + \left(1 + x\right)}{y}}}}{y + x} \]

   10. Taylor expanded in y around 0 75.6%

       \[\leadsto \frac{\frac{\color{blue}{1}}{\frac{y + \left(1 + x\right)}{y}}}{y + x} \]

   if -0.110000000000000001 < x < -6.6000000000000004e-171

   1. Initial program 87.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 92.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 92.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 92.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 92.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 92.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 92.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 72.7%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + x}} \]
   5. Step-by-step derivation

      1. +-commutative 72.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + 1}} \]

   6. Simplified 72.7%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x + 1}} \]

   7. Taylor expanded in x around 0 72.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{y} \]

   if -6.6000000000000004e-171 < x

   1. Initial program 65.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 82.5%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 82.5%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 71.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 82.5%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 82.5%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 82.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Step-by-step derivation

      1. associate-*r/ 66.0%

         \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

      2. fma-udef 56.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 56.0%

         \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. associate-+r+ 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

      6. *-commutative 65.9%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

      7. frac-times 86.4%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

      8. *-commutative 86.4%

         \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      9. associate-/r* 99.7%

         \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      10. associate-*r/ 99.9%

          \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

      11. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

      12. associate-+l+ 99.9%

          \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

      13. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

      14. +-commutative 99.9%

          \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

   6. Taylor expanded in x around 0 59.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{1 + y}}}{y + x} \]
   7. Step-by-step derivation

      1. +-commutative 59.6%

         \[\leadsto \frac{\frac{x}{\color{blue}{y + 1}}}{y + x} \]

   8. Simplified 59.6%

      \[\leadsto \frac{\color{blue}{\frac{x}{y + 1}}}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.11:\\ \;\;\;\;\frac{\frac{1}{\frac{y + \left(1 + x\right)}{y}}}{y + x}\\ \mathbf{elif}\;x \leq -6.6 \cdot 10^{-171}:\\ \;\;\;\;y \cdot \frac{x}{\left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]

Alternative 15: 59.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{1 + x}}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.05e-97)
   (/ 1.0 (/ x (/ y (+ 1.0 x))))
   (if (or (<= x -1.9e-137) (not (<= x -6.6e-171)))
     (/ x (+ y (* y y)))
     (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-97) {
		tmp = 1.0 / (x / (y / (1.0 + x)));
	} else if ((x <= -1.9e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = 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 <= (-1.05d-97)) then
        tmp = 1.0d0 / (x / (y / (1.0d0 + x)))
    else if ((x <= (-1.9d-137)) .or. (.not. (x <= (-6.6d-171)))) then
        tmp = x / (y + (y * y))
    else
        tmp = y / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.05e-97) {
		tmp = 1.0 / (x / (y / (1.0 + x)));
	} else if ((x <= -1.9e-137) || !(x <= -6.6e-171)) {
		tmp = x / (y + (y * y));
	} else {
		tmp = y / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.05e-97:
		tmp = 1.0 / (x / (y / (1.0 + x)))
	elif (x <= -1.9e-137) or not (x <= -6.6e-171):
		tmp = x / (y + (y * y))
	else:
		tmp = y / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.05e-97)
		tmp = Float64(1.0 / Float64(x / Float64(y / Float64(1.0 + x))));
	elseif ((x <= -1.9e-137) || !(x <= -6.6e-171))
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(y / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.05e-97)
		tmp = 1.0 / (x / (y / (1.0 + x)));
	elseif ((x <= -1.9e-137) || ~((x <= -6.6e-171)))
		tmp = x / (y + (y * y));
	else
		tmp = y / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.05e-97], N[(1.0 / N[(x / N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -1.9e-137], N[Not[LessEqual[x, -6.6e-171]], $MachinePrecision]], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0500000000000001e-97

   1. Initial program 68.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 89.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 89.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 89.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 89.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 89.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 69.9%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. clear-num 69.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + x\right) \cdot x}{y}}} \]

      2. inv-pow 69.9%

         \[\leadsto \color{blue}{{\left(\frac{\left(1 + x\right) \cdot x}{y}\right)}^{-1}} \]

      3. *-commutative 69.9%

         \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(1 + x\right)}}{y}\right)}^{-1} \]

   6. Applied egg-rr 69.9%

      \[\leadsto \color{blue}{{\left(\frac{x \cdot \left(1 + x\right)}{y}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 69.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(1 + x\right)}{y}}} \]

      2. associate-/l* 69.7%

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{y}{1 + x}}}} \]

   8. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{y}{1 + x}}}} \]

   if -1.0500000000000001e-97 < x < -1.89999999999999999e-137 or -6.6000000000000004e-171 < x

   1. Initial program 67.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 87.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 87.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 87.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 87.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 87.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 61.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 61.6%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 61.6%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 61.6%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   if -1.89999999999999999e-137 < x < -6.6000000000000004e-171

   1. Initial program 67.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 76.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 76.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 76.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 76.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 76.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 76.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 46.0%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 46.0%

      \[\leadsto \frac{y}{\color{blue}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{\frac{x}{\frac{y}{1 + x}}}\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-137} \lor \neg \left(x \leq -6.6 \cdot 10^{-171}\right):\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x}\\ \end{array} \]

Alternative 16: 55.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 120:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x x))))
   (if (<= y -1.6e-129)
     t_0
     (if (<= y 1.25e-129)
       (/ 1.0 (/ x y))
       (if (<= y 120.0) t_0 (/ (/ x y) y))))))

C

double code(double x, double y) {
	double t_0 = y / (x * x);
	double tmp;
	if (y <= -1.6e-129) {
		tmp = t_0;
	} else if (y <= 1.25e-129) {
		tmp = 1.0 / (x / y);
	} else if (y <= 120.0) {
		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 <= (-1.6d-129)) then
        tmp = t_0
    else if (y <= 1.25d-129) then
        tmp = 1.0d0 / (x / y)
    else if (y <= 120.0d0) 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 <= -1.6e-129) {
		tmp = t_0;
	} else if (y <= 1.25e-129) {
		tmp = 1.0 / (x / y);
	} else if (y <= 120.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * x)
	tmp = 0
	if y <= -1.6e-129:
		tmp = t_0
	elif y <= 1.25e-129:
		tmp = 1.0 / (x / y)
	elif y <= 120.0:
		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 <= -1.6e-129)
		tmp = t_0;
	elseif (y <= 1.25e-129)
		tmp = Float64(1.0 / Float64(x / y));
	elseif (y <= 120.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * x);
	tmp = 0.0;
	if (y <= -1.6e-129)
		tmp = t_0;
	elseif (y <= 1.25e-129)
		tmp = 1.0 / (x / y);
	elseif (y <= 120.0)
		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, -1.6e-129], t$95$0, If[LessEqual[y, 1.25e-129], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 120.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6000000000000001e-129 or 1.25000000000000007e-129 < y < 120

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 91.4%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 91.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 59.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 91.4%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 91.4%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 91.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 36.5%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 36.5%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   6. Simplified 36.5%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1.6000000000000001e-129 < y < 1.25000000000000007e-129

   1. Initial program 58.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 74.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 83.1%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. clear-num 82.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + x\right) \cdot x}{y}}} \]

      2. inv-pow 82.9%

         \[\leadsto \color{blue}{{\left(\frac{\left(1 + x\right) \cdot x}{y}\right)}^{-1}} \]

      3. *-commutative 82.9%

         \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(1 + x\right)}}{y}\right)}^{-1} \]

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\frac{x \cdot \left(1 + x\right)}{y}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 82.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(1 + x\right)}{y}}} \]

      2. associate-/l* 82.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{y}{1 + x}}}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{y}{1 + x}}}} \]

   9. Taylor expanded in x around 0 76.3%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{y}}} \]

   if 120 < y

   1. Initial program 58.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 80.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 80.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 70.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 80.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 81.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 68.1%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 68.1%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 68.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot y} \]

      2. times-frac 70.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]

   8. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*l/ 70.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{y}} \]

      2. *-lft-identity 70.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   10. Simplified 70.8%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{-129}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-129}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 120:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 17: 55.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{x}}{x}\\ \mathbf{if}\;y \leq -2.45 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 120:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y x) x)))
   (if (<= y -2.45e-127)
     t_0
     (if (<= y 7e-130) (/ 1.0 (/ x y)) (if (<= y 120.0) t_0 (/ (/ x y) y))))))

C

double code(double x, double y) {
	double t_0 = (y / x) / x;
	double tmp;
	if (y <= -2.45e-127) {
		tmp = t_0;
	} else if (y <= 7e-130) {
		tmp = 1.0 / (x / y);
	} else if (y <= 120.0) {
		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 <= (-2.45d-127)) then
        tmp = t_0
    else if (y <= 7d-130) then
        tmp = 1.0d0 / (x / y)
    else if (y <= 120.0d0) 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 <= -2.45e-127) {
		tmp = t_0;
	} else if (y <= 7e-130) {
		tmp = 1.0 / (x / y);
	} else if (y <= 120.0) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) / x
	tmp = 0
	if y <= -2.45e-127:
		tmp = t_0
	elif y <= 7e-130:
		tmp = 1.0 / (x / y)
	elif y <= 120.0:
		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 <= -2.45e-127)
		tmp = t_0;
	elseif (y <= 7e-130)
		tmp = Float64(1.0 / Float64(x / y));
	elseif (y <= 120.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) / x;
	tmp = 0.0;
	if (y <= -2.45e-127)
		tmp = t_0;
	elseif (y <= 7e-130)
		tmp = 1.0 / (x / y);
	elseif (y <= 120.0)
		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, -2.45e-127], t$95$0, If[LessEqual[y, 7e-130], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 120.0], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.45e-127 or 6.9999999999999998e-130 < y < 120

   1. Initial program 78.8%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 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 x around inf 50.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x}} \]

   5. Taylor expanded in x around inf 37.1%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x} \]
   6. Step-by-step derivation

      1. associate-*l/ 37.1%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{y}{x}}{x}} \]

      2. *-un-lft-identity 37.1%

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{x} \]

   7. Applied egg-rr 37.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x}} \]

   if -2.45e-127 < y < 6.9999999999999998e-130

   1. Initial program 58.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 74.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 74.8%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 83.1%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. clear-num 82.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + x\right) \cdot x}{y}}} \]

      2. inv-pow 82.9%

         \[\leadsto \color{blue}{{\left(\frac{\left(1 + x\right) \cdot x}{y}\right)}^{-1}} \]

      3. *-commutative 82.9%

         \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(1 + x\right)}}{y}\right)}^{-1} \]

   6. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{{\left(\frac{x \cdot \left(1 + x\right)}{y}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 82.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(1 + x\right)}{y}}} \]

      2. associate-/l* 82.9%

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{y}{1 + x}}}} \]

   8. Simplified 82.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{y}{1 + x}}}} \]

   9. Taylor expanded in x around 0 76.3%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{y}}} \]

   if 120 < y

   1. Initial program 58.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 80.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 80.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 70.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 80.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 81.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 81.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 68.1%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 68.1%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 68.1%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 68.1%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot y} \]

      2. times-frac 70.7%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]

   8. Applied egg-rr 70.7%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y}} \]
   9. Step-by-step derivation

      1. associate-*l/ 70.8%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y}}{y}} \]

      2. *-lft-identity 70.8%

         \[\leadsto \frac{\color{blue}{\frac{x}{y}}}{y} \]

   10. Simplified 70.8%

       \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.45 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-130}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{elif}\;y \leq 120:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 18: 45.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-217}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -3.3e-217) (- (/ y x) y) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -3.3e-217) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-3.3d-217)) then
        tmp = (y / x) - y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -3.3e-217) {
		tmp = (y / x) - y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -3.3e-217:
		tmp = (y / x) - y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -3.3e-217)
		tmp = Float64(Float64(y / x) - y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -3.3e-217)
		tmp = (y / x) - y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -3.3e-217], N[(N[(y / x), $MachinePrecision] - y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 82.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 82.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 34.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 82.8%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 82.8%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 82.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in x around inf 70.2%

      \[\leadsto \color{blue}{\frac{y}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 70.2%

         \[\leadsto \frac{y}{\color{blue}{x \cdot x}} \]

   6. Simplified 70.2%

      \[\leadsto \color{blue}{\frac{y}{x \cdot x}} \]

   if -1 < x < -3.29999999999999993e-217

   1. Initial program 78.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.9%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 84.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 84.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 84.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 84.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 32.5%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   5. Taylor expanded in x around 0 32.5%

      \[\leadsto \color{blue}{\frac{y}{x} + -1 \cdot y} \]
   6. Step-by-step derivation

      1. neg-mul-1 32.5%

         \[\leadsto \frac{y}{x} + \color{blue}{\left(-y\right)} \]

      2. unsub-neg 32.5%

         \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{\frac{y}{x} - y} \]

   if -3.29999999999999993e-217 < x

   1. Initial program 66.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 83.9%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 83.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 73.5%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 84.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 84.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 84.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 42.3%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 42.3%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 42.3%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{-217}:\\ \;\;\;\;\frac{y}{x} - y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 19: 44.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8.6e-18) (/ 1.0 (/ x y)) (/ x (* y y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8.6e-18) {
		tmp = 1.0 / (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 <= 8.6d-18) then
        tmp = 1.0d0 / (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 <= 8.6e-18) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8.6e-18:
		tmp = 1.0 / (x / y)
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8.6e-18)
		tmp = Float64(1.0 / 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 <= 8.6e-18)
		tmp = 1.0 / (x / y);
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8.6e-18], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.6000000000000005e-18

   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 87.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 87.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 87.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 87.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 87.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around 0 56.8%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
   5. Step-by-step derivation

      1. clear-num 56.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + x\right) \cdot x}{y}}} \]

      2. inv-pow 56.7%

         \[\leadsto \color{blue}{{\left(\frac{\left(1 + x\right) \cdot x}{y}\right)}^{-1}} \]

      3. *-commutative 56.7%

         \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(1 + x\right)}}{y}\right)}^{-1} \]

   6. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{{\left(\frac{x \cdot \left(1 + x\right)}{y}\right)}^{-1}} \]
   7. Step-by-step derivation

      1. unpow-1 56.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(1 + x\right)}{y}}} \]

      2. associate-/l* 57.1%

         \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{y}{1 + x}}}} \]

   8. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{y}{1 + x}}}} \]

   9. Taylor expanded in x around 0 37.3%

      \[\leadsto \frac{1}{\frac{x}{\color{blue}{y}}} \]

   if 8.6000000000000005e-18 < y

   1. Initial program 61.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 82.6%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 82.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 69.1%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 82.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 82.6%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 82.6%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 62.9%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 62.9%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 62.9%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 20: 27.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{x}{y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 (/ x y)))

C

double code(double x, double y) {
	return 1.0 / (x / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (x / y)
end function

Java

public static double code(double x, double y) {
	return 1.0 / (x / y);
}

Python

def code(x, y):
	return 1.0 / (x / y)

Julia

function code(x, y)
	return Float64(1.0 / Float64(x / y))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (x / y);
end

Wolfram

code[x_, y_] := N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 87.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.5%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around 0 49.3%

   \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]
5. Step-by-step derivation

   1. clear-num 49.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(1 + x\right) \cdot x}{y}}} \]

   2. inv-pow 49.2%

      \[\leadsto \color{blue}{{\left(\frac{\left(1 + x\right) \cdot x}{y}\right)}^{-1}} \]

   3. *-commutative 49.2%

      \[\leadsto {\left(\frac{\color{blue}{x \cdot \left(1 + x\right)}}{y}\right)}^{-1} \]

6. Applied egg-rr 49.2%

   \[\leadsto \color{blue}{{\left(\frac{x \cdot \left(1 + x\right)}{y}\right)}^{-1}} \]
7. Step-by-step derivation

   1. unpow-1 49.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot \left(1 + x\right)}{y}}} \]

   2. associate-/l* 49.3%

      \[\leadsto \frac{1}{\color{blue}{\frac{x}{\frac{y}{1 + x}}}} \]

8. Simplified 49.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{x}{\frac{y}{1 + x}}}} \]

9. Taylor expanded in x around 0 28.0%

   \[\leadsto \frac{1}{\frac{x}{\color{blue}{y}}} \]

10. Final simplification 28.0%

    \[\leadsto \frac{1}{\frac{x}{y}} \]

Alternative 21: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 y))

C

double code(double x, double y) {
	return 1.0 / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / y
end function

Java

public static double code(double x, double y) {
	return 1.0 / y;
}

Python

def code(x, y):
	return 1.0 / y

Julia

function code(x, y)
	return Float64(1.0 / y)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / y;
end

Wolfram

code[x_, y_] := N[(1.0 / y), $MachinePrecision]

Derivation

1. Initial program 67.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 87.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.5%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around inf 44.2%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{x}} \]

5. Taylor expanded in x around 0 4.0%

   \[\leadsto \color{blue}{\frac{1}{y}} \]

6. Final simplification 4.0%

   \[\leadsto \frac{1}{y} \]

Alternative 22: 26.5% 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 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 87.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 87.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 87.5%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in y around 0 49.3%

   \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

5. Taylor expanded in x around 0 27.8%

   \[\leadsto \frac{y}{\color{blue}{x}} \]

6. Final simplification 27.8%

   \[\leadsto \frac{y}{x} \]

Alternative 23: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.8%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-*r/ 83.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   2. *-commutative 83.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   3. distribute-rgt1-in 63.2%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. fma-def 83.5%

      \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

   5. cube-unmult 83.5%

      \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

3. Simplified 83.5%

   \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Step-by-step derivation

   1. associate-*r/ 67.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   2. fma-udef 52.9%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.9%

      \[\leadsto \frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 67.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. associate-+r+ 67.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + \left(y + 1\right)\right)} \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)} \]

   6. *-commutative 67.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. frac-times 87.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   8. *-commutative 87.5%

      \[\leadsto \color{blue}{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

   9. associate-/r* 99.7%

      \[\leadsto \frac{y}{x + \left(y + 1\right)} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

   10. associate-*r/ 99.8%

       \[\leadsto \color{blue}{\frac{\frac{y}{x + \left(y + 1\right)} \cdot \frac{x}{x + y}}{x + y}} \]

   11. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{\left(y + 1\right) + x}} \cdot \frac{x}{x + y}}{x + y} \]

   12. associate-+l+ 99.8%

       \[\leadsto \frac{\frac{y}{\color{blue}{y + \left(1 + x\right)}} \cdot \frac{x}{x + y}}{x + y} \]

   13. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{\color{blue}{y + x}}}{x + y} \]

   14. +-commutative 99.8%

       \[\leadsto \frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{\color{blue}{y + x}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\frac{y}{y + \left(1 + x\right)} \cdot \frac{x}{y + x}}{y + x}} \]

6. Taylor expanded in y around 0 50.2%

   \[\leadsto \frac{\color{blue}{\frac{y}{1 + x}}}{y + x} \]

7. Taylor expanded in x around 0 3.5%

   \[\leadsto \color{blue}{1} \]

8. Final simplification 3.5%

   \[\leadsto 1 \]

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 2023208 
(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))))