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

Percentage Accurate: 68.3% → 99.8%

Time: 16.1s

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: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 71.5%

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

   1. times-frac 88.9%

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

   2. associate-+r+ 88.9%

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

   3. associate-*l/ 82.4%

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

   4. times-frac 99.8%

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

   5. +-commutative 99.8%

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

   6. +-commutative 99.8%

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

   7. associate-+l+ 99.8%

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

   8. +-commutative 99.8%

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

3. Applied egg-rr 99.8%

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

4. Final simplification 99.8%

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

Alternative 2: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 1.38 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{t_0}}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 1.38e-179)
     (/ (/ y x) (+ x (+ y 1.0)))
     (if (<= y 0.00025)
       (* y (/ x (* (+ x y) (* (+ x y) t_0))))
       (* (/ (/ y t_0) (+ x y)) (/ x y))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.38e-179) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * (x / ((x + y) * ((x + y) * t_0)));
	} else {
		tmp = ((y / t_0) / (x + y)) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 1.38d-179) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 0.00025d0) then
        tmp = y * (x / ((x + y) * ((x + y) * t_0)))
    else
        tmp = ((y / t_0) / (x + y)) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 1.38e-179) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * (x / ((x + y) * ((x + y) * t_0)));
	} else {
		tmp = ((y / t_0) / (x + y)) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 1.38e-179:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 0.00025:
		tmp = y * (x / ((x + y) * ((x + y) * t_0)))
	else:
		tmp = ((y / t_0) / (x + y)) * (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 1.38e-179)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 0.00025)
		tmp = Float64(y * Float64(x / Float64(Float64(x + y) * Float64(Float64(x + y) * t_0))));
	else
		tmp = Float64(Float64(Float64(y / t_0) / Float64(x + y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 1.38e-179)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 0.00025)
		tmp = y * (x / ((x + y) * ((x + y) * t_0)));
	else
		tmp = ((y / t_0) / (x + y)) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.38e-179], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00025], N[(y * N[(x / N[(N[(x + y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3800000000000001e-179

   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 90.1%

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

      2. /-rgt-identity 90.1%

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

      3. associate-/l/ 90.1%

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

      4. *-lft-identity 90.1%

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

      5. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around inf 63.4%

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

      1. expm1-log1p-u 62.8%

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

      2. expm1-udef 41.1%

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

      3. frac-times 41.1%

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

      4. *-un-lft-identity 41.1%

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

   6. Applied egg-rr 41.1%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.3%

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

      3. associate-/r* 63.4%

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

   8. Simplified 63.4%

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

   if 1.3800000000000001e-179 < y < 2.5000000000000001e-4

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

      2. associate-*r/ 96.8%

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

      3. *-commutative 96.8%

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

      4. distribute-rgt1-in 79.7%

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

      5. fma-def 96.8%

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

      6. cube-unmult 96.9%

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

   3. Simplified 96.9%

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

      1. fma-udef 79.8%

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

      2. cube-mult 79.7%

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

      3. distribute-rgt1-in 96.8%

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

      4. associate-+r+ 96.8%

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

      5. associate-*r* 96.8%

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

      6. +-commutative 96.8%

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

      7. associate-+l+ 96.8%

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

      8. +-commutative 96.8%

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

      9. +-commutative 96.8%

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

   5. Applied egg-rr 96.8%

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

   if 2.5000000000000001e-4 < y

   1. Initial program 55.4%

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

      1. times-frac 79.6%

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

      2. associate-+r+ 79.6%

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

      3. associate-*l/ 79.6%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 85.3%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.38 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 3: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 1.38 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{y}{x}}{t_0}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 1.38e-179)
     (/ (/ y x) t_0)
     (if (<= y 1.6e+149)
       (* (/ x (* (+ x y) (+ x y))) (/ y t_0))
       (/ (/ x (+ x y)) (+ (+ y 1.0) (* x 2.0)))))))

C

double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.38e-179) {
		tmp = (y / x) / t_0;
	} else if (y <= 1.6e+149) {
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x + (y + 1.0d0)
    if (y <= 1.38d-179) then
        tmp = (y / x) / t_0
    else if (y <= 1.6d+149) then
        tmp = (x / ((x + y) * (x + y))) * (y / t_0)
    else
        tmp = (x / (x + y)) / ((y + 1.0d0) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 1.38e-179) {
		tmp = (y / x) / t_0;
	} else if (y <= 1.6e+149) {
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 1.38e-179:
		tmp = (y / x) / t_0
	elif y <= 1.6e+149:
		tmp = (x / ((x + y) * (x + y))) * (y / t_0)
	else:
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 1.38e-179)
		tmp = Float64(Float64(y / x) / t_0);
	elseif (y <= 1.6e+149)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / t_0));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(y + 1.0) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 1.38e-179)
		tmp = (y / x) / t_0;
	elseif (y <= 1.6e+149)
		tmp = (x / ((x + y) * (x + y))) * (y / t_0);
	else
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.38e-179], N[(N[(y / x), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y, 1.6e+149], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3800000000000001e-179

   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 90.1%

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

      2. /-rgt-identity 90.1%

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

      3. associate-/l/ 90.1%

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

      4. *-lft-identity 90.1%

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

      5. associate-+l+ 90.1%

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

   3. Simplified 90.1%

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

   4. Taylor expanded in x around inf 63.4%

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

      1. expm1-log1p-u 62.8%

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

      2. expm1-udef 41.1%

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

      3. frac-times 41.1%

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

      4. *-un-lft-identity 41.1%

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

   6. Applied egg-rr 41.1%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 63.3%

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

      3. associate-/r* 63.4%

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

   8. Simplified 63.4%

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

   if 1.3800000000000001e-179 < y < 1.6000000000000001e149

   1. Initial program 77.7%

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

      1. times-frac 98.0%

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

      2. /-rgt-identity 98.0%

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

      3. associate-/l/ 98.0%

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

      4. *-lft-identity 98.0%

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

      5. associate-+l+ 98.0%

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

   3. Simplified 98.0%

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

   if 1.6000000000000001e149 < y

   1. Initial program 52.1%

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

      1. times-frac 68.5%

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

      2. associate-+r+ 68.5%

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

      3. associate-*l/ 68.5%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. associate-+r+ 99.7%

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

      3. +-commutative 99.7%

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

      4. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/l* 99.7%

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

       2. +-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. associate-+r+ 99.7%

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

       5. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 91.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.38 \cdot 10^{-179}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]

Alternative 4: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x + y}\\ \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{t_0}{y + 1}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x y))))
   (if (<= y 4.5e-124)
     (/ (/ y x) (+ x (+ y 1.0)))
     (if (<= y 1.3e-68)
       (- (/ x y) x)
       (if (<= y 4e-54)
         (/ y (* x (+ x 1.0)))
         (if (<= y 5e-8)
           (/ t_0 (+ y 1.0))
           (if (<= y 1.6e+149) (/ x (* (+ x y) (+ x y))) (/ t_0 y))))))))

C

double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 4e-54) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5e-8) {
		tmp = t_0 / (y + 1.0);
	} else if (y <= 1.6e+149) {
		tmp = x / ((x + y) * (x + y));
	} else {
		tmp = t_0 / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (x + y)
    if (y <= 4.5d-124) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 1.3d-68) then
        tmp = (x / y) - x
    else if (y <= 4d-54) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 5d-8) then
        tmp = t_0 / (y + 1.0d0)
    else if (y <= 1.6d+149) then
        tmp = x / ((x + y) * (x + y))
    else
        tmp = t_0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (x + y);
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 4e-54) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 5e-8) {
		tmp = t_0 / (y + 1.0);
	} else if (y <= 1.6e+149) {
		tmp = x / ((x + y) * (x + y));
	} else {
		tmp = t_0 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (x + y)
	tmp = 0
	if y <= 4.5e-124:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 1.3e-68:
		tmp = (x / y) - x
	elif y <= 4e-54:
		tmp = y / (x * (x + 1.0))
	elif y <= 5e-8:
		tmp = t_0 / (y + 1.0)
	elif y <= 1.6e+149:
		tmp = x / ((x + y) * (x + y))
	else:
		tmp = t_0 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(x + y))
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 1.3e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 4e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 5e-8)
		tmp = Float64(t_0 / Float64(y + 1.0));
	elseif (y <= 1.6e+149)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(x + y)));
	else
		tmp = Float64(t_0 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (x + y);
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 1.3e-68)
		tmp = (x / y) - x;
	elseif (y <= 4e-54)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 5e-8)
		tmp = t_0 / (y + 1.0);
	elseif (y <= 1.6e+149)
		tmp = x / ((x + y) * (x + y));
	else
		tmp = t_0 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.5e-124], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 4e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e-8], N[(t$95$0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+149], N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / y), $MachinePrecision]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 65.7%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 40.7%

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

      3. frac-times 40.7%

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

      4. *-un-lft-identity 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 65.5%

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

      3. associate-/r* 65.7%

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

   8. Simplified 65.7%

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

   if 4.4999999999999996e-124 < y < 1.2999999999999999e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

   9. Simplified 83.6%

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

   if 1.2999999999999999e-68 < y < 4.0000000000000001e-54

   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 99.6%

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

      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 y around 0 85.2%

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

   if 4.0000000000000001e-54 < y < 4.9999999999999998e-8

   1. Initial program 93.0%

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

      1. times-frac 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 82.9%

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

      1. +-commutative 82.9%

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

   6. Simplified 82.9%

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

      1. *-commutative 82.9%

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

      2. clear-num 82.7%

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

      3. associate-/r* 61.0%

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

      4. +-commutative 61.0%

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

      5. +-commutative 61.0%

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

      6. frac-times 61.1%

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

      7. *-un-lft-identity 61.1%

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

      8. +-commutative 61.1%

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

      9. +-commutative 61.1%

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

   8. Applied egg-rr 61.1%

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

   9. Taylor expanded in x around 0 48.9%

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

       1. +-commutative 48.9%

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

   11. Simplified 48.9%

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

   if 4.9999999999999998e-8 < y < 1.6000000000000001e149

   1. Initial program 62.8%

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

      1. times-frac 95.8%

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

      2. /-rgt-identity 95.8%

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

      3. associate-/l/ 95.8%

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

      4. *-lft-identity 95.8%

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

      5. associate-+l+ 95.8%

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

   3. Simplified 95.8%

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

   4. Taylor expanded in y around inf 75.0%

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

   if 1.6000000000000001e149 < y

   1. Initial program 52.1%

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

      1. times-frac 68.5%

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

      2. /-rgt-identity 68.5%

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

      3. associate-/l/ 68.5%

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

      4. *-lft-identity 68.5%

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

      5. associate-+l+ 68.5%

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

   3. Simplified 68.5%

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

   4. Taylor expanded in x around 0 68.5%

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

      1. +-commutative 68.5%

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

   6. Simplified 68.5%

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

      1. *-commutative 68.5%

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

      2. clear-num 68.5%

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

      3. associate-/r* 90.0%

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

      4. +-commutative 90.0%

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

      5. +-commutative 90.0%

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

      6. frac-times 90.0%

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

      7. *-un-lft-identity 90.0%

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

      8. +-commutative 90.0%

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

      9. +-commutative 90.0%

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

   8. Applied egg-rr 90.0%

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

   9. Taylor expanded in y around inf 89.9%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \]

Alternative 5: 66.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* (+ x y) (+ x y)))))
   (if (<= y 1.8e-126)
     (/ (/ y x) (+ x (+ y 1.0)))
     (if (<= y 0.00025)
       (* y t_0)
       (if (<= y 1.6e+149) t_0 (/ (/ x (+ x y)) (+ (+ y 1.0) (* x 2.0))))))))

C

double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * t_0;
	} else if (y <= 1.6e+149) {
		tmp = t_0;
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / ((x + y) * (x + y))
    if (y <= 1.8d-126) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 0.00025d0) then
        tmp = y * t_0
    else if (y <= 1.6d+149) then
        tmp = t_0
    else
        tmp = (x / (x + y)) / ((y + 1.0d0) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (y <= 1.8e-126) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * t_0;
	} else if (y <= 1.6e+149) {
		tmp = t_0;
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / ((x + y) * (x + y))
	tmp = 0
	if y <= 1.8e-126:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 0.00025:
		tmp = y * t_0
	elif y <= 1.6e+149:
		tmp = t_0
	else:
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(Float64(x + y) * Float64(x + y)))
	tmp = 0.0
	if (y <= 1.8e-126)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 0.00025)
		tmp = Float64(y * t_0);
	elseif (y <= 1.6e+149)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(y + 1.0) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / ((x + y) * (x + y));
	tmp = 0.0;
	if (y <= 1.8e-126)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 0.00025)
		tmp = y * t_0;
	elseif (y <= 1.6e+149)
		tmp = t_0;
	else
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.8e-126], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00025], N[(y * t$95$0), $MachinePrecision], If[LessEqual[y, 1.6e+149], t$95$0, N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.8e-126

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 65.7%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 40.7%

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

      3. frac-times 40.7%

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

      4. *-un-lft-identity 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 65.5%

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

      3. associate-/r* 65.7%

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

   8. Simplified 65.7%

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

   if 1.8e-126 < y < 2.5000000000000001e-4

   1. Initial program 85.9%

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

      1. times-frac 99.6%

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

      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 90.4%

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

      1. +-commutative 90.4%

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

   6. Simplified 90.4%

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

   7. Taylor expanded in y around 0 87.2%

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

   if 2.5000000000000001e-4 < y < 1.6000000000000001e149

   1. Initial program 60.2%

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

      1. times-frac 95.6%

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

      2. /-rgt-identity 95.6%

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

      3. associate-/l/ 95.6%

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

      4. *-lft-identity 95.6%

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

      5. associate-+l+ 95.6%

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

   3. Simplified 95.6%

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

   4. Taylor expanded in y around inf 76.4%

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

   if 1.6000000000000001e149 < y

   1. Initial program 52.1%

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

      1. times-frac 68.5%

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

      2. associate-+r+ 68.5%

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

      3. associate-*l/ 68.5%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. associate-+r+ 99.7%

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

      3. +-commutative 99.7%

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

      4. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/l* 99.7%

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

       2. +-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. associate-+r+ 99.7%

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

       5. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 91.3%

       \[\leadsto \frac{\frac{x}{y + x}}{\color{blue}{2 \cdot x + \left(1 + y\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]

Alternative 6: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-125)
   (/ (/ y x) (+ x (+ y 1.0)))
   (if (<= y 4.9e-26)
     (* y (/ x (* (+ x y) (+ x y))))
     (* (/ (/ y (+ y (+ x 1.0))) (+ x y)) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-125) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 4.9e-26) {
		tmp = y * (x / ((x + y) * (x + y)));
	} else {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.2d-125) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 4.9d-26) then
        tmp = y * (x / ((x + y) * (x + y)))
    else
        tmp = ((y / (y + (x + 1.0d0))) / (x + y)) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-125) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 4.9e-26) {
		tmp = y * (x / ((x + y) * (x + y)));
	} else {
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e-125:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 4.9e-26:
		tmp = y * (x / ((x + y) * (x + y)))
	else:
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.2e-125)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 4.9e-26)
		tmp = Float64(y * Float64(x / Float64(Float64(x + y) * Float64(x + y))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + Float64(x + 1.0))) / Float64(x + y)) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-125)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 4.9e-26)
		tmp = y * (x / ((x + y) * (x + y)));
	else
		tmp = ((y / (y + (x + 1.0))) / (x + y)) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.2e-125], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.9e-26], N[(y * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.2e-125

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 65.7%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 40.7%

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

      3. frac-times 40.7%

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

      4. *-un-lft-identity 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 65.5%

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

      3. associate-/r* 65.7%

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

   8. Simplified 65.7%

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

   if 4.2e-125 < y < 4.8999999999999999e-26

   1. Initial program 85.7%

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

      1. times-frac 99.6%

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

      2. /-rgt-identity 99.6%

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

      3. associate-/l/ 99.6%

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

      4. *-lft-identity 99.6%

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

      5. associate-+l+ 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 91.7%

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

      1. +-commutative 91.7%

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

   6. Simplified 91.7%

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

   7. Taylor expanded in y around 0 91.7%

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

   if 4.8999999999999999e-26 < y

   1. Initial program 58.4%

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

      1. times-frac 81.5%

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

      2. associate-+r+ 81.5%

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

      3. associate-*l/ 81.5%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

   4. Taylor expanded in x around 0 85.4%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-125}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{-26}:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + \left(x + 1\right)}}{x + y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7: 67.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9.5e-127)
   (/ (/ y x) (+ x (+ y 1.0)))
   (if (<= y 1.6e+149)
     (* (/ x (* (+ x y) (+ x y))) (/ y (+ y 1.0)))
     (/ (/ x (+ x y)) (+ (+ y 1.0) (* x 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-127) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.6e+149) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.5d-127) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 1.6d+149) then
        tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0d0))
    else
        tmp = (x / (x + y)) / ((y + 1.0d0) + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.5e-127) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.6e+149) {
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	} else {
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.5e-127:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 1.6e+149:
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0))
	else:
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.5e-127)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 1.6e+149)
		tmp = Float64(Float64(x / Float64(Float64(x + y) * Float64(x + y))) * Float64(y / Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(y + 1.0) + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.5e-127)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 1.6e+149)
		tmp = (x / ((x + y) * (x + y))) * (y / (y + 1.0));
	else
		tmp = (x / (x + y)) / ((y + 1.0) + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.5e-127], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+149], N[(N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.4999999999999997e-127

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 65.7%

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

      1. expm1-log1p-u 65.1%

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

      2. expm1-udef 40.7%

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

      3. frac-times 40.7%

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

      4. *-un-lft-identity 40.7%

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

   6. Applied egg-rr 40.7%

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

      1. expm1-def 65.0%

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

      2. expm1-log1p 65.5%

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

      3. associate-/r* 65.7%

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

   8. Simplified 65.7%

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

   if 9.4999999999999997e-127 < y < 1.6000000000000001e149

   1. Initial program 73.9%

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

      1. times-frac 97.7%

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

      2. /-rgt-identity 97.7%

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

      3. associate-/l/ 97.7%

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

      4. *-lft-identity 97.7%

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

      5. associate-+l+ 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in x around 0 84.7%

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

      1. +-commutative 84.7%

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

   6. Simplified 84.7%

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

   if 1.6000000000000001e149 < y

   1. Initial program 52.1%

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

      1. times-frac 68.5%

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

      2. associate-+r+ 68.5%

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

      3. associate-*l/ 68.5%

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

      4. times-frac 99.7%

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

      5. +-commutative 99.7%

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

      6. +-commutative 99.7%

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

      7. associate-+l+ 99.7%

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

      8. +-commutative 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. clear-num 99.7%

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

      2. associate-+r+ 99.7%

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

      3. +-commutative 99.7%

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

      4. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*r/ 99.8%

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

      2. +-commutative 99.8%

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

      3. clear-num 99.8%

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

      4. +-commutative 99.8%

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

      5. +-commutative 99.8%

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

   9. Applied egg-rr 99.8%

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

       1. associate-/l* 99.7%

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

       2. +-commutative 99.7%

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

       3. +-commutative 99.7%

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

       4. associate-+r+ 99.7%

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

       5. +-commutative 99.7%

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

   11. Simplified 99.7%

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

   12. Taylor expanded in y around inf 91.3%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(y + 1\right) + x \cdot 2}\\ \end{array} \]

Alternative 8: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - x\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -7.4 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) x)) (t_1 (/ y (* x x))))
   (if (<= y -7.4e-110)
     t_1
     (if (<= y 2.2e-203)
       (* y (/ 1.0 x))
       (if (<= y 4.5e-124)
         t_1
         (if (<= y 2.85e-68)
           t_0
           (if (<= y 2.6e-54) t_1 (if (<= y 0.75) t_0 (/ x (* y y))))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -7.4e-110) {
		tmp = t_1;
	} else if (y <= 2.2e-203) {
		tmp = y * (1.0 / x);
	} else if (y <= 4.5e-124) {
		tmp = t_1;
	} else if (y <= 2.85e-68) {
		tmp = t_0;
	} else if (y <= 2.6e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (x / y) - x
    t_1 = y / (x * x)
    if (y <= (-7.4d-110)) then
        tmp = t_1
    else if (y <= 2.2d-203) then
        tmp = y * (1.0d0 / x)
    else if (y <= 4.5d-124) then
        tmp = t_1
    else if (y <= 2.85d-68) then
        tmp = t_0
    else if (y <= 2.6d-54) then
        tmp = t_1
    else if (y <= 0.75d0) 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 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -7.4e-110) {
		tmp = t_1;
	} else if (y <= 2.2e-203) {
		tmp = y * (1.0 / x);
	} else if (y <= 4.5e-124) {
		tmp = t_1;
	} else if (y <= 2.85e-68) {
		tmp = t_0;
	} else if (y <= 2.6e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - x
	t_1 = y / (x * x)
	tmp = 0
	if y <= -7.4e-110:
		tmp = t_1
	elif y <= 2.2e-203:
		tmp = y * (1.0 / x)
	elif y <= 4.5e-124:
		tmp = t_1
	elif y <= 2.85e-68:
		tmp = t_0
	elif y <= 2.6e-54:
		tmp = t_1
	elif y <= 0.75:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - x)
	t_1 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -7.4e-110)
		tmp = t_1;
	elseif (y <= 2.2e-203)
		tmp = Float64(y * Float64(1.0 / x));
	elseif (y <= 4.5e-124)
		tmp = t_1;
	elseif (y <= 2.85e-68)
		tmp = t_0;
	elseif (y <= 2.6e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - x;
	t_1 = y / (x * x);
	tmp = 0.0;
	if (y <= -7.4e-110)
		tmp = t_1;
	elseif (y <= 2.2e-203)
		tmp = y * (1.0 / x);
	elseif (y <= 4.5e-124)
		tmp = t_1;
	elseif (y <= 2.85e-68)
		tmp = t_0;
	elseif (y <= 2.6e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.4e-110], t$95$1, If[LessEqual[y, 2.2e-203], N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e-124], t$95$1, If[LessEqual[y, 2.85e-68], t$95$0, If[LessEqual[y, 2.6e-54], t$95$1, If[LessEqual[y, 0.75], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.40000000000000032e-110 or 2.2e-203 < y < 4.4999999999999996e-124 or 2.8500000000000001e-68 < y < 2.60000000000000002e-54

   1. Initial program 81.3%

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

      1. associate-*r/ 86.3%

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

      2. *-commutative 86.3%

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

      3. distribute-rgt1-in 52.2%

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

      4. fma-def 86.3%

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

      5. cube-unmult 86.4%

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

   3. Simplified 86.4%

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

   4. Taylor expanded in x around inf 40.5%

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

      1. unpow2 40.5%

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

   6. Simplified 40.5%

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

   if -7.40000000000000032e-110 < y < 2.2e-203

   1. Initial program 68.9%

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

      1. times-frac 84.2%

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

      2. /-rgt-identity 84.2%

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

      3. associate-/l/ 84.2%

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

      4. *-lft-identity 84.2%

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

      5. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 73.3%

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

      1. +-commutative 73.3%

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

   6. Simplified 73.3%

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

   7. Taylor expanded in y around 0 73.3%

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

   8. Taylor expanded in x around inf 81.3%

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

   if 4.4999999999999996e-124 < y < 2.8500000000000001e-68 or 2.60000000000000002e-54 < y < 0.75

   1. Initial program 82.6%

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

      1. times-frac 99.6%

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

      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 56.7%

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

      1. distribute-rgt-in 56.8%

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

      2. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in y around 0 55.2%

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

      1. neg-mul-1 55.2%

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

      2. unsub-neg 55.2%

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

   9. Simplified 55.2%

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

   if 0.75 < y

   1. Initial program 55.4%

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

      1. associate-*r/ 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt1-in 66.6%

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

      4. fma-def 70.9%

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

      5. cube-unmult 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in y around inf 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-203}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 9: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - x\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{-105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) x)) (t_1 (/ y (* x x))))
   (if (<= y -4.4e-105)
     t_1
     (if (<= y 4.1e-203)
       (/ (- y) (- x))
       (if (<= y 4.2e-124)
         t_1
         (if (<= y 2.95e-68)
           t_0
           (if (<= y 2.45e-54) t_1 (if (<= y 0.75) t_0 (/ x (* y y))))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -4.4e-105) {
		tmp = t_1;
	} else if (y <= 4.1e-203) {
		tmp = -y / -x;
	} else if (y <= 4.2e-124) {
		tmp = t_1;
	} else if (y <= 2.95e-68) {
		tmp = t_0;
	} else if (y <= 2.45e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (x / y) - x
    t_1 = y / (x * x)
    if (y <= (-4.4d-105)) then
        tmp = t_1
    else if (y <= 4.1d-203) then
        tmp = -y / -x
    else if (y <= 4.2d-124) then
        tmp = t_1
    else if (y <= 2.95d-68) then
        tmp = t_0
    else if (y <= 2.45d-54) then
        tmp = t_1
    else if (y <= 0.75d0) 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 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -4.4e-105) {
		tmp = t_1;
	} else if (y <= 4.1e-203) {
		tmp = -y / -x;
	} else if (y <= 4.2e-124) {
		tmp = t_1;
	} else if (y <= 2.95e-68) {
		tmp = t_0;
	} else if (y <= 2.45e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		tmp = t_0;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - x
	t_1 = y / (x * x)
	tmp = 0
	if y <= -4.4e-105:
		tmp = t_1
	elif y <= 4.1e-203:
		tmp = -y / -x
	elif y <= 4.2e-124:
		tmp = t_1
	elif y <= 2.95e-68:
		tmp = t_0
	elif y <= 2.45e-54:
		tmp = t_1
	elif y <= 0.75:
		tmp = t_0
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - x)
	t_1 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -4.4e-105)
		tmp = t_1;
	elseif (y <= 4.1e-203)
		tmp = Float64(Float64(-y) / Float64(-x));
	elseif (y <= 4.2e-124)
		tmp = t_1;
	elseif (y <= 2.95e-68)
		tmp = t_0;
	elseif (y <= 2.45e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - x;
	t_1 = y / (x * x);
	tmp = 0.0;
	if (y <= -4.4e-105)
		tmp = t_1;
	elseif (y <= 4.1e-203)
		tmp = -y / -x;
	elseif (y <= 4.2e-124)
		tmp = t_1;
	elseif (y <= 2.95e-68)
		tmp = t_0;
	elseif (y <= 2.45e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.4e-105], t$95$1, If[LessEqual[y, 4.1e-203], N[((-y) / (-x)), $MachinePrecision], If[LessEqual[y, 4.2e-124], t$95$1, If[LessEqual[y, 2.95e-68], t$95$0, If[LessEqual[y, 2.45e-54], t$95$1, If[LessEqual[y, 0.75], t$95$0, N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.40000000000000008e-105 or 4.09999999999999981e-203 < y < 4.2000000000000002e-124 or 2.95e-68 < y < 2.4500000000000001e-54

   1. Initial program 81.1%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt1-in 51.7%

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

      4. fma-def 86.2%

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

      5. cube-unmult 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around inf 39.8%

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

      1. unpow2 39.8%

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

   6. Simplified 39.8%

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

   if -4.40000000000000008e-105 < y < 4.09999999999999981e-203

   1. Initial program 69.3%

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

      1. times-frac 84.4%

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

      2. /-rgt-identity 84.4%

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

      3. associate-/l/ 84.4%

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

      4. *-lft-identity 84.4%

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

      5. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. +-commutative 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 72.2%

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

   8. Taylor expanded in x around inf 80.2%

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

      1. associate-*l/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

      3. frac-2neg 80.3%

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

   10. Applied egg-rr 80.3%

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

   if 4.2000000000000002e-124 < y < 2.95e-68 or 2.4500000000000001e-54 < y < 0.75

   1. Initial program 82.6%

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

      1. times-frac 99.6%

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

      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 56.7%

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

      1. distribute-rgt-in 56.8%

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

      2. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in y around 0 55.2%

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

      1. neg-mul-1 55.2%

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

      2. unsub-neg 55.2%

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

   9. Simplified 55.2%

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

   if 0.75 < y

   1. Initial program 55.4%

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

      1. associate-*r/ 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt1-in 66.6%

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

      4. fma-def 70.9%

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

      5. cube-unmult 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in y around inf 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{-203}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 10: 53.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} - x\\ t_1 := \frac{y}{x \cdot x}\\ \mathbf{if}\;y \leq -3.9 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-205}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (- (/ x y) x)) (t_1 (/ y (* x x))))
   (if (<= y -3.9e-101)
     t_1
     (if (<= y 1.85e-205)
       (/ (- y) (- x))
       (if (<= y 4.5e-124)
         t_1
         (if (<= y 1.45e-68)
           t_0
           (if (<= y 3.4e-54) t_1 (if (<= y 0.75) t_0 (/ (/ x y) y)))))))))

C

double code(double x, double y) {
	double t_0 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -3.9e-101) {
		tmp = t_1;
	} else if (y <= 1.85e-205) {
		tmp = -y / -x;
	} else if (y <= 4.5e-124) {
		tmp = t_1;
	} else if (y <= 1.45e-68) {
		tmp = t_0;
	} else if (y <= 3.4e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		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) :: t_1
    real(8) :: tmp
    t_0 = (x / y) - x
    t_1 = y / (x * x)
    if (y <= (-3.9d-101)) then
        tmp = t_1
    else if (y <= 1.85d-205) then
        tmp = -y / -x
    else if (y <= 4.5d-124) then
        tmp = t_1
    else if (y <= 1.45d-68) then
        tmp = t_0
    else if (y <= 3.4d-54) then
        tmp = t_1
    else if (y <= 0.75d0) 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 = (x / y) - x;
	double t_1 = y / (x * x);
	double tmp;
	if (y <= -3.9e-101) {
		tmp = t_1;
	} else if (y <= 1.85e-205) {
		tmp = -y / -x;
	} else if (y <= 4.5e-124) {
		tmp = t_1;
	} else if (y <= 1.45e-68) {
		tmp = t_0;
	} else if (y <= 3.4e-54) {
		tmp = t_1;
	} else if (y <= 0.75) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) - x
	t_1 = y / (x * x)
	tmp = 0
	if y <= -3.9e-101:
		tmp = t_1
	elif y <= 1.85e-205:
		tmp = -y / -x
	elif y <= 4.5e-124:
		tmp = t_1
	elif y <= 1.45e-68:
		tmp = t_0
	elif y <= 3.4e-54:
		tmp = t_1
	elif y <= 0.75:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) - x)
	t_1 = Float64(y / Float64(x * x))
	tmp = 0.0
	if (y <= -3.9e-101)
		tmp = t_1;
	elseif (y <= 1.85e-205)
		tmp = Float64(Float64(-y) / Float64(-x));
	elseif (y <= 4.5e-124)
		tmp = t_1;
	elseif (y <= 1.45e-68)
		tmp = t_0;
	elseif (y <= 3.4e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) - x;
	t_1 = y / (x * x);
	tmp = 0.0;
	if (y <= -3.9e-101)
		tmp = t_1;
	elseif (y <= 1.85e-205)
		tmp = -y / -x;
	elseif (y <= 4.5e-124)
		tmp = t_1;
	elseif (y <= 1.45e-68)
		tmp = t_0;
	elseif (y <= 3.4e-54)
		tmp = t_1;
	elseif (y <= 0.75)
		tmp = t_0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$1 = N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.9e-101], t$95$1, If[LessEqual[y, 1.85e-205], N[((-y) / (-x)), $MachinePrecision], If[LessEqual[y, 4.5e-124], t$95$1, If[LessEqual[y, 1.45e-68], t$95$0, If[LessEqual[y, 3.4e-54], t$95$1, If[LessEqual[y, 0.75], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.90000000000000015e-101 or 1.85e-205 < y < 4.4999999999999996e-124 or 1.45e-68 < y < 3.39999999999999987e-54

   1. Initial program 81.1%

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

      1. associate-*r/ 86.2%

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

      2. *-commutative 86.2%

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

      3. distribute-rgt1-in 51.7%

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

      4. fma-def 86.2%

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

      5. cube-unmult 86.2%

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

   3. Simplified 86.2%

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

   4. Taylor expanded in x around inf 39.8%

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

      1. unpow2 39.8%

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

   6. Simplified 39.8%

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

   if -3.90000000000000015e-101 < y < 1.85e-205

   1. Initial program 69.3%

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

      1. times-frac 84.4%

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

      2. /-rgt-identity 84.4%

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

      3. associate-/l/ 84.4%

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

      4. *-lft-identity 84.4%

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

      5. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. +-commutative 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 72.2%

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

   8. Taylor expanded in x around inf 80.2%

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

      1. associate-*l/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

      3. frac-2neg 80.3%

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

   10. Applied egg-rr 80.3%

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

   if 4.4999999999999996e-124 < y < 1.45e-68 or 3.39999999999999987e-54 < y < 0.75

   1. Initial program 82.6%

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

      1. times-frac 99.6%

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

      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 56.7%

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

      1. distribute-rgt-in 56.8%

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

      2. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in y around 0 55.2%

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

      1. neg-mul-1 55.2%

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

      2. unsub-neg 55.2%

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

   9. Simplified 55.2%

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

   if 0.75 < y

   1. Initial program 55.4%

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

      1. associate-*r/ 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt1-in 66.6%

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

      4. fma-def 70.9%

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

      5. cube-unmult 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in y around inf 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. times-frac 75.4%

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

   8. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   10. Simplified 75.5%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.9 \cdot 10^{-101}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-205}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 11: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x} \cdot \frac{1}{x}\\ t_1 := \frac{x}{y} - x\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-204}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ y x) (/ 1.0 x))) (t_1 (- (/ x y) x)))
   (if (<= y -7.6e-106)
     t_0
     (if (<= y 1.75e-204)
       (/ (- y) (- x))
       (if (<= y 4.5e-124)
         t_0
         (if (<= y 1.3e-68)
           t_1
           (if (<= y 2.05e-54)
             (/ y (* x x))
             (if (<= y 0.75) t_1 (/ (/ x y) y)))))))))

C

double code(double x, double y) {
	double t_0 = (y / x) * (1.0 / x);
	double t_1 = (x / y) - x;
	double tmp;
	if (y <= -7.6e-106) {
		tmp = t_0;
	} else if (y <= 1.75e-204) {
		tmp = -y / -x;
	} else if (y <= 4.5e-124) {
		tmp = t_0;
	} else if (y <= 1.3e-68) {
		tmp = t_1;
	} else if (y <= 2.05e-54) {
		tmp = y / (x * x);
	} else if (y <= 0.75) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = (y / x) * (1.0d0 / x)
    t_1 = (x / y) - x
    if (y <= (-7.6d-106)) then
        tmp = t_0
    else if (y <= 1.75d-204) then
        tmp = -y / -x
    else if (y <= 4.5d-124) then
        tmp = t_0
    else if (y <= 1.3d-68) then
        tmp = t_1
    else if (y <= 2.05d-54) then
        tmp = y / (x * x)
    else if (y <= 0.75d0) then
        tmp = t_1
    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) * (1.0 / x);
	double t_1 = (x / y) - x;
	double tmp;
	if (y <= -7.6e-106) {
		tmp = t_0;
	} else if (y <= 1.75e-204) {
		tmp = -y / -x;
	} else if (y <= 4.5e-124) {
		tmp = t_0;
	} else if (y <= 1.3e-68) {
		tmp = t_1;
	} else if (y <= 2.05e-54) {
		tmp = y / (x * x);
	} else if (y <= 0.75) {
		tmp = t_1;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / x) * (1.0 / x)
	t_1 = (x / y) - x
	tmp = 0
	if y <= -7.6e-106:
		tmp = t_0
	elif y <= 1.75e-204:
		tmp = -y / -x
	elif y <= 4.5e-124:
		tmp = t_0
	elif y <= 1.3e-68:
		tmp = t_1
	elif y <= 2.05e-54:
		tmp = y / (x * x)
	elif y <= 0.75:
		tmp = t_1
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / x) * Float64(1.0 / x))
	t_1 = Float64(Float64(x / y) - x)
	tmp = 0.0
	if (y <= -7.6e-106)
		tmp = t_0;
	elseif (y <= 1.75e-204)
		tmp = Float64(Float64(-y) / Float64(-x));
	elseif (y <= 4.5e-124)
		tmp = t_0;
	elseif (y <= 1.3e-68)
		tmp = t_1;
	elseif (y <= 2.05e-54)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 0.75)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / x) * (1.0 / x);
	t_1 = (x / y) - x;
	tmp = 0.0;
	if (y <= -7.6e-106)
		tmp = t_0;
	elseif (y <= 1.75e-204)
		tmp = -y / -x;
	elseif (y <= 4.5e-124)
		tmp = t_0;
	elseif (y <= 1.3e-68)
		tmp = t_1;
	elseif (y <= 2.05e-54)
		tmp = y / (x * x);
	elseif (y <= 0.75)
		tmp = t_1;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[y, -7.6e-106], t$95$0, If[LessEqual[y, 1.75e-204], N[((-y) / (-x)), $MachinePrecision], If[LessEqual[y, 4.5e-124], t$95$0, If[LessEqual[y, 1.3e-68], t$95$1, If[LessEqual[y, 2.05e-54], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.75], t$95$1, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -7.5999999999999999e-106 or 1.75000000000000013e-204 < y < 4.4999999999999996e-124

   1. Initial program 79.9%

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

      1. times-frac 95.4%

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

      2. /-rgt-identity 95.4%

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

      3. associate-/l/ 95.4%

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

      4. *-lft-identity 95.4%

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

      5. associate-+l+ 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around inf 46.3%

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

   5. Taylor expanded in x around inf 39.3%

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

   if -7.5999999999999999e-106 < y < 1.75000000000000013e-204

   1. Initial program 69.3%

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

      1. times-frac 84.4%

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

      2. /-rgt-identity 84.4%

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

      3. associate-/l/ 84.4%

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

      4. *-lft-identity 84.4%

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

      5. associate-+l+ 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. +-commutative 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in y around 0 72.2%

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

   8. Taylor expanded in x around inf 80.2%

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

      1. associate-*l/ 80.3%

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

      2. *-un-lft-identity 80.3%

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

      3. frac-2neg 80.3%

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

   10. Applied egg-rr 80.3%

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

   if 4.4999999999999996e-124 < y < 1.2999999999999999e-68 or 2.05e-54 < y < 0.75

   1. Initial program 82.6%

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

      1. times-frac 99.6%

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

      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 56.7%

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

      1. distribute-rgt-in 56.8%

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

      2. *-lft-identity 56.8%

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

   6. Simplified 56.8%

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

   7. Taylor expanded in y around 0 55.2%

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

      1. neg-mul-1 55.2%

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

      2. unsub-neg 55.2%

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

   9. Simplified 55.2%

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

   if 1.2999999999999999e-68 < y < 2.05e-54

   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. associate-*r/ 99.7%

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

      2. *-commutative 99.7%

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

      3. distribute-rgt1-in 49.7%

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

      4. fma-def 99.7%

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

      5. cube-unmult 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 83.6%

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

      1. unpow2 83.6%

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

   6. Simplified 83.6%

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

   if 0.75 < y

   1. Initial program 55.4%

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

      1. associate-*r/ 71.0%

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

      2. *-commutative 71.0%

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

      3. distribute-rgt1-in 66.6%

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

      4. fma-def 70.9%

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

      5. cube-unmult 71.0%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in y around inf 63.0%

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

      1. unpow2 63.0%

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

   6. Simplified 63.0%

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

      1. *-un-lft-identity 63.0%

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

      2. times-frac 75.4%

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

   8. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   10. Simplified 75.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-204}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 12: 60.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y + y \cdot y}\\ \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ y (* y y)))))
   (if (<= x -180000000000.0)
     (/ y (* x x))
     (if (<= x -2.35e-42)
       t_0
       (if (<= x -4e-130)
         (/ (- y) (- x))
         (if (<= x 9.5e-27) t_0 (/ (/ x y) y)))))))

C

double code(double x, double y) {
	double t_0 = x / (y + (y * y));
	double tmp;
	if (x <= -180000000000.0) {
		tmp = y / (x * x);
	} else if (x <= -2.35e-42) {
		tmp = t_0;
	} else if (x <= -4e-130) {
		tmp = -y / -x;
	} else if (x <= 9.5e-27) {
		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 = x / (y + (y * y))
    if (x <= (-180000000000.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-2.35d-42)) then
        tmp = t_0
    else if (x <= (-4d-130)) then
        tmp = -y / -x
    else if (x <= 9.5d-27) 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 = x / (y + (y * y));
	double tmp;
	if (x <= -180000000000.0) {
		tmp = y / (x * x);
	} else if (x <= -2.35e-42) {
		tmp = t_0;
	} else if (x <= -4e-130) {
		tmp = -y / -x;
	} else if (x <= 9.5e-27) {
		tmp = t_0;
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y + (y * y))
	tmp = 0
	if x <= -180000000000.0:
		tmp = y / (x * x)
	elif x <= -2.35e-42:
		tmp = t_0
	elif x <= -4e-130:
		tmp = -y / -x
	elif x <= 9.5e-27:
		tmp = t_0
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y + Float64(y * y)))
	tmp = 0.0
	if (x <= -180000000000.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -2.35e-42)
		tmp = t_0;
	elseif (x <= -4e-130)
		tmp = Float64(Float64(-y) / Float64(-x));
	elseif (x <= 9.5e-27)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y + (y * y));
	tmp = 0.0;
	if (x <= -180000000000.0)
		tmp = y / (x * x);
	elseif (x <= -2.35e-42)
		tmp = t_0;
	elseif (x <= -4e-130)
		tmp = -y / -x;
	elseif (x <= 9.5e-27)
		tmp = t_0;
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -180000000000.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.35e-42], t$95$0, If[LessEqual[x, -4e-130], N[((-y) / (-x)), $MachinePrecision], If[LessEqual[x, 9.5e-27], t$95$0, N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8e11

   1. Initial program 66.0%

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

      1. associate-*r/ 74.6%

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

      2. *-commutative 74.6%

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

      3. distribute-rgt1-in 28.2%

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

      4. fma-def 74.6%

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

      5. cube-unmult 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in x around inf 77.1%

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

      1. unpow2 77.1%

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

   6. Simplified 77.1%

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

   if -1.8e11 < x < -2.35e-42 or -4.0000000000000003e-130 < x < 9.50000000000000037e-27

   1. Initial program 75.5%

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

      1. times-frac 88.9%

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

      2. /-rgt-identity 88.9%

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

      3. associate-/l/ 88.9%

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

      4. *-lft-identity 88.9%

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

      5. associate-+l+ 88.9%

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

   3. Simplified 88.9%

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

   4. Taylor expanded in x around 0 79.3%

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

      1. distribute-rgt-in 79.3%

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

      2. *-lft-identity 79.3%

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

   6. Simplified 79.3%

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

   if -2.35e-42 < x < -4.0000000000000003e-130

   1. Initial program 73.9%

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

      1. times-frac 99.5%

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

      2. /-rgt-identity 99.5%

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

      3. associate-/l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 99.5%

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

      1. +-commutative 99.5%

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

   6. Simplified 99.5%

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

   7. Taylor expanded in y around 0 95.1%

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

   8. Taylor expanded in x around inf 47.6%

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

      1. associate-*l/ 47.7%

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

      2. *-un-lft-identity 47.7%

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

      3. frac-2neg 47.7%

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

   10. Applied egg-rr 47.7%

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

   if 9.50000000000000037e-27 < x

   1. Initial program 70.8%

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

      1. associate-*r/ 79.2%

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

      2. *-commutative 79.2%

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

      3. distribute-rgt1-in 71.9%

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

      4. fma-def 79.2%

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

      5. cube-unmult 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in y around inf 21.3%

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

      1. unpow2 21.3%

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

   6. Simplified 21.3%

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

      1. *-un-lft-identity 21.3%

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

      2. times-frac 23.4%

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

   8. Applied egg-rr 23.4%

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

      1. associate-*l/ 23.4%

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

      2. *-lft-identity 23.4%

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

   10. Simplified 23.4%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -180000000000:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -2.35 \cdot 10^{-42}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-130}:\\ \;\;\;\;\frac{-y}{-x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 13: 60.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{if}\;y \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 10^{+16}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (* x (+ x 1.0)))))
   (if (<= y 3.3e-124)
     t_0
     (if (<= y 1.35e-68)
       (- (/ x y) x)
       (if (<= y 5.2e-44)
         t_0
         (if (<= y 1e+16) (/ x (+ y (* y y))) (/ (/ x y) y)))))))

C

double code(double x, double y) {
	double t_0 = y / (x * (x + 1.0));
	double tmp;
	if (y <= 3.3e-124) {
		tmp = t_0;
	} else if (y <= 1.35e-68) {
		tmp = (x / y) - x;
	} else if (y <= 5.2e-44) {
		tmp = t_0;
	} else if (y <= 1e+16) {
		tmp = x / (y + (y * 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) :: t_0
    real(8) :: tmp
    t_0 = y / (x * (x + 1.0d0))
    if (y <= 3.3d-124) then
        tmp = t_0
    else if (y <= 1.35d-68) then
        tmp = (x / y) - x
    else if (y <= 5.2d-44) then
        tmp = t_0
    else if (y <= 1d+16) then
        tmp = x / (y + (y * y))
    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 + 1.0));
	double tmp;
	if (y <= 3.3e-124) {
		tmp = t_0;
	} else if (y <= 1.35e-68) {
		tmp = (x / y) - x;
	} else if (y <= 5.2e-44) {
		tmp = t_0;
	} else if (y <= 1e+16) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x * (x + 1.0))
	tmp = 0
	if y <= 3.3e-124:
		tmp = t_0
	elif y <= 1.35e-68:
		tmp = (x / y) - x
	elif y <= 5.2e-44:
		tmp = t_0
	elif y <= 1e+16:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x * Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 3.3e-124)
		tmp = t_0;
	elseif (y <= 1.35e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 5.2e-44)
		tmp = t_0;
	elseif (y <= 1e+16)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x * (x + 1.0));
	tmp = 0.0;
	if (y <= 3.3e-124)
		tmp = t_0;
	elseif (y <= 1.35e-68)
		tmp = (x / y) - x;
	elseif (y <= 5.2e-44)
		tmp = t_0;
	elseif (y <= 1e+16)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.3e-124], t$95$0, If[LessEqual[y, 1.35e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 5.2e-44], t$95$0, If[LessEqual[y, 1e+16], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 3.29999999999999984e-124 or 1.3500000000000001e-68 < y < 5.1999999999999996e-44

   1. Initial program 77.0%

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

      1. times-frac 91.4%

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

      2. /-rgt-identity 91.4%

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

      3. associate-/l/ 91.4%

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

      4. *-lft-identity 91.4%

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

      5. associate-+l+ 91.4%

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

   3. Simplified 91.4%

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

   4. Taylor expanded in y around 0 65.0%

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

   if 3.29999999999999984e-124 < y < 1.3500000000000001e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

   9. Simplified 83.6%

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

   if 5.1999999999999996e-44 < y < 1e16

   1. Initial program 93.6%

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

      1. times-frac 99.4%

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

      2. /-rgt-identity 99.4%

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

      3. associate-/l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. distribute-rgt-in 60.7%

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

      2. *-lft-identity 60.7%

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

   6. Simplified 60.7%

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

   if 1e16 < y

   1. Initial program 54.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. distribute-rgt1-in 65.6%

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

      4. fma-def 70.0%

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

      5. cube-unmult 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in y around inf 62.6%

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

      1. unpow2 62.6%

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

   6. Simplified 62.6%

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

      1. *-un-lft-identity 62.6%

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

      2. times-frac 75.4%

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

   8. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   10. Simplified 75.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 10^{+16}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 14: 60.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-124)
   (/ y (+ x (* x x)))
   (if (<= y 1.3e-68)
     (- (/ x y) x)
     (if (<= y 7e-44)
       (/ y (* x (+ x 1.0)))
       (if (<= y 2e+17) (/ x (+ y (* y y))) (/ (/ x y) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = y / (x + (x * x));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 7e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2e+17) {
		tmp = x / (y + (y * 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 <= 4.5d-124) then
        tmp = y / (x + (x * x))
    else if (y <= 1.3d-68) then
        tmp = (x / y) - x
    else if (y <= 7d-44) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 2d+17) then
        tmp = x / (y + (y * 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 <= 4.5e-124) {
		tmp = y / (x + (x * x));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 7e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2e+17) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-124:
		tmp = y / (x + (x * x))
	elif y <= 1.3e-68:
		tmp = (x / y) - x
	elif y <= 7e-44:
		tmp = y / (x * (x + 1.0))
	elif y <= 2e+17:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	elseif (y <= 1.3e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 7e-44)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 2e+17)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = y / (x + (x * x));
	elseif (y <= 1.3e-68)
		tmp = (x / y) - x;
	elseif (y <= 7e-44)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 2e+17)
		tmp = x / (y + (y * y));
	else
		tmp = (x / y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-124], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 7e-44], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2e+17], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. associate-+r+ 90.7%

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

      3. associate-*l/ 82.2%

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

      4. times-frac 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. distribute-lft-in 64.2%

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

      3. *-rgt-identity 64.2%

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

   6. Simplified 64.2%

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

   if 4.4999999999999996e-124 < y < 1.2999999999999999e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

   9. Simplified 83.6%

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

   if 1.2999999999999999e-68 < y < 6.9999999999999995e-44

   1. Initial program 97.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 y around 0 76.5%

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

   if 6.9999999999999995e-44 < y < 2e17

   1. Initial program 93.6%

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

      1. times-frac 99.4%

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

      2. /-rgt-identity 99.4%

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

      3. associate-/l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. distribute-rgt-in 60.7%

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

      2. *-lft-identity 60.7%

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

   6. Simplified 60.7%

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

   if 2e17 < y

   1. Initial program 54.0%

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

      1. associate-*r/ 70.0%

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

      2. *-commutative 70.0%

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

      3. distribute-rgt1-in 65.6%

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

      4. fma-def 70.0%

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

      5. cube-unmult 70.1%

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

   3. Simplified 70.1%

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

   4. Taylor expanded in y around inf 62.6%

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

      1. unpow2 62.6%

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

   6. Simplified 62.6%

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

      1. *-un-lft-identity 62.6%

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

      2. times-frac 75.4%

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

   8. Applied egg-rr 75.4%

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

      1. associate-*l/ 75.5%

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

      2. *-lft-identity 75.5%

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

   10. Simplified 75.5%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y}\\ \end{array} \]

Alternative 15: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-124)
   (/ y (+ x (* x x)))
   (if (<= y 6.1e-68)
     (- (/ x y) x)
     (if (<= y 3.8e-44)
       (/ y (* x (+ x 1.0)))
       (if (<= y 4.5e+15) (/ x (+ y (* y y))) (/ (/ x (+ x y)) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = y / (x + (x * x));
	} else if (y <= 6.1e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.8e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4.5e+15) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (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 <= 4.5d-124) then
        tmp = y / (x + (x * x))
    else if (y <= 6.1d-68) then
        tmp = (x / y) - x
    else if (y <= 3.8d-44) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 4.5d+15) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / (x + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = y / (x + (x * x));
	} else if (y <= 6.1e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.8e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 4.5e+15) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (x + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-124:
		tmp = y / (x + (x * x))
	elif y <= 6.1e-68:
		tmp = (x / y) - x
	elif y <= 3.8e-44:
		tmp = y / (x * (x + 1.0))
	elif y <= 4.5e+15:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / (x + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(y / Float64(x + Float64(x * x)));
	elseif (y <= 6.1e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 3.8e-44)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 4.5e+15)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = y / (x + (x * x));
	elseif (y <= 6.1e-68)
		tmp = (x / y) - x;
	elseif (y <= 3.8e-44)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 4.5e+15)
		tmp = x / (y + (y * y));
	else
		tmp = (x / (x + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-124], N[(y / N[(x + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.1e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 3.8e-44], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+15], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. associate-+r+ 90.7%

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

      3. associate-*l/ 82.2%

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

      4. times-frac 99.9%

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

      5. +-commutative 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+l+ 99.9%

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

      8. +-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 64.1%

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

      1. *-commutative 64.1%

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

      2. distribute-lft-in 64.2%

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

      3. *-rgt-identity 64.2%

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

   6. Simplified 64.2%

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

   if 4.4999999999999996e-124 < y < 6.1e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

   9. Simplified 83.6%

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

   if 6.1e-68 < y < 3.8000000000000001e-44

   1. Initial program 97.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 y around 0 76.5%

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

   if 3.8000000000000001e-44 < y < 4.5e15

   1. Initial program 93.6%

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

      1. times-frac 99.4%

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

      2. /-rgt-identity 99.4%

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

      3. associate-/l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. distribute-rgt-in 60.7%

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

      2. *-lft-identity 60.7%

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

   6. Simplified 60.7%

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

   if 4.5e15 < y

   1. Initial program 54.0%

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

      1. times-frac 79.0%

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

      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 x around 0 71.6%

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

      1. +-commutative 71.6%

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

   6. Simplified 71.6%

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

      1. *-commutative 71.6%

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

      2. clear-num 71.6%

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

      3. associate-/r* 76.4%

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

      4. +-commutative 76.4%

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

      5. +-commutative 76.4%

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

      6. frac-times 76.4%

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

      7. *-un-lft-identity 76.4%

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

      8. +-commutative 76.4%

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

      9. +-commutative 76.4%

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

   8. Applied egg-rr 76.4%

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

   9. Taylor expanded in y around inf 75.9%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{y}{x + x \cdot x}\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \]

Alternative 16: 61.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-124)
   (* (/ 1.0 x) (/ y (+ x 1.0)))
   (if (<= y 1.3e-68)
     (- (/ x y) x)
     (if (<= y 5e-44)
       (/ y (* x (+ x 1.0)))
       (if (<= y 2.7e+15) (/ x (+ y (* y y))) (/ (/ x (+ x y)) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (1.0 / x) * (y / (x + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 5e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2.7e+15) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (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 <= 4.5d-124) then
        tmp = (1.0d0 / x) * (y / (x + 1.0d0))
    else if (y <= 1.3d-68) then
        tmp = (x / y) - x
    else if (y <= 5d-44) then
        tmp = y / (x * (x + 1.0d0))
    else if (y <= 2.7d+15) then
        tmp = x / (y + (y * y))
    else
        tmp = (x / (x + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (1.0 / x) * (y / (x + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 5e-44) {
		tmp = y / (x * (x + 1.0));
	} else if (y <= 2.7e+15) {
		tmp = x / (y + (y * y));
	} else {
		tmp = (x / (x + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-124:
		tmp = (1.0 / x) * (y / (x + 1.0))
	elif y <= 1.3e-68:
		tmp = (x / y) - x
	elif y <= 5e-44:
		tmp = y / (x * (x + 1.0))
	elif y <= 2.7e+15:
		tmp = x / (y + (y * y))
	else:
		tmp = (x / (x + y)) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + 1.0)));
	elseif (y <= 1.3e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 5e-44)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	elseif (y <= 2.7e+15)
		tmp = Float64(x / Float64(y + Float64(y * y)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = (1.0 / x) * (y / (x + 1.0));
	elseif (y <= 1.3e-68)
		tmp = (x / y) - x;
	elseif (y <= 5e-44)
		tmp = y / (x * (x + 1.0));
	elseif (y <= 2.7e+15)
		tmp = x / (y + (y * y));
	else
		tmp = (x / (x + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-124], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 5e-44], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.7e+15], N[(x / N[(y + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

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

      1. times-frac 90.7%

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

      2. /-rgt-identity 90.7%

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

      3. associate-/l/ 90.7%

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

      4. *-lft-identity 90.7%

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

      5. associate-+l+ 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in x around inf 65.7%

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

   5. Taylor expanded in y around 0 65.4%

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

      1. +-commutative 65.4%

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

   7. Simplified 65.4%

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

   if 4.4999999999999996e-124 < y < 1.2999999999999999e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

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

      1. distribute-rgt-in 83.6%

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

      2. *-lft-identity 83.6%

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

   6. Simplified 83.6%

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

   7. Taylor expanded in y around 0 83.6%

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

      1. neg-mul-1 83.6%

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

      2. unsub-neg 83.6%

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

   9. Simplified 83.6%

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

   if 1.2999999999999999e-68 < y < 5.00000000000000039e-44

   1. Initial program 97.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 y around 0 76.5%

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

   if 5.00000000000000039e-44 < y < 2.7e15

   1. Initial program 93.6%

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

      1. times-frac 99.4%

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

      2. /-rgt-identity 99.4%

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

      3. associate-/l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. associate-+l+ 99.4%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 60.5%

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

      1. distribute-rgt-in 60.7%

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

      2. *-lft-identity 60.7%

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

   6. Simplified 60.7%

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

   if 2.7e15 < y

   1. Initial program 54.0%

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

      1. times-frac 79.0%

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

      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 x around 0 71.6%

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

      1. +-commutative 71.6%

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

   6. Simplified 71.6%

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

      1. *-commutative 71.6%

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

      2. clear-num 71.6%

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

      3. associate-/r* 76.4%

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

      4. +-commutative 76.4%

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

      5. +-commutative 76.4%

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

      6. frac-times 76.4%

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

      7. *-un-lft-identity 76.4%

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

      8. +-commutative 76.4%

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

      9. +-commutative 76.4%

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

   8. Applied egg-rr 76.4%

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

   9. Taylor expanded in y around inf 75.9%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\frac{x}{y + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \]

Alternative 17: 61.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-124)
   (* (/ 1.0 x) (/ y (+ x 1.0)))
   (if (<= y 1.55e-68)
     (- (/ x y) x)
     (if (<= y 3.5e-54) (/ y (* x (+ x 1.0))) (/ (/ x (+ x y)) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (1.0 / x) * (y / (x + 1.0));
	} else if (y <= 1.55e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-124) then
        tmp = (1.0d0 / x) * (y / (x + 1.0d0))
    else if (y <= 1.55d-68) then
        tmp = (x / y) - x
    else if (y <= 3.5d-54) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (1.0 / x) * (y / (x + 1.0));
	} else if (y <= 1.55e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-124:
		tmp = (1.0 / x) * (y / (x + 1.0))
	elif y <= 1.55e-68:
		tmp = (x / y) - x
	elif y <= 3.5e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(Float64(1.0 / x) * Float64(y / Float64(x + 1.0)));
	elseif (y <= 1.55e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 3.5e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = (1.0 / x) * (y / (x + 1.0));
	elseif (y <= 1.55e-68)
		tmp = (x / y) - x;
	elseif (y <= 3.5e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-124], N[(N[(1.0 / x), $MachinePrecision] * N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.55e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 3.5e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]

   5. Taylor expanded in y around 0 65.4%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{y}{1 + x}} \]
   6. Step-by-step derivation

      1. +-commutative 65.4%

         \[\leadsto \frac{1}{x} \cdot \frac{y}{\color{blue}{x + 1}} \]

   7. Simplified 65.4%

      \[\leadsto \frac{1}{x} \cdot \color{blue}{\frac{y}{x + 1}} \]

   if 4.4999999999999996e-124 < y < 1.55e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 83.6%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 83.6%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 83.6%

      \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 83.6%

         \[\leadsto \frac{x}{y} + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 83.6%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 1.55e-68 < y < 3.49999999999999982e-54

   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 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      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 y around 0 85.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 3.49999999999999982e-54 < y

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 75.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 75.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 75.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]
   7. Step-by-step derivation

      1. *-commutative 75.2%

         \[\leadsto \color{blue}{\frac{y}{y + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      2. clear-num 75.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + 1}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      3. associate-/r* 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. +-commutative 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \]

      5. +-commutative 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \]

      6. frac-times 73.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + 1}{y} \cdot \left(y + x\right)}} \]

      7. *-un-lft-identity 73.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      8. +-commutative 73.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      9. +-commutative 73.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \color{blue}{\left(x + y\right)}} \]

   8. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \left(x + y\right)}} \]

   9. Taylor expanded in x around 0 70.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 70.5%

          \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   11. Simplified 70.5%

       \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{y}{x + 1}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 18: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e-124)
   (/ (/ y x) (+ x (+ y 1.0)))
   (if (<= y 1.3e-68)
     (- (/ x y) x)
     (if (<= y 3.5e-54) (/ y (* x (+ x 1.0))) (/ (/ x (+ x y)) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.5d-124) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 1.3d-68) then
        tmp = (x / y) - x
    else if (y <= 3.5d-54) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (x + y)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e-124) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 1.3e-68) {
		tmp = (x / y) - x;
	} else if (y <= 3.5e-54) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (x + y)) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e-124:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 1.3e-68:
		tmp = (x / y) - x
	elif y <= 3.5e-54:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (x + y)) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e-124)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 1.3e-68)
		tmp = Float64(Float64(x / y) - x);
	elseif (y <= 3.5e-54)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e-124)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 1.3e-68)
		tmp = (x / y) - x;
	elseif (y <= 3.5e-54)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (x + y)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e-124], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.3e-68], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[y, 3.5e-54], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.4999999999999996e-124

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 65.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 40.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\right)} - 1} \]

      3. frac-times 40.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot y}{x \cdot \left(x + \left(y + 1\right)\right)}}\right)} - 1 \]

      4. *-un-lft-identity 40.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{x \cdot \left(x + \left(y + 1\right)\right)}\right)} - 1 \]

   6. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 65.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\right)\right)} \]

      2. expm1-log1p 65.5%

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}} \]

      3. associate-/r* 65.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + \left(y + 1\right)}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + \left(y + 1\right)}} \]

   if 4.4999999999999996e-124 < y < 1.2999999999999999e-68

   1. Initial program 47.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   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 x around 0 83.6%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 83.6%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 83.6%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 83.6%

      \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 83.6%

         \[\leadsto \frac{x}{y} + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 83.6%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 83.6%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 1.2999999999999999e-68 < y < 3.49999999999999982e-54

   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 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      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 y around 0 85.2%

      \[\leadsto \color{blue}{\frac{y}{\left(1 + x\right) \cdot x}} \]

   if 3.49999999999999982e-54 < y

   1. Initial program 64.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 84.1%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 84.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 75.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 75.2%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 75.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]
   7. Step-by-step derivation

      1. *-commutative 75.2%

         \[\leadsto \color{blue}{\frac{y}{y + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      2. clear-num 75.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + 1}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      3. associate-/r* 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. +-commutative 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \]

      5. +-commutative 73.3%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \]

      6. frac-times 73.3%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + 1}{y} \cdot \left(y + x\right)}} \]

      7. *-un-lft-identity 73.3%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      8. +-commutative 73.3%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      9. +-commutative 73.3%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \color{blue}{\left(x + y\right)}} \]

   8. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \left(x + y\right)}} \]

   9. Taylor expanded in x around 0 70.5%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{1 + y}} \]
   10. Step-by-step derivation

       1. +-commutative 70.5%

          \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]

   11. Simplified 70.5%

       \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y + 1}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y + 1}\\ \end{array} \]

Alternative 19: 66.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{if}\;y \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot t_0\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* (+ x y) (+ x y)))))
   (if (<= y 6e-127)
     (/ (/ y x) (+ x (+ y 1.0)))
     (if (<= y 0.00025)
       (* y t_0)
       (if (<= y 1.15e+148) t_0 (/ (/ x (+ x y)) y))))))

C

double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (y <= 6e-127) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * t_0;
	} else if (y <= 1.15e+148) {
		tmp = t_0;
	} else {
		tmp = (x / (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 = x / ((x + y) * (x + y))
    if (y <= 6d-127) then
        tmp = (y / x) / (x + (y + 1.0d0))
    else if (y <= 0.00025d0) then
        tmp = y * t_0
    else if (y <= 1.15d+148) then
        tmp = t_0
    else
        tmp = (x / (x + y)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / ((x + y) * (x + y));
	double tmp;
	if (y <= 6e-127) {
		tmp = (y / x) / (x + (y + 1.0));
	} else if (y <= 0.00025) {
		tmp = y * t_0;
	} else if (y <= 1.15e+148) {
		tmp = t_0;
	} else {
		tmp = (x / (x + y)) / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / ((x + y) * (x + y))
	tmp = 0
	if y <= 6e-127:
		tmp = (y / x) / (x + (y + 1.0))
	elif y <= 0.00025:
		tmp = y * t_0
	elif y <= 1.15e+148:
		tmp = t_0
	else:
		tmp = (x / (x + y)) / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(Float64(x + y) * Float64(x + y)))
	tmp = 0.0
	if (y <= 6e-127)
		tmp = Float64(Float64(y / x) / Float64(x + Float64(y + 1.0)));
	elseif (y <= 0.00025)
		tmp = Float64(y * t_0);
	elseif (y <= 1.15e+148)
		tmp = t_0;
	else
		tmp = Float64(Float64(x / Float64(x + y)) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / ((x + y) * (x + y));
	tmp = 0.0;
	if (y <= 6e-127)
		tmp = (y / x) / (x + (y + 1.0));
	elseif (y <= 0.00025)
		tmp = y * t_0;
	elseif (y <= 1.15e+148)
		tmp = t_0;
	else
		tmp = (x / (x + y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 6e-127], N[(N[(y / x), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.00025], N[(y * t$95$0), $MachinePrecision], If[LessEqual[y, 1.15e+148], t$95$0, N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < 6.00000000000000017e-127

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around inf 65.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]
   5. Step-by-step derivation

      1. expm1-log1p-u 65.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\right)\right)} \]

      2. expm1-udef 40.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{x} \cdot \frac{y}{x + \left(y + 1\right)}\right)} - 1} \]

      3. frac-times 40.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot y}{x \cdot \left(x + \left(y + 1\right)\right)}}\right)} - 1 \]

      4. *-un-lft-identity 40.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{y}}{x \cdot \left(x + \left(y + 1\right)\right)}\right)} - 1 \]

   6. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 65.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}\right)\right)} \]

      2. expm1-log1p 65.5%

         \[\leadsto \color{blue}{\frac{y}{x \cdot \left(x + \left(y + 1\right)\right)}} \]

      3. associate-/r* 65.7%

         \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + \left(y + 1\right)}} \]

   8. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{\frac{y}{x}}{x + \left(y + 1\right)}} \]

   if 6.00000000000000017e-127 < y < 2.5000000000000001e-4

   1. Initial program 85.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      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 90.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 90.4%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 90.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]

   7. Taylor expanded in y around 0 87.2%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{y} \]

   if 2.5000000000000001e-4 < y < 1.15e148

   1. Initial program 60.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 95.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 95.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 95.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 95.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 95.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in y around inf 76.4%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{1} \]

   if 1.15e148 < y

   1. Initial program 52.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 68.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 68.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 68.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 68.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 68.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 68.5%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 68.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 68.5%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 68.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]
   7. Step-by-step derivation

      1. *-commutative 68.5%

         \[\leadsto \color{blue}{\frac{y}{y + 1} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}} \]

      2. clear-num 68.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{y + 1}{y}}} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \]

      3. associate-/r* 90.0%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \color{blue}{\frac{\frac{x}{x + y}}{x + y}} \]

      4. +-commutative 90.0%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{\color{blue}{y + x}}}{x + y} \]

      5. +-commutative 90.0%

         \[\leadsto \frac{1}{\frac{y + 1}{y}} \cdot \frac{\frac{x}{y + x}}{\color{blue}{y + x}} \]

      6. frac-times 90.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \frac{x}{y + x}}{\frac{y + 1}{y} \cdot \left(y + x\right)}} \]

      7. *-un-lft-identity 90.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{y + x}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      8. +-commutative 90.0%

         \[\leadsto \frac{\frac{x}{\color{blue}{x + y}}}{\frac{y + 1}{y} \cdot \left(y + x\right)} \]

      9. +-commutative 90.0%

         \[\leadsto \frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \color{blue}{\left(x + y\right)}} \]

   8. Applied egg-rr 90.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{x + y}}{\frac{y + 1}{y} \cdot \left(x + y\right)}} \]

   9. Taylor expanded in y around inf 89.9%

      \[\leadsto \frac{\frac{x}{x + y}}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + \left(y + 1\right)}\\ \mathbf{elif}\;y \leq 0.00025:\\ \;\;\;\;y \cdot \frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+148}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(x + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{y}\\ \end{array} \]

Alternative 20: 45.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-124)
   (* y (/ 1.0 x))
   (if (<= y 0.75) (- (/ x y) x) (/ x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-124) {
		tmp = y * (1.0 / x);
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-124) then
        tmp = y * (1.0d0 / x)
    else if (y <= 0.75d0) then
        tmp = (x / y) - x
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-124) {
		tmp = y * (1.0 / x);
	} else if (y <= 0.75) {
		tmp = (x / y) - x;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-124:
		tmp = y * (1.0 / x)
	elif y <= 0.75:
		tmp = (x / y) - x
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-124)
		tmp = Float64(y * Float64(1.0 / x));
	elseif (y <= 0.75)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-124)
		tmp = y * (1.0 / x);
	elseif (y <= 0.75)
		tmp = (x / y) - x;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-124], N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.75], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.99999999999999973e-124

   1. Initial program 75.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 90.7%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 90.7%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 75.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 75.1%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 75.1%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]

   7. Taylor expanded in y around 0 71.5%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{y} \]

   8. Taylor expanded in x around inf 42.2%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot y \]

   if 3.99999999999999973e-124 < y < 0.75

   1. Initial program 85.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 99.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      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 47.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 47.3%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 47.3%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 47.3%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 46.0%

      \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 46.0%

         \[\leadsto \frac{x}{y} + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 46.0%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 46.0%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   if 0.75 < y

   1. Initial program 55.4%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. *-commutative 71.0%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      3. distribute-rgt1-in 66.6%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. fma-def 70.9%

         \[\leadsto x \cdot \frac{y}{\color{blue}{\mathsf{fma}\left(x + y, x + y, \left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)\right)}} \]

      5. cube-unmult 71.0%

         \[\leadsto x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(x + y\right)}^{3}}\right)} \]

   3. Simplified 71.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]

   4. Taylor expanded in y around inf 63.0%

      \[\leadsto \color{blue}{\frac{x}{{y}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 63.0%

         \[\leadsto \frac{x}{\color{blue}{y \cdot y}} \]

   6. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{elif}\;y \leq 0.75:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \]

Alternative 21: 23.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -7.2e-136) (* y (/ 1.0 x)) (- (/ x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-136) {
		tmp = y * (1.0 / x);
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-7.2d-136)) then
        tmp = y * (1.0d0 / x)
    else
        tmp = (x / y) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -7.2e-136) {
		tmp = y * (1.0 / x);
	} else {
		tmp = (x / y) - x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -7.2e-136:
		tmp = y * (1.0 / x)
	else:
		tmp = (x / y) - x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -7.2e-136)
		tmp = Float64(y * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -7.2e-136)
		tmp = y * (1.0 / x);
	else
		tmp = (x / y) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -7.2e-136], N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.1999999999999996e-136

   1. Initial program 70.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 89.5%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      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 x around 0 70.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
   5. Step-by-step derivation

      1. +-commutative 70.9%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

   6. Simplified 70.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]

   7. Taylor expanded in y around 0 65.3%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{y} \]

   8. Taylor expanded in x around inf 35.9%

      \[\leadsto \color{blue}{\frac{1}{x}} \cdot y \]

   if -7.1999999999999996e-136 < x

   1. Initial program 72.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. times-frac 88.6%

         \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

      2. /-rgt-identity 88.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

      3. associate-/l/ 88.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

      4. *-lft-identity 88.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

      5. associate-+l+ 88.6%

         \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

   3. Simplified 88.6%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

   4. Taylor expanded in x around 0 53.8%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   5. Step-by-step derivation

      1. distribute-rgt-in 53.9%

         \[\leadsto \frac{x}{\color{blue}{1 \cdot y + y \cdot y}} \]

      2. *-lft-identity 53.9%

         \[\leadsto \frac{x}{\color{blue}{y} + y \cdot y} \]

   6. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{x}{y + y \cdot y}} \]

   7. Taylor expanded in y around 0 17.9%

      \[\leadsto \color{blue}{\frac{x}{y} + -1 \cdot x} \]
   8. Step-by-step derivation

      1. neg-mul-1 17.9%

         \[\leadsto \frac{x}{y} + \color{blue}{\left(-x\right)} \]

      2. unsub-neg 17.9%

         \[\leadsto \color{blue}{\frac{x}{y} - x} \]

   9. Simplified 17.9%

      \[\leadsto \color{blue}{\frac{x}{y} - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.2 \cdot 10^{-136}:\\ \;\;\;\;y \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - x\\ \end{array} \]

Alternative 22: 26.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (/ 1.0 x)))

C

double code(double x, double y) {
	return y * (1.0 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * (1.0d0 / x)
end function

Java

public static double code(double x, double y) {
	return y * (1.0 / x);
}

Python

def code(x, y):
	return y * (1.0 / x)

Julia

function code(x, y)
	return Float64(y * Float64(1.0 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = y * (1.0 / x);
end

Wolfram

code[x_, y_] := N[(y * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.9%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around 0 76.3%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 + y}} \]
5. Step-by-step derivation

   1. +-commutative 76.3%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{y + 1}} \]

6. Simplified 76.3%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{y + 1}} \]

7. Taylor expanded in y around 0 69.3%

   \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{y} \]

8. Taylor expanded in x around inf 29.7%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot y \]

9. Final simplification 29.7%

   \[\leadsto y \cdot \frac{1}{x} \]

Alternative 23: 4.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 1.0 x))

C

double code(double x, double y) {
	return 1.0 / x;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

public static double code(double x, double y) {
	return 1.0 / x;
}

Python

def code(x, y):
	return 1.0 / x

Julia

function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 71.5%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. times-frac 88.9%

      \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\left(x + y\right) + 1}} \]

   2. /-rgt-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{\frac{y}{\left(x + y\right) + 1}}{1}} \]

   3. associate-/l/ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \color{blue}{\frac{y}{1 \cdot \left(\left(x + y\right) + 1\right)}} \]

   4. *-lft-identity 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{\left(x + y\right) + 1}} \]

   5. associate-+l+ 88.9%

      \[\leadsto \frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 88.9%

   \[\leadsto \color{blue}{\frac{x}{\left(x + y\right) \cdot \left(x + y\right)} \cdot \frac{y}{x + \left(y + 1\right)}} \]

4. Taylor expanded in x around inf 53.8%

   \[\leadsto \color{blue}{\frac{1}{x}} \cdot \frac{y}{x + \left(y + 1\right)} \]

5. Taylor expanded in y around inf 4.2%

   \[\leadsto \color{blue}{\frac{1}{x}} \]

6. Final simplification 4.2%

   \[\leadsto \frac{1}{x} \]

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 2023196 
(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))))