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

Percentage Accurate: 69.2% → 99.8%

Time: 14.0s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.7%

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

   1. *-commutative 64.7%

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

   2. associate-*l* 64.7%

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

   3. times-frac 91.0%

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

   4. +-commutative 91.0%

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

   5. distribute-lft-in 90.9%

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

   6. *-rgt-identity 90.9%

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

   7. pow2 90.9%

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

4. Applied egg-rr 90.9%

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

   1. associate-*r/ 91.0%

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

   2. +-commutative 91.0%

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

   3. +-commutative 91.0%

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

   4. +-commutative 91.0%

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

6. Simplified 91.0%

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

   1. unpow2 91.0%

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

8. Applied egg-rr 91.0%

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

   1. distribute-rgt1-in 91.0%

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

   2. times-frac 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;x \leq -0.034:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.6e+68)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= x -0.034)
     (* x (/ y (* (+ x (+ y 1.0)) (* (+ y x) (+ y x)))))
     (/ (* (/ y (+ y x)) (/ x (+ y 1.0))) (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+68) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (x <= -0.034) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.6d+68)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (x <= (-0.034d0)) then
        tmp = x * (y / ((x + (y + 1.0d0)) * ((y + x) * (y + x))))
    else
        tmp = ((y / (y + x)) * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.6e+68) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (x <= -0.034) {
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.6e+68:
		tmp = (y / (x + 1.0)) / (y + x)
	elif x <= -0.034:
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))))
	else:
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.6e+68)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (x <= -0.034)
		tmp = Float64(x * Float64(y / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) * Float64(y + x)))));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.6e+68)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (x <= -0.034)
		tmp = x * (y / ((x + (y + 1.0)) * ((y + x) * (y + x))));
	else
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.6e+68], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -0.034], N[(x * N[(y / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.6e68

   1. Initial program 55.5%

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

   3. Taylor expanded in x around inf 55.5%

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

   4. Taylor expanded in x around inf 55.5%

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

      1. associate-/l* 74.2%

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

      2. associate-*l* 74.2%

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

      3. +-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. associate-*r/ 55.5%

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

      2. associate-*r* 55.5%

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

      3. *-commutative 55.5%

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

      4. +-commutative 55.5%

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

      5. times-frac 79.8%

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

      6. +-commutative 79.8%

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

      7. associate-*l/ 79.8%

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

      8. times-frac 85.6%

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

      9. *-inverses 85.6%

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

   8. Simplified 85.6%

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

   if -5.6e68 < x < -0.034000000000000002

   1. Initial program 65.2%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   if -0.034000000000000002 < x

   1. Initial program 67.0%

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

   3. Taylor expanded in x around 0 64.0%

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

      1. *-un-lft-identity 64.0%

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

      2. associate-*l* 64.0%

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

      3. +-commutative 64.0%

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

      4. times-frac 63.9%

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

      5. *-commutative 63.9%

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

      6. +-commutative 63.9%

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

   5. Applied egg-rr 63.9%

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

      1. +-commutative 63.9%

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

      2. *-commutative 63.9%

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

      3. times-frac 84.3%

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

      4. +-commutative 84.3%

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

   7. Simplified 84.3%

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

      1. *-commutative 84.3%

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

      2. +-commutative 84.3%

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

      3. div-inv 84.5%

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

      4. frac-times 64.0%

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

      5. *-commutative 64.0%

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

      6. +-commutative 64.0%

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

      7. times-frac 84.5%

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

   9. Applied egg-rr 84.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;x \leq -0.034:\\ \;\;\;\;x \cdot \frac{y}{\left(x + \left(y + 1\right)\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(y \cdot \frac{x}{y + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -3.5e-12)
   (/ (/ y (+ x 1.0)) (+ y x))
   (if (<= x 9.5e-269)
     (* (/ 1.0 (+ y x)) (* y (/ x (+ y x))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-12) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (x <= 9.5e-269) {
		tmp = (1.0 / (y + x)) * (y * (x / (y + x)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.5d-12)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else if (x <= 9.5d-269) then
        tmp = (1.0d0 / (y + x)) * (y * (x / (y + x)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.5e-12) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else if (x <= 9.5e-269) {
		tmp = (1.0 / (y + x)) * (y * (x / (y + x)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.5e-12:
		tmp = (y / (x + 1.0)) / (y + x)
	elif x <= 9.5e-269:
		tmp = (1.0 / (y + x)) * (y * (x / (y + x)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.5e-12)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	elseif (x <= 9.5e-269)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y * Float64(x / Float64(y + x))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.5e-12)
		tmp = (y / (x + 1.0)) / (y + x);
	elseif (x <= 9.5e-269)
		tmp = (1.0 / (y + x)) * (y * (x / (y + x)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.5e-12], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.5e-269], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.5e-12

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 57.2%

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

   4. Taylor expanded in x around inf 52.8%

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

      1. associate-/l* 69.7%

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

      2. associate-*l* 69.8%

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

      3. +-commutative 69.8%

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

   6. Applied egg-rr 69.8%

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

      1. associate-*r/ 52.8%

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

      2. associate-*r* 52.8%

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

      3. *-commutative 52.8%

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

      4. +-commutative 52.8%

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

      5. times-frac 75.0%

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

      6. +-commutative 75.0%

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

      7. associate-*l/ 75.1%

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

      8. times-frac 75.0%

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

      9. *-inverses 75.0%

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

   8. Simplified 75.0%

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

   if -3.5e-12 < x < 9.5000000000000006e-269

   1. Initial program 63.9%

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

   3. Taylor expanded in x around 0 63.9%

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

      1. *-un-lft-identity 63.9%

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

      2. associate-*l* 63.9%

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

      3. +-commutative 63.9%

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

      4. times-frac 72.4%

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

      5. *-commutative 72.4%

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

      6. +-commutative 72.4%

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

   5. Applied egg-rr 72.4%

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

      1. +-commutative 72.4%

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

      2. *-commutative 72.4%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in y around 0 79.1%

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

   if 9.5000000000000006e-269 < x

   1. Initial program 68.4%

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

      1. associate-/l* 80.6%

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

      2. associate-+l+ 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in x around 0 43.7%

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

      1. +-commutative 43.7%

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

   7. Simplified 43.7%

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

      1. *-un-lft-identity 43.7%

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

      2. *-commutative 43.7%

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

      3. times-frac 48.5%

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

   9. Applied egg-rr 48.5%

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

       1. *-commutative 48.5%

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

   11. Simplified 48.5%

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

       1. un-div-inv 48.6%

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

   13. Applied egg-rr 48.6%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(y \cdot \frac{x}{y + x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) + \left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-119)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.8e+154)
     (/ x (+ (+ y x) (* (+ y x) (+ y x))))
     (* (/ x (+ y 1.0)) (/ 1.0 (+ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-119) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+154) {
		tmp = x / ((y + x) + ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) * (1.0 / (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-119) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.8d+154) then
        tmp = x / ((y + x) + ((y + x) * (y + x)))
    else
        tmp = (x / (y + 1.0d0)) * (1.0d0 / (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-119) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+154) {
		tmp = x / ((y + x) + ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + 1.0)) * (1.0 / (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-119:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.8e+154:
		tmp = x / ((y + x) + ((y + x) * (y + x)))
	else:
		tmp = (x / (y + 1.0)) * (1.0 / (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-119)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.8e+154)
		tmp = Float64(x / Float64(Float64(y + x) + Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) * Float64(1.0 / Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-119)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.8e+154)
		tmp = x / ((y + x) + ((y + x) * (y + x)));
	else
		tmp = (x / (y + 1.0)) * (1.0 / (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-119], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+154], N[(x / N[(N[(y + x), $MachinePrecision] + N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.1e-119

   1. Initial program 66.6%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around 0 55.1%

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

      1. associate-/r* 55.8%

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

      2. +-commutative 55.8%

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

   7. Simplified 55.8%

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

   if 2.1e-119 < y < 1.8e154

   1. Initial program 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l* 65.7%

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

      3. times-frac 94.0%

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

      4. +-commutative 94.0%

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

      5. distribute-lft-in 93.9%

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

      6. *-rgt-identity 93.9%

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

      7. pow2 93.9%

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

   4. Applied egg-rr 93.9%

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

      1. associate-*r/ 94.0%

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

      2. +-commutative 94.0%

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

      3. +-commutative 94.0%

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

      4. +-commutative 94.0%

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

   6. Simplified 94.0%

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

      1. unpow2 94.0%

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

   8. Applied egg-rr 94.0%

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

   9. Taylor expanded in y around inf 79.8%

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

   if 1.8e154 < y

   1. Initial program 51.7%

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

   3. Taylor expanded in x around 0 51.7%

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

      1. *-un-lft-identity 51.7%

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

      2. associate-*l* 51.7%

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

      3. +-commutative 51.7%

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

      4. times-frac 51.7%

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

      5. *-commutative 51.7%

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

      6. +-commutative 51.7%

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

   5. Applied egg-rr 51.7%

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

      1. +-commutative 51.7%

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

      2. *-commutative 51.7%

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

      3. times-frac 76.3%

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

      4. +-commutative 76.3%

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

   7. Simplified 76.3%

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

   8. Taylor expanded in x around 0 76.1%

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

      1. +-commutative 76.1%

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

   10. Simplified 76.1%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{x}{\left(y + x\right) + \left(y + x\right) \cdot \left(y + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y + 1} \cdot \frac{1}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -230000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -230000.0)
   (/ (/ y (+ x 1.0)) (+ y x))
   (/ (* (/ y (+ y x)) (/ x (+ y 1.0))) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -230000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-230000.0d0)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = ((y / (y + x)) * (x / (y + 1.0d0))) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -230000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -230000.0:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -230000.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(y + 1.0))) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -230000.0)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = ((y / (y + x)) * (x / (y + 1.0))) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -230000.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3e5

   1. Initial program 57.8%

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

   3. Taylor expanded in x around inf 57.8%

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

   4. Taylor expanded in x around inf 57.7%

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

      1. associate-/l* 76.3%

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

      2. associate-*l* 76.3%

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

      3. +-commutative 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-*r/ 57.7%

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

      2. associate-*r* 57.7%

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

      3. *-commutative 57.7%

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

      4. +-commutative 57.7%

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

      5. times-frac 82.1%

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

      6. +-commutative 82.1%

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

      7. associate-*l/ 82.2%

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

      8. times-frac 82.1%

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

      9. *-inverses 82.1%

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

   8. Simplified 82.1%

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

   if -2.3e5 < x

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0 63.9%

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

      1. *-un-lft-identity 63.9%

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

      2. associate-*l* 63.9%

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

      3. +-commutative 63.9%

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

      4. times-frac 64.3%

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

      5. *-commutative 64.3%

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

      6. +-commutative 64.3%

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

   5. Applied egg-rr 64.3%

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

      1. +-commutative 64.3%

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

      2. *-commutative 64.3%

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

      3. times-frac 84.5%

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

      4. +-commutative 84.5%

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

   7. Simplified 84.5%

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

      1. *-commutative 84.5%

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

      2. +-commutative 84.5%

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

      3. div-inv 84.6%

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

      4. frac-times 64.3%

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

      5. *-commutative 64.3%

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

      6. +-commutative 64.3%

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

      7. times-frac 84.6%

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

   9. Applied egg-rr 84.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -230000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -28500000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -28500000.0)
   (/ (/ y (+ x 1.0)) (+ y x))
   (* (/ y (+ y x)) (/ (/ x (+ y x)) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -28500000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (y / (y + x)) * ((x / (y + x)) / (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-28500000.0d0)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (y / (y + x)) * ((x / (y + x)) / (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -28500000.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (y / (y + x)) * ((x / (y + x)) / (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -28500000.0:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (y / (y + x)) * ((x / (y + x)) / (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -28500000.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -28500000.0)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (y / (y + x)) * ((x / (y + x)) / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -28500000.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.85e7

   1. Initial program 57.8%

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

   3. Taylor expanded in x around inf 57.8%

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

   4. Taylor expanded in x around inf 57.7%

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

      1. associate-/l* 76.3%

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

      2. associate-*l* 76.3%

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

      3. +-commutative 76.3%

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

   6. Applied egg-rr 76.3%

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

      1. associate-*r/ 57.7%

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

      2. associate-*r* 57.7%

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

      3. *-commutative 57.7%

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

      4. +-commutative 57.7%

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

      5. times-frac 82.1%

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

      6. +-commutative 82.1%

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

      7. associate-*l/ 82.2%

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

      8. times-frac 82.1%

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

      9. *-inverses 82.1%

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

   8. Simplified 82.1%

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

   if -2.85e7 < x

   1. Initial program 66.9%

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

   3. Taylor expanded in x around 0 63.9%

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

      1. *-commutative 63.9%

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

      2. associate-*l* 63.9%

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

      3. +-commutative 63.9%

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

      4. times-frac 83.4%

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

      5. +-commutative 83.4%

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

   5. Applied egg-rr 83.4%

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

      1. +-commutative 83.4%

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

      2. associate-/r* 84.6%

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

      3. +-commutative 84.6%

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

   7. Simplified 84.6%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -28500000:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -18000 \lor \neg \left(y \leq 1.95 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y -18000.0) (not (<= y 1.95e-118)))
   (/ x (* y (+ x 1.0)))
   (/ 1.0 (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if ((y <= -18000.0) || !(y <= 1.95e-118)) {
		tmp = x / (y * (x + 1.0));
	} else {
		tmp = 1.0 / (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 <= (-18000.0d0)) .or. (.not. (y <= 1.95d-118))) then
        tmp = x / (y * (x + 1.0d0))
    else
        tmp = 1.0d0 / (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= -18000.0) || !(y <= 1.95e-118)) {
		tmp = x / (y * (x + 1.0));
	} else {
		tmp = 1.0 / (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= -18000.0) or not (y <= 1.95e-118):
		tmp = x / (y * (x + 1.0))
	else:
		tmp = 1.0 / (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= -18000.0) || !(y <= 1.95e-118))
		tmp = Float64(x / Float64(y * Float64(x + 1.0)));
	else
		tmp = Float64(1.0 / Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= -18000.0) || ~((y <= 1.95e-118)))
		tmp = x / (y * (x + 1.0));
	else
		tmp = 1.0 / (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, -18000.0], N[Not[LessEqual[y, 1.95e-118]], $MachinePrecision]], N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -18000 or 1.95e-118 < y

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 48.9%

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

   4. Taylor expanded in y around inf 47.2%

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

      1. +-commutative 47.2%

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

   6. Simplified 47.2%

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

   if -18000 < y < 1.95e-118

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0 58.9%

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

   4. Taylor expanded in y around 0 56.7%

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

      1. clear-num 57.0%

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

      2. inv-pow 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. unpow-1 57.0%

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

   8. Simplified 57.0%

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

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -18000 \lor \neg \left(y \leq 1.95 \cdot 10^{-118}\right):\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -11000:\\ \;\;\;\;\frac{x}{y \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -11000.0)
   (/ x (* y (+ x 1.0)))
   (if (<= y 2.1e-118) (/ 1.0 (/ x y)) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = x / (y * (x + 1.0));
	} else if (y <= 2.1e-118) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= (-11000.0d0)) then
        tmp = x / (y * (x + 1.0d0))
    else if (y <= 2.1d-118) then
        tmp = 1.0d0 / (x / y)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -11000.0) {
		tmp = x / (y * (x + 1.0));
	} else if (y <= 2.1e-118) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -11000.0:
		tmp = x / (y * (x + 1.0))
	elif y <= 2.1e-118:
		tmp = 1.0 / (x / y)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -11000.0)
		tmp = Float64(x / Float64(y * Float64(x + 1.0)));
	elseif (y <= 2.1e-118)
		tmp = Float64(1.0 / Float64(x / y));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -11000.0)
		tmp = x / (y * (x + 1.0));
	elseif (y <= 2.1e-118)
		tmp = 1.0 / (x / y);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -11000.0], N[(x / N[(y * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.1e-118], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -11000

   1. Initial program 56.6%

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

   3. Taylor expanded in x around inf 42.2%

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

   4. Taylor expanded in y around inf 40.2%

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

      1. +-commutative 40.2%

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

   6. Simplified 40.2%

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

   if -11000 < y < 2.1e-118

   1. Initial program 74.4%

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

   3. Taylor expanded in x around 0 58.9%

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

   4. Taylor expanded in y around 0 56.7%

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

      1. clear-num 57.0%

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

      2. inv-pow 57.0%

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

   6. Applied egg-rr 57.0%

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

      1. unpow-1 57.0%

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

   8. Simplified 57.0%

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

   if 2.1e-118 < y

   1. Initial program 60.8%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. +-commutative 61.7%

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

   7. Simplified 61.7%

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

Alternative 9: 62.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1450:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1450.0) (/ (/ y (+ x 1.0)) (+ y x)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1450.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1450.0d0)) then
        tmp = (y / (x + 1.0d0)) / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1450.0) {
		tmp = (y / (x + 1.0)) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1450.0:
		tmp = (y / (x + 1.0)) / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1450.0)
		tmp = Float64(Float64(y / Float64(x + 1.0)) / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1450.0)
		tmp = (y / (x + 1.0)) / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1450.0], N[(N[(y / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1450

   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. Add Preprocessing

   3. Taylor expanded in x around inf 58.4%

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

   4. Taylor expanded in x around inf 56.9%

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

      1. associate-/l* 75.1%

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

      2. associate-*l* 75.2%

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

      3. +-commutative 75.2%

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

   6. Applied egg-rr 75.2%

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

      1. associate-*r/ 56.8%

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

      2. associate-*r* 56.9%

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

      3. *-commutative 56.9%

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

      4. +-commutative 56.9%

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

      5. times-frac 80.8%

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

      6. +-commutative 80.8%

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

      7. associate-*l/ 80.9%

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

      8. times-frac 80.8%

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

      9. *-inverses 80.8%

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

   8. Simplified 80.8%

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

   if -1450 < x

   1. Initial program 66.7%

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

      1. associate-/l* 77.0%

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

      2. associate-+l+ 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in x around 0 56.5%

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

      1. +-commutative 56.5%

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

   7. Simplified 56.5%

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

      1. *-un-lft-identity 56.5%

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

      2. *-commutative 56.5%

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

      3. times-frac 59.5%

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

   9. Applied egg-rr 59.5%

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

       1. *-commutative 59.5%

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

   11. Simplified 59.5%

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

       1. un-div-inv 59.6%

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

   13. Applied egg-rr 59.6%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1450:\\ \;\;\;\;\frac{\frac{y}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -500000.0) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -500000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-500000.0d0)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -500000.0) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -500000.0:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -500000.0)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -500000.0)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -500000.0], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5e5

   1. Initial program 57.8%

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

      1. associate-/l* 76.3%

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

      2. associate-+l+ 76.3%

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

   3. Simplified 76.3%

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

   5. Taylor expanded in y around 0 75.7%

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

      1. associate-/r* 81.7%

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

      2. +-commutative 81.7%

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

   7. Simplified 81.7%

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

   if -5e5 < x

   1. Initial program 66.9%

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

      1. associate-/l* 77.1%

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

      2. associate-+l+ 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in x around 0 56.7%

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

      1. +-commutative 56.7%

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

   7. Simplified 56.7%

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

      1. *-un-lft-identity 56.7%

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

      2. *-commutative 56.7%

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

      3. times-frac 59.7%

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

   9. Applied egg-rr 59.7%

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

       1. *-commutative 59.7%

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

   11. Simplified 59.7%

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

       1. un-div-inv 59.8%

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

   13. Applied egg-rr 59.8%

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

Alternative 11: 60.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-118) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-118) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-118:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-118)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-118)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-118], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e-118

   1. Initial program 66.6%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around 0 55.1%

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

   if 2.1e-118 < y

   1. Initial program 60.8%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. +-commutative 61.7%

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

   7. Simplified 61.7%

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

      1. *-un-lft-identity 61.7%

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

      2. *-commutative 61.7%

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

      3. times-frac 62.0%

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

   9. Applied egg-rr 62.0%

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

       1. *-commutative 62.0%

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

   11. Simplified 62.0%

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

       1. un-div-inv 62.1%

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

   13. Applied egg-rr 62.1%

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

4. Final simplification 57.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.1e-118) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.1d-118) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.1e-118) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.1e-118:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.1e-118)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.1e-118)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.1e-118], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1e-118

   1. Initial program 66.6%

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

      1. associate-/l* 75.7%

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

      2. associate-+l+ 75.7%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in y around 0 55.1%

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

   if 2.1e-118 < y

   1. Initial program 60.8%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. +-commutative 61.7%

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

   7. Simplified 61.7%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-118) (/ 1.0 (/ x y)) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-118) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-118) then
        tmp = 1.0d0 / (x / y)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-118) {
		tmp = 1.0 / (x / y);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-118:
		tmp = 1.0 / (x / y)
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-118)
		tmp = Float64(1.0 / Float64(x / y));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-118)
		tmp = 1.0 / (x / y);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-118], N[(1.0 / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999997e-118

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0 56.8%

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

   4. Taylor expanded in y around 0 32.9%

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

      1. clear-num 33.1%

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

      2. inv-pow 33.1%

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

   6. Applied egg-rr 33.1%

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

      1. unpow-1 33.1%

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

   8. Simplified 33.1%

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

   if 1.99999999999999997e-118 < y

   1. Initial program 60.8%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. +-commutative 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in y around 0 38.5%

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

Alternative 14: 34.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e-118) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e-118) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2d-118) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2e-118) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e-118:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e-118)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e-118)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e-118], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999997e-118

   1. Initial program 66.6%

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

   3. Taylor expanded in x around 0 56.8%

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

   4. Taylor expanded in y around 0 32.9%

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

   if 1.99999999999999997e-118 < y

   1. Initial program 60.8%

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

      1. associate-/l* 79.4%

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

      2. associate-+l+ 79.4%

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

   3. Simplified 79.4%

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

   5. Taylor expanded in x around 0 61.7%

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

      1. +-commutative 61.7%

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

   7. Simplified 61.7%

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

   8. Taylor expanded in y around 0 38.5%

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

Alternative 15: 26.5% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.7%

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

   1. associate-/l* 76.9%

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

   2. associate-+l+ 76.9%

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

3. Simplified 76.9%

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

5. Taylor expanded in x around 0 46.9%

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

   1. +-commutative 46.9%

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

7. Simplified 46.9%

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

8. Taylor expanded in y around 0 26.2%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
9. Add Preprocessing

Alternative 16: 3.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 64.7%

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

3. Taylor expanded in x around inf 58.5%

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

4. Taylor expanded in x around inf 34.2%

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

5. Taylor expanded in x around 0 3.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 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 2024170 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

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