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

Percentage Accurate: 68.8% → 99.8%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.8% 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

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

FPCore

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

C

double code(double x, double y) {
	return ((x / (x + y)) * (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 / (x + y)) * (y / (x + y))) / (x + (y + 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

   1. times-frac 89.5%

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

   2. associate-*r/ 89.5%

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

   3. pow2 89.5%

      \[\leadsto \frac{\frac{x}{\color{blue}{{\left(x + y\right)}^{2}}} \cdot y}{\left(x + y\right) + 1} \]

   4. associate-+l+ 89.5%

      \[\leadsto \frac{\frac{x}{{\left(x + y\right)}^{2}} \cdot y}{\color{blue}{x + \left(y + 1\right)}} \]

4. Applied egg-rr 89.5%

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

   1. pow2 89.5%

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

   2. associate-*l/ 73.7%

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

   3. times-frac 99.8%

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

   4. +-commutative 99.8%

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

   5. +-commutative 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 67.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.5e+70)
   (* (/ y (+ x y)) (/ 1.0 x))
   (if (<= x -6.8e-161)
     (/ (* x y) (* (* (+ x y) (+ x y)) (+ 1.0 (+ x y))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.5e+70) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -6.8e-161) {
		tmp = (x * y) / (((x + y) * (x + y)) * (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 (x <= (-5.5d+70)) then
        tmp = (y / (x + y)) * (1.0d0 / x)
    else if (x <= (-6.8d-161)) then
        tmp = (x * y) / (((x + y) * (x + y)) * (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 (x <= -5.5e+70) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -6.8e-161) {
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + (x + y)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.5e+70:
		tmp = (y / (x + y)) * (1.0 / x)
	elif x <= -6.8e-161:
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + (x + y)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.5e+70)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / x));
	elseif (x <= -6.8e-161)
		tmp = Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(1.0 + Float64(x + y))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.5e+70)
		tmp = (y / (x + y)) * (1.0 / x);
	elseif (x <= -6.8e-161)
		tmp = (x * y) / (((x + y) * (x + y)) * (1.0 + (x + y)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.5e+70], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.8e-161], N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x + y), $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 < -5.49999999999999986e70

   1. Initial program 39.7%

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

      1. *-commutative 39.7%

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

      2. associate-*l* 39.7%

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

      3. times-frac 81.7%

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

      4. +-commutative 81.7%

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

      5. distribute-lft-in 81.7%

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

      6. *-rgt-identity 81.7%

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

      7. pow2 81.7%

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

   4. Applied egg-rr 81.7%

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

   5. Taylor expanded in x around inf 76.1%

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

   if -5.49999999999999986e70 < x < -6.79999999999999964e-161

   1. Initial program 90.8%

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

   if -6.79999999999999964e-161 < x

   1. Initial program 69.1%

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

   3. Taylor expanded in x around 0 51.7%

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

      1. div-inv 51.7%

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

      2. +-commutative 51.7%

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

   5. Applied egg-rr 51.7%

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

      1. associate-*r/ 51.7%

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

      2. *-rgt-identity 51.7%

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

      3. associate-/r* 54.0%

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

   7. Simplified 54.0%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{+70}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(1 + \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6 \cdot 10^{+164}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x + \left(y + \left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -6e+164)
   (* (/ y (+ x y)) (/ 1.0 x))
   (if (<= x -5.05e-138)
     (/ y (+ x (+ y (* (+ x y) (+ x y)))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6e+164) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / (x + (y + ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-6d+164)) then
        tmp = (y / (x + y)) * (1.0d0 / x)
    else if (x <= (-5.05d-138)) then
        tmp = y / (x + (y + ((x + y) * (x + y))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6e+164) {
		tmp = (y / (x + y)) * (1.0 / x);
	} else if (x <= -5.05e-138) {
		tmp = y / (x + (y + ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6e+164:
		tmp = (y / (x + y)) * (1.0 / x)
	elif x <= -5.05e-138:
		tmp = y / (x + (y + ((x + y) * (x + y))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6e+164)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(1.0 / x));
	elseif (x <= -5.05e-138)
		tmp = Float64(y / Float64(x + Float64(y + Float64(Float64(x + y) * Float64(x + y)))));
	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 <= -6e+164)
		tmp = (y / (x + y)) * (1.0 / x);
	elseif (x <= -5.05e-138)
		tmp = y / (x + (y + ((x + y) * (x + y))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6e+164], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -5.05e-138], N[(y / N[(x + N[(y + N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $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 < -6.00000000000000001e164

   1. Initial program 42.1%

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

      1. *-commutative 42.1%

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

      2. associate-*l* 42.1%

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

      3. times-frac 72.2%

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

      4. +-commutative 72.2%

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

      5. distribute-lft-in 72.2%

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

      6. *-rgt-identity 72.2%

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

      7. pow2 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in x around inf 87.0%

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

   if -6.00000000000000001e164 < x < -5.0499999999999997e-138

   1. Initial program 77.2%

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

      1. *-commutative 77.2%

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

      2. associate-*l* 77.2%

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

      3. times-frac 98.4%

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

      4. +-commutative 98.4%

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

      5. distribute-lft-in 98.4%

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

      6. *-rgt-identity 98.4%

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

      7. pow2 98.4%

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

   4. Applied egg-rr 98.4%

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

      1. associate-*r/ 98.4%

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

      2. associate-+l+ 98.4%

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

   6. Applied egg-rr 98.4%

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

      1. unpow2 98.4%

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

   8. Applied egg-rr 98.4%

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

   9. Taylor expanded in y around 0 73.9%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 53.5%

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

      1. div-inv 53.4%

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

      2. +-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. associate-/r* 55.7%

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

   7. Simplified 55.7%

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

Alternative 4: 63.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{x + \left(y + \left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e-175)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.8e+155)
     (/ x (+ x (+ y (* (+ x y) (+ x y)))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-175) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+155) {
		tmp = x / (x + (y + ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1d-175) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.8d+155) then
        tmp = x / (x + (y + ((x + y) * (x + y))))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e-175) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.8e+155) {
		tmp = x / (x + (y + ((x + y) * (x + y))));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-175:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.8e+155:
		tmp = x / (x + (y + ((x + y) * (x + y))))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-175)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.8e+155)
		tmp = Float64(x / Float64(x + Float64(y + Float64(Float64(x + y) * Float64(x + y)))));
	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 <= 1e-175)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.8e+155)
		tmp = x / (x + (y + ((x + y) * (x + y))));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-175], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e+155], N[(x / N[(x + N[(y + N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1e-175

   1. Initial program 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-*l* 69.4%

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

      3. times-frac 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-lft-in 96.6%

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

      6. *-rgt-identity 96.6%

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

      7. pow2 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. associate-*r/ 96.6%

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

      2. associate-+l+ 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in y around 0 58.7%

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

      1. *-rgt-identity 58.7%

         \[\leadsto \frac{y}{\color{blue}{x \cdot 1} + {x}^{2}} \]

      2. unpow2 58.7%

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

      3. distribute-lft-in 58.7%

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

      4. associate-/r* 60.2%

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

   9. Simplified 60.2%

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

   if 1e-175 < y < 1.80000000000000004e155

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      2. associate-*l* 68.6%

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

      3. times-frac 92.2%

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

      4. +-commutative 92.2%

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

      5. distribute-lft-in 92.2%

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

      6. *-rgt-identity 92.2%

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

      7. pow2 92.2%

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

   4. Applied egg-rr 92.2%

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

      1. associate-*r/ 92.2%

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

      2. associate-+l+ 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. unpow2 92.2%

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

   8. Applied egg-rr 92.2%

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

   9. Taylor expanded in y around inf 65.0%

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

   if 1.80000000000000004e155 < y

   1. Initial program 67.1%

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

   3. Taylor expanded in x around 0 82.3%

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

      1. div-inv 82.3%

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

      2. +-commutative 82.3%

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

   5. Applied egg-rr 82.3%

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

      1. associate-*r/ 82.3%

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

      2. *-rgt-identity 82.3%

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

      3. associate-/r* 96.8%

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

   7. Simplified 96.8%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-175}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+155}:\\ \;\;\;\;\frac{x}{x + \left(y + \left(x + y\right) \cdot \left(x + y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.05e-138) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.05d-138)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.05e-138:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.05e-138)
		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 <= -5.05e-138)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.05e-138], 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 < -5.0499999999999997e-138

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.4%

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

      3. times-frac 90.3%

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

      4. +-commutative 90.3%

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

      5. distribute-lft-in 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. pow2 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

   6. Applied egg-rr 90.3%

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

   7. Taylor expanded in y around 0 55.3%

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

      1. *-rgt-identity 55.3%

         \[\leadsto \frac{y}{\color{blue}{x \cdot 1} + {x}^{2}} \]

      2. unpow2 55.3%

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

      3. distribute-lft-in 55.4%

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

      4. associate-/r* 59.8%

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

   9. Simplified 59.8%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 53.5%

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

      1. div-inv 53.4%

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

      2. +-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. associate-/r* 55.7%

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

   7. Simplified 55.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.05e-138], 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 x < -5.0499999999999997e-138

   1. Initial program 66.4%

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

   3. Taylor expanded in y around 0 55.4%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 53.5%

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

      1. div-inv 53.4%

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

      2. +-commutative 53.4%

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

   5. Applied egg-rr 53.4%

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

      1. associate-*r/ 53.5%

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

      2. *-rgt-identity 53.5%

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

      3. associate-/r* 55.7%

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

   7. Simplified 55.7%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.05e-138) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.05d-138)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.05e-138:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.05e-138], 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 x < -5.0499999999999997e-138

   1. Initial program 66.4%

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

   3. Taylor expanded in y around 0 55.4%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 53.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-183) (/ y x) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-183) {
		tmp = 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 (y <= 1.9d-183) then
        tmp = 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 (y <= 1.9e-183) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-183:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-183)
		tmp = Float64(y / x);
	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 <= 1.9e-183)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-183], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8999999999999998e-183

   1. Initial program 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*l* 69.8%

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

      3. times-frac 96.6%

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

      4. +-commutative 96.6%

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

      5. distribute-lft-in 96.6%

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

      6. *-rgt-identity 96.6%

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

      7. pow2 96.6%

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

   4. Applied egg-rr 96.6%

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

      1. associate-*r/ 96.6%

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

      2. associate-+l+ 96.6%

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

   6. Applied egg-rr 96.6%

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

   7. Taylor expanded in y around 0 59.1%

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

      1. *-rgt-identity 59.1%

         \[\leadsto \frac{y}{\color{blue}{x \cdot 1} + {x}^{2}} \]

      2. unpow2 59.1%

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

      3. distribute-lft-in 59.1%

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

      4. associate-/r* 60.5%

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

   9. Simplified 60.5%

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

   10. Taylor expanded in x around 0 40.0%

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

   if 1.8999999999999998e-183 < y

   1. Initial program 67.4%

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

   3. Taylor expanded in x around 0 57.9%

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

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-183}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.05 \cdot 10^{-138}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -5.05e-138) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.05e-138) {
		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 (x <= (-5.05d-138)) 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 (x <= -5.05e-138) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.05e-138:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.05e-138)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.05e-138)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.05e-138], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.0499999999999997e-138

   1. Initial program 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-*l* 66.4%

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

      3. times-frac 90.3%

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

      4. +-commutative 90.3%

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

      5. distribute-lft-in 90.3%

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

      6. *-rgt-identity 90.3%

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

      7. pow2 90.3%

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

   4. Applied egg-rr 90.3%

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

      1. associate-*r/ 90.3%

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

      2. associate-+l+ 90.3%

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

   6. Applied egg-rr 90.3%

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

   7. Taylor expanded in y around 0 55.3%

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

      1. *-rgt-identity 55.3%

         \[\leadsto \frac{y}{\color{blue}{x \cdot 1} + {x}^{2}} \]

      2. unpow2 55.3%

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

      3. distribute-lft-in 55.4%

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

      4. associate-/r* 59.8%

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

   9. Simplified 59.8%

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

   10. Taylor expanded in x around 0 28.8%

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

   if -5.0499999999999997e-138 < x

   1. Initial program 70.2%

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

   3. Taylor expanded in x around 0 53.5%

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

   4. Taylor expanded in y around 0 35.8%

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

Alternative 10: 27.4% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.8%

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

3. Taylor expanded in x around 0 47.7%

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

4. Taylor expanded in y around 0 25.6%

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

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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