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

Percentage Accurate: 69.3% → 99.7%

Time: 46.0s

Alternatives: 19

Speedup: 0.9×

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: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. associate-+r+ 72.0%

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

   2. *-un-lft-identity 72.0%

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

   3. associate-*l* 72.0%

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

   4. times-frac 78.2%

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

   5. +-commutative 78.2%

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

   6. *-commutative 78.2%

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

   7. +-commutative 78.2%

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

   8. +-commutative 78.2%

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

   9. associate-+l+ 78.2%

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

4. Applied egg-rr 78.2%

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

   1. *-commutative 78.2%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 84.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y + x}\\ \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{t\_0}{\frac{y + x}{x}}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{x + \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ y x))))
   (if (<= x -1.55e+97)
     (/ (/ t_0 (/ (+ y x) x)) (/ x y))
     (if (<= x -1.5e-32)
       (* x (/ (/ y (* (+ y x) (+ y x))) (+ x (+ 1.0 y))))
       (* t_0 (* (/ x (+ y x)) (/ y (+ 1.0 y))))))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double tmp;
	if (x <= -1.55e+97) {
		tmp = (t_0 / ((y + x) / x)) / (x / y);
	} else if (x <= -1.5e-32) {
		tmp = x * ((y / ((y + x) * (y + x))) / (x + (1.0 + y)));
	} else {
		tmp = t_0 * ((x / (y + x)) * (y / (1.0 + y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (y + x)
    if (x <= (-1.55d+97)) then
        tmp = (t_0 / ((y + x) / x)) / (x / y)
    else if (x <= (-1.5d-32)) then
        tmp = x * ((y / ((y + x) * (y + x))) / (x + (1.0d0 + y)))
    else
        tmp = t_0 * ((x / (y + x)) * (y / (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double tmp;
	if (x <= -1.55e+97) {
		tmp = (t_0 / ((y + x) / x)) / (x / y);
	} else if (x <= -1.5e-32) {
		tmp = x * ((y / ((y + x) * (y + x))) / (x + (1.0 + y)));
	} else {
		tmp = t_0 * ((x / (y + x)) * (y / (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (y + x)
	tmp = 0
	if x <= -1.55e+97:
		tmp = (t_0 / ((y + x) / x)) / (x / y)
	elif x <= -1.5e-32:
		tmp = x * ((y / ((y + x) * (y + x))) / (x + (1.0 + y)))
	else:
		tmp = t_0 * ((x / (y + x)) * (y / (1.0 + y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(y + x))
	tmp = 0.0
	if (x <= -1.55e+97)
		tmp = Float64(Float64(t_0 / Float64(Float64(y + x) / x)) / Float64(x / y));
	elseif (x <= -1.5e-32)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(y + x))) / Float64(x + Float64(1.0 + y))));
	else
		tmp = Float64(t_0 * Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (y + x);
	tmp = 0.0;
	if (x <= -1.55e+97)
		tmp = (t_0 / ((y + x) / x)) / (x / y);
	elseif (x <= -1.5e-32)
		tmp = x * ((y / ((y + x) * (y + x))) / (x + (1.0 + y)));
	else
		tmp = t_0 * ((x / (y + x)) * (y / (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.55e+97], N[(N[(t$95$0 / N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.5e-32], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.54999999999999991e97

   1. Initial program 65.5%

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

      1. associate-+r+ 65.5%

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

      2. *-un-lft-identity 65.5%

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

      3. associate-*l* 65.5%

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

      4. times-frac 73.0%

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

      5. +-commutative 73.0%

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

      6. *-commutative 73.0%

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

      7. +-commutative 73.0%

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

      8. +-commutative 73.0%

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

      9. associate-+l+ 73.0%

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

   4. Applied egg-rr 73.0%

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

      1. *-commutative 73.0%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. frac-times 73.0%

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

      3. *-commutative 73.0%

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

      4. clear-num 73.0%

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

      5. frac-times 72.3%

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

      6. metadata-eval 72.3%

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

      7. *-commutative 72.3%

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

      8. times-frac 96.2%

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

      9. +-commutative 96.2%

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

   8. Applied egg-rr 96.2%

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

      1. associate-/l/ 99.8%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around inf 90.9%

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

   if -1.54999999999999991e97 < x < -1.5e-32

   1. Initial program 78.4%

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

      1. associate-/r* 87.2%

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

      2. +-commutative 87.2%

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

      3. +-commutative 87.2%

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

      4. associate-/l* 92.2%

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

      5. +-commutative 92.2%

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

      6. associate-*r/ 85.1%

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

      7. +-commutative 85.1%

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

      8. +-commutative 85.1%

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

      9. +-commutative 85.1%

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

      10. associate-+l+ 85.1%

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

   3. Simplified 85.1%

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

   if -1.5e-32 < x

   1. Initial program 72.6%

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

      1. associate-+r+ 72.6%

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

      2. *-un-lft-identity 72.6%

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

      3. associate-*l* 72.7%

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

      4. times-frac 78.2%

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

      5. +-commutative 78.2%

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

      6. *-commutative 78.2%

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

      7. +-commutative 78.2%

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

      8. +-commutative 78.2%

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

      9. associate-+l+ 78.2%

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

   4. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 84.4%

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

      1. +-commutative 84.4%

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

   9. Simplified 84.4%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{\frac{1}{y + x}}{\frac{y + x}{x}}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{x + \left(1 + y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(1 + x\right)\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{1 + x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{t\_0}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ 1.0 x))))
   (if (<= y 7.2e-145)
     (* (/ 1.0 (+ y x)) (/ y (+ 1.0 x)))
     (if (<= y 3.2e+157) (* x (/ (/ 1.0 t_0) (+ y x))) (/ (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	} else if (y <= 3.2e+157) {
		tmp = x * ((1.0 / t_0) / (y + x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (1.0d0 + x)
    if (y <= 7.2d-145) then
        tmp = (1.0d0 / (y + x)) * (y / (1.0d0 + x))
    else if (y <= 3.2d+157) then
        tmp = x * ((1.0d0 / t_0) / (y + x))
    else
        tmp = (x / y) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (1.0 + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	} else if (y <= 3.2e+157) {
		tmp = x * ((1.0 / t_0) / (y + x));
	} else {
		tmp = (x / y) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (1.0 + x)
	tmp = 0
	if y <= 7.2e-145:
		tmp = (1.0 / (y + x)) * (y / (1.0 + x))
	elif y <= 3.2e+157:
		tmp = x * ((1.0 / t_0) / (y + x))
	else:
		tmp = (x / y) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (y <= 7.2e-145)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(1.0 + x)));
	elseif (y <= 3.2e+157)
		tmp = Float64(x * Float64(Float64(1.0 / t_0) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (1.0 + x);
	tmp = 0.0;
	if (y <= 7.2e-145)
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	elseif (y <= 3.2e+157)
		tmp = x * ((1.0 / t_0) / (y + x));
	else
		tmp = (x / y) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-145], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.2e+157], N[(x * N[(N[(1.0 / t$95$0), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000001e-145

   1. Initial program 70.8%

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

      1. associate-+r+ 70.8%

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

      2. *-un-lft-identity 70.8%

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

      3. associate-*l* 70.9%

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

      4. times-frac 75.4%

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

      5. +-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. +-commutative 75.4%

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

      8. +-commutative 75.4%

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

      9. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. *-commutative 75.4%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 55.4%

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

      1. +-commutative 55.4%

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

   9. Simplified 55.4%

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

   if 7.2000000000000001e-145 < y < 3.1999999999999999e157

   1. Initial program 77.6%

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

      1. associate-+r+ 77.6%

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

      2. *-un-lft-identity 77.6%

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

      3. associate-*l* 77.6%

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

      4. times-frac 89.6%

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

      5. +-commutative 89.6%

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

      6. *-commutative 89.6%

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

      7. +-commutative 89.6%

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

      8. +-commutative 89.6%

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

      9. associate-+l+ 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. associate-*r* 99.8%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. frac-times 94.0%

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

      5. *-un-lft-identity 94.0%

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

      6. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/r/ 88.7%

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

      2. associate-/r* 88.8%

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

      3. +-commutative 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in y around inf 76.8%

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

   if 3.1999999999999999e157 < y

   1. Initial program 63.2%

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

      1. associate-/r* 63.2%

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

      2. +-commutative 63.2%

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

      3. +-commutative 63.2%

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

      4. associate-/l* 85.2%

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

      5. +-commutative 85.2%

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

      6. associate-*r/ 85.2%

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

      7. +-commutative 85.2%

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

      8. +-commutative 85.2%

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

      9. +-commutative 85.2%

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

      10. associate-+l+ 85.2%

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

   3. Simplified 85.2%

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

   5. Taylor expanded in y around inf 85.4%

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

      1. add0 85.4%

         \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{y}}{x + \left(y + 1\right)} + 0} \]

      2. associate-/l/ 85.2%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      3. un-div-inv 85.2%

         \[\leadsto \color{blue}{\frac{x}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      4. associate-+r+ 85.2%

         \[\leadsto \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot y} + 0 \]

      5. +-commutative 85.2%

         \[\leadsto \frac{x}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot y} + 0 \]

      6. associate-+r+ 85.2%

         \[\leadsto \frac{x}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot y} + 0 \]

      7. +-commutative 85.2%

         \[\leadsto \frac{x}{\left(y + \color{blue}{\left(1 + x\right)}\right) \cdot y} + 0 \]

   7. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(1 + x\right)\right) \cdot y} + 0} \]
   8. Step-by-step derivation

      1. associate-/l/ 96.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + \left(1 + x\right)}} + 0 \]

      2. add0 96.1%

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

      3. +-commutative 96.1%

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

   9. Simplified 96.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{1 + x}\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 64.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y + x}\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;t\_0 \cdot \frac{y}{1 + x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{y + x}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ y x))))
   (if (<= y 7.2e-145)
     (* t_0 (/ y (+ 1.0 x)))
     (if (<= y 8.5e+157)
       (* x (/ (/ 1.0 (+ y (+ 1.0 x))) (+ y x)))
       (/ t_0 (/ (+ y x) x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = t_0 * (y / (1.0 + x));
	} else if (y <= 8.5e+157) {
		tmp = x * ((1.0 / (y + (1.0 + x))) / (y + x));
	} else {
		tmp = t_0 / ((y + x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (y + x)
    if (y <= 7.2d-145) then
        tmp = t_0 * (y / (1.0d0 + x))
    else if (y <= 8.5d+157) then
        tmp = x * ((1.0d0 / (y + (1.0d0 + x))) / (y + x))
    else
        tmp = t_0 / ((y + x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = t_0 * (y / (1.0 + x));
	} else if (y <= 8.5e+157) {
		tmp = x * ((1.0 / (y + (1.0 + x))) / (y + x));
	} else {
		tmp = t_0 / ((y + x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (y + x)
	tmp = 0
	if y <= 7.2e-145:
		tmp = t_0 * (y / (1.0 + x))
	elif y <= 8.5e+157:
		tmp = x * ((1.0 / (y + (1.0 + x))) / (y + x))
	else:
		tmp = t_0 / ((y + x) / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(y + x))
	tmp = 0.0
	if (y <= 7.2e-145)
		tmp = Float64(t_0 * Float64(y / Float64(1.0 + x)));
	elseif (y <= 8.5e+157)
		tmp = Float64(x * Float64(Float64(1.0 / Float64(y + Float64(1.0 + x))) / Float64(y + x)));
	else
		tmp = Float64(t_0 / Float64(Float64(y + x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (y + x);
	tmp = 0.0;
	if (y <= 7.2e-145)
		tmp = t_0 * (y / (1.0 + x));
	elseif (y <= 8.5e+157)
		tmp = x * ((1.0 / (y + (1.0 + x))) / (y + x));
	else
		tmp = t_0 / ((y + x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-145], N[(t$95$0 * N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.5e+157], N[(x * N[(N[(1.0 / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000001e-145

   1. Initial program 70.8%

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

      1. associate-+r+ 70.8%

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

      2. *-un-lft-identity 70.8%

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

      3. associate-*l* 70.9%

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

      4. times-frac 75.4%

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

      5. +-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. +-commutative 75.4%

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

      8. +-commutative 75.4%

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

      9. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. *-commutative 75.4%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in y around 0 55.4%

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

      1. +-commutative 55.4%

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

   9. Simplified 55.4%

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

   if 7.2000000000000001e-145 < y < 8.4999999999999998e157

   1. Initial program 77.6%

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

      1. associate-+r+ 77.6%

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

      2. *-un-lft-identity 77.6%

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

      3. associate-*l* 77.6%

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

      4. times-frac 89.6%

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

      5. +-commutative 89.6%

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

      6. *-commutative 89.6%

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

      7. +-commutative 89.6%

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

      8. +-commutative 89.6%

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

      9. associate-+l+ 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. associate-*r* 99.8%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. frac-times 94.0%

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

      5. *-un-lft-identity 94.0%

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

      6. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/r/ 88.7%

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

      2. associate-/r* 88.8%

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

      3. +-commutative 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in y around inf 76.8%

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

   if 8.4999999999999998e157 < y

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-un-lft-identity 63.2%

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

      3. associate-*l* 63.2%

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

      4. times-frac 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. associate-+l+ 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. frac-times 63.2%

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

      3. *-commutative 63.2%

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

      4. clear-num 63.2%

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

      5. frac-times 63.2%

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

      6. metadata-eval 63.2%

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

      7. *-commutative 63.2%

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

      8. times-frac 96.5%

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

      9. +-commutative 96.5%

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

   8. Applied egg-rr 96.5%

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

      1. associate-/l/ 99.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in y around inf 96.1%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{1 + x}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y + x}\\ t_1 := y + \left(1 + x\right)\\ \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;t\_0 \cdot \frac{y}{t\_1}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{t\_1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\frac{y + x}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ y x))) (t_1 (+ y (+ 1.0 x))))
   (if (<= y 7.2e-145)
     (* t_0 (/ y t_1))
     (if (<= y 4.3e+157)
       (* x (/ (/ 1.0 t_1) (+ y x)))
       (/ t_0 (/ (+ y x) x))))))

C

double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double t_1 = y + (1.0 + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = t_0 * (y / t_1);
	} else if (y <= 4.3e+157) {
		tmp = x * ((1.0 / t_1) / (y + x));
	} else {
		tmp = t_0 / ((y + x) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (y + x)
    t_1 = y + (1.0d0 + x)
    if (y <= 7.2d-145) then
        tmp = t_0 * (y / t_1)
    else if (y <= 4.3d+157) then
        tmp = x * ((1.0d0 / t_1) / (y + x))
    else
        tmp = t_0 / ((y + x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 / (y + x);
	double t_1 = y + (1.0 + x);
	double tmp;
	if (y <= 7.2e-145) {
		tmp = t_0 * (y / t_1);
	} else if (y <= 4.3e+157) {
		tmp = x * ((1.0 / t_1) / (y + x));
	} else {
		tmp = t_0 / ((y + x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 / (y + x)
	t_1 = y + (1.0 + x)
	tmp = 0
	if y <= 7.2e-145:
		tmp = t_0 * (y / t_1)
	elif y <= 4.3e+157:
		tmp = x * ((1.0 / t_1) / (y + x))
	else:
		tmp = t_0 / ((y + x) / x)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 / Float64(y + x))
	t_1 = Float64(y + Float64(1.0 + x))
	tmp = 0.0
	if (y <= 7.2e-145)
		tmp = Float64(t_0 * Float64(y / t_1));
	elseif (y <= 4.3e+157)
		tmp = Float64(x * Float64(Float64(1.0 / t_1) / Float64(y + x)));
	else
		tmp = Float64(t_0 / Float64(Float64(y + x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 / (y + x);
	t_1 = y + (1.0 + x);
	tmp = 0.0;
	if (y <= 7.2e-145)
		tmp = t_0 * (y / t_1);
	elseif (y <= 4.3e+157)
		tmp = x * ((1.0 / t_1) / (y + x));
	else
		tmp = t_0 / ((y + x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 7.2e-145], N[(t$95$0 * N[(y / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+157], N[(x * N[(N[(1.0 / t$95$1), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.2000000000000001e-145

   1. Initial program 70.8%

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

      1. associate-+r+ 70.8%

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

      2. *-un-lft-identity 70.8%

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

      3. associate-*l* 70.9%

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

      4. times-frac 75.4%

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

      5. +-commutative 75.4%

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

      6. *-commutative 75.4%

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

      7. +-commutative 75.4%

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

      8. +-commutative 75.4%

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

      9. associate-+l+ 75.4%

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

   4. Applied egg-rr 75.4%

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

      1. *-commutative 75.4%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

   6. Simplified 99.6%

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

   7. Taylor expanded in x around inf 56.1%

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

   if 7.2000000000000001e-145 < y < 4.3e157

   1. Initial program 77.6%

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

      1. associate-+r+ 77.6%

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

      2. *-un-lft-identity 77.6%

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

      3. associate-*l* 77.6%

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

      4. times-frac 89.6%

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

      5. +-commutative 89.6%

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

      6. *-commutative 89.6%

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

      7. +-commutative 89.6%

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

      8. +-commutative 89.6%

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

      9. associate-+l+ 89.6%

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

   4. Applied egg-rr 89.6%

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

      1. *-commutative 89.6%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. associate-*r* 99.8%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. frac-times 94.0%

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

      5. *-un-lft-identity 94.0%

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

      6. +-commutative 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. associate-/r/ 88.7%

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

      2. associate-/r* 88.8%

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

      3. +-commutative 88.8%

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

   10. Simplified 88.8%

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

   11. Taylor expanded in y around inf 76.8%

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

   if 4.3e157 < y

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-un-lft-identity 63.2%

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

      3. associate-*l* 63.2%

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

      4. times-frac 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. associate-+l+ 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. frac-times 63.2%

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

      3. *-commutative 63.2%

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

      4. clear-num 63.2%

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

      5. frac-times 63.2%

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

      6. metadata-eval 63.2%

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

      7. *-commutative 63.2%

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

      8. times-frac 96.5%

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

      9. +-commutative 96.5%

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

   8. Applied egg-rr 96.5%

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

      1. associate-/l/ 99.9%

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

      2. associate-/r* 99.9%

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

      3. +-commutative 99.9%

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

   10. Simplified 99.9%

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

   11. Taylor expanded in y around inf 96.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+157}:\\ \;\;\;\;x \cdot \frac{\frac{1}{y + \left(1 + x\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y + x}}{\frac{y + x}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+157)
   (* (/ y (+ y x)) (/ x (* (+ y x) (+ y (+ 1.0 x)))))
   (* (/ 1.0 (+ y x)) (* (/ x (+ y x)) (/ y (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+157) {
		tmp = (y / (y + x)) * (x / ((y + x) * (y + (1.0 + x))));
	} else {
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + 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 <= 2.6d+157) then
        tmp = (y / (y + x)) * (x / ((y + x) * (y + (1.0d0 + x))))
    else
        tmp = (1.0d0 / (y + x)) * ((x / (y + x)) * (y / (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+157) {
		tmp = (y / (y + x)) * (x / ((y + x) * (y + (1.0 + x))));
	} else {
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+157:
		tmp = (y / (y + x)) * (x / ((y + x) * (y + (1.0 + x))))
	else:
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+157)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * Float64(y + Float64(1.0 + x)))));
	else
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+157)
		tmp = (y / (y + x)) * (x / ((y + x) * (y + (1.0 + x))));
	else
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+157], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000011e157

   1. Initial program 72.9%

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

      1. associate-+r+ 72.9%

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

      2. *-commutative 72.9%

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

      3. associate-*l* 73.0%

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

      4. times-frac 95.1%

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

      5. +-commutative 95.1%

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

      6. +-commutative 95.1%

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

      7. +-commutative 95.1%

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

      8. associate-+l+ 95.1%

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

   4. Applied egg-rr 95.1%

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

   if 2.60000000000000011e157 < y

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-un-lft-identity 63.2%

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

      3. associate-*l* 63.2%

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

      4. times-frac 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. associate-+l+ 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 96.2%

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

      1. +-commutative 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{\frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.6e+157)
   (/ (/ y (* (+ y x) (+ y (+ 1.0 x)))) (/ (+ y x) x))
   (* (/ 1.0 (+ y x)) (* (/ x (+ y x)) (/ y (+ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+157) {
		tmp = (y / ((y + x) * (y + (1.0 + x)))) / ((y + x) / x);
	} else {
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + 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 <= 2.6d+157) then
        tmp = (y / ((y + x) * (y + (1.0d0 + x)))) / ((y + x) / x)
    else
        tmp = (1.0d0 / (y + x)) * ((x / (y + x)) * (y / (1.0d0 + y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.6e+157) {
		tmp = (y / ((y + x) * (y + (1.0 + x)))) / ((y + x) / x);
	} else {
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.6e+157:
		tmp = (y / ((y + x) * (y + (1.0 + x)))) / ((y + x) / x)
	else:
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.6e+157)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(1.0 + x)))) / Float64(Float64(y + x) / x));
	else
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(1.0 + y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.6e+157)
		tmp = (y / ((y + x) * (y + (1.0 + x)))) / ((y + x) / x);
	else
		tmp = (1.0 / (y + x)) * ((x / (y + x)) * (y / (1.0 + y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.6e+157], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.60000000000000011e157

   1. Initial program 72.9%

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

      1. associate-+r+ 72.9%

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

      2. *-un-lft-identity 72.9%

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

      3. associate-*l* 73.0%

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

      4. times-frac 79.8%

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

      5. +-commutative 79.8%

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

      6. *-commutative 79.8%

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

      7. +-commutative 79.8%

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

      8. +-commutative 79.8%

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

      9. associate-+l+ 79.8%

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

   4. Applied egg-rr 79.8%

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

      1. *-commutative 79.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

      1. associate-*r* 99.7%

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

      2. clear-num 99.7%

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

      3. un-div-inv 99.7%

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

      4. frac-times 95.1%

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

      5. *-un-lft-identity 95.1%

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

      6. +-commutative 95.1%

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

   8. Applied egg-rr 95.1%

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

   if 2.60000000000000011e157 < y

   1. Initial program 63.2%

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

      1. associate-+r+ 63.2%

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

      2. *-un-lft-identity 63.2%

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

      3. associate-*l* 63.2%

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

      4. times-frac 63.2%

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

      5. +-commutative 63.2%

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

      6. *-commutative 63.2%

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

      7. +-commutative 63.2%

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

      8. +-commutative 63.2%

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

      9. associate-+l+ 63.2%

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

   4. Applied egg-rr 63.2%

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

      1. *-commutative 63.2%

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

      2. times-frac 100.0%

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

      3. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in x around 0 96.2%

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

      1. +-commutative 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(1 + x\right)\right)}}{\frac{y + x}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + x} \cdot \left(\frac{x}{y + x} \cdot \frac{y}{1 + y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -360000000000:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(1 + y\right)}}{\frac{y + x}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -360000000000.0)
   (* (/ 1.0 (+ y x)) (/ y (+ y (+ 1.0 x))))
   (/ (/ y (* (+ y x) (+ 1.0 y))) (/ (+ y x) x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -360000000000.0) {
		tmp = (1.0 / (y + x)) * (y / (y + (1.0 + x)));
	} else {
		tmp = (y / ((y + x) * (1.0 + y))) / ((y + x) / 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 <= (-360000000000.0d0)) then
        tmp = (1.0d0 / (y + x)) * (y / (y + (1.0d0 + x)))
    else
        tmp = (y / ((y + x) * (1.0d0 + y))) / ((y + x) / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -360000000000.0) {
		tmp = (1.0 / (y + x)) * (y / (y + (1.0 + x)));
	} else {
		tmp = (y / ((y + x) * (1.0 + y))) / ((y + x) / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -360000000000.0:
		tmp = (1.0 / (y + x)) * (y / (y + (1.0 + x)))
	else:
		tmp = (y / ((y + x) * (1.0 + y))) / ((y + x) / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -360000000000.0)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(y + Float64(1.0 + x))));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(1.0 + y))) / Float64(Float64(y + x) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -360000000000.0)
		tmp = (1.0 / (y + x)) * (y / (y + (1.0 + x)));
	else
		tmp = (y / ((y + x) * (1.0 + y))) / ((y + x) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -360000000000.0], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.6e11

   1. Initial program 65.8%

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

      1. associate-+r+ 65.8%

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

      2. *-un-lft-identity 65.8%

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

      3. associate-*l* 65.8%

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

      4. times-frac 74.6%

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

      5. +-commutative 74.6%

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

      6. *-commutative 74.6%

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

      7. +-commutative 74.6%

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

      8. +-commutative 74.6%

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

      9. associate-+l+ 74.6%

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

   4. Applied egg-rr 74.6%

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

      1. *-commutative 74.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 84.7%

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

   if -3.6e11 < x

   1. Initial program 73.8%

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

      1. associate-+r+ 73.8%

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

      2. *-un-lft-identity 73.8%

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

      3. associate-*l* 73.8%

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

      4. times-frac 79.2%

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

      5. +-commutative 79.2%

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

      6. *-commutative 79.2%

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

      7. +-commutative 79.2%

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

      8. +-commutative 79.2%

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

      9. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 84.3%

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

      1. +-commutative 84.3%

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

   9. Simplified 84.3%

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

       1. associate-*r* 84.3%

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

       2. clear-num 84.2%

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

       3. un-div-inv 84.3%

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

       4. frac-times 84.3%

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

       5. *-un-lft-identity 84.3%

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

       6. +-commutative 84.3%

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

   11. Applied egg-rr 84.3%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -360000000000:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{y + \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(1 + y\right)}}{\frac{y + x}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{x}\\ \mathbf{if}\;x \leq -7000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y + x}}{t\_0}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(1 + y\right)}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ y x) x)))
   (if (<= x -7000000000.0)
     (/ (/ (/ 1.0 (+ y x)) t_0) (/ x y))
     (/ (/ y (* (+ y x) (+ 1.0 y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (y + x) / x;
	double tmp;
	if (x <= -7000000000.0) {
		tmp = ((1.0 / (y + x)) / t_0) / (x / y);
	} else {
		tmp = (y / ((y + x) * (1.0 + y))) / t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / x
    if (x <= (-7000000000.0d0)) then
        tmp = ((1.0d0 / (y + x)) / t_0) / (x / y)
    else
        tmp = (y / ((y + x) * (1.0d0 + y))) / t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y + x) / x;
	double tmp;
	if (x <= -7000000000.0) {
		tmp = ((1.0 / (y + x)) / t_0) / (x / y);
	} else {
		tmp = (y / ((y + x) * (1.0 + y))) / t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y + x) / x
	tmp = 0
	if x <= -7000000000.0:
		tmp = ((1.0 / (y + x)) / t_0) / (x / y)
	else:
		tmp = (y / ((y + x) * (1.0 + y))) / t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y + x) / x)
	tmp = 0.0
	if (x <= -7000000000.0)
		tmp = Float64(Float64(Float64(1.0 / Float64(y + x)) / t_0) / Float64(x / y));
	else
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(1.0 + y))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y + x) / x;
	tmp = 0.0;
	if (x <= -7000000000.0)
		tmp = ((1.0 / (y + x)) / t_0) / (x / y);
	else
		tmp = (y / ((y + x) * (1.0 + y))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[x, -7000000000.0], N[(N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -7e9

   1. Initial program 65.8%

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

      1. associate-+r+ 65.8%

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

      2. *-un-lft-identity 65.8%

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

      3. associate-*l* 65.8%

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

      4. times-frac 74.6%

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

      5. +-commutative 74.6%

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

      6. *-commutative 74.6%

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

      7. +-commutative 74.6%

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

      8. +-commutative 74.6%

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

      9. associate-+l+ 74.6%

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

   4. Applied egg-rr 74.6%

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

      1. *-commutative 74.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. frac-times 74.6%

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

      3. *-commutative 74.6%

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

      4. clear-num 74.6%

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

      5. frac-times 73.1%

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

      6. metadata-eval 73.1%

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

      7. *-commutative 73.1%

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

      8. times-frac 96.0%

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

      9. +-commutative 96.0%

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

   8. Applied egg-rr 96.0%

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

      1. associate-/l/ 99.7%

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

      2. associate-/r* 99.7%

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

      3. +-commutative 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 86.3%

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

   if -7e9 < x

   1. Initial program 73.8%

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

      1. associate-+r+ 73.8%

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

      2. *-un-lft-identity 73.8%

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

      3. associate-*l* 73.8%

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

      4. times-frac 79.2%

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

      5. +-commutative 79.2%

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

      6. *-commutative 79.2%

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

      7. +-commutative 79.2%

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

      8. +-commutative 79.2%

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

      9. associate-+l+ 79.2%

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

   4. Applied egg-rr 79.2%

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

      1. *-commutative 79.2%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in x around 0 84.3%

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

      1. +-commutative 84.3%

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

   9. Simplified 84.3%

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

       1. associate-*r* 84.3%

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

       2. clear-num 84.2%

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

       3. un-div-inv 84.3%

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

       4. frac-times 84.3%

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

       5. *-un-lft-identity 84.3%

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

       6. +-commutative 84.3%

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

   11. Applied egg-rr 84.3%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7000000000:\\ \;\;\;\;\frac{\frac{\frac{1}{y + x}}{\frac{y + x}{x}}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{\left(y + x\right) \cdot \left(1 + y\right)}}{\frac{y + x}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-154) {
		tmp = y * (1.0 / (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 <= (-1.0d0)) then
        tmp = (1.0d0 / x) * (y / x)
    else if (x <= (-1.8d-154)) then
        tmp = y * (1.0d0 / (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 <= -1.0) {
		tmp = (1.0 / x) * (y / x);
	} else if (x <= -1.8e-154) {
		tmp = y * (1.0 / (y + x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 67.5%

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

      1. associate-+r+ 67.5%

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

      2. *-un-lft-identity 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. +-commutative 75.9%

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

      8. +-commutative 75.9%

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

      9. associate-+l+ 75.9%

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

   4. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 81.5%

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

   8. Taylor expanded in y around 0 81.1%

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

   if -1 < x < -1.8000000000000001e-154

   1. Initial program 76.4%

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

      1. associate-+r+ 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. associate-*l* 76.4%

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

      4. times-frac 81.0%

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

      5. +-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. +-commutative 81.0%

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

      8. +-commutative 81.0%

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

      9. associate-+l+ 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. *-commutative 81.0%

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

      2. times-frac 99.4%

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

      3. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

   9. Simplified 96.9%

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

   10. Taylor expanded in y around 0 44.7%

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

   if -1.8000000000000001e-154 < x

   1. Initial program 72.8%

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

      1. associate-/r* 78.2%

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

      2. +-commutative 78.2%

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

      3. +-commutative 78.2%

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

      4. associate-/l* 88.3%

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

      5. +-commutative 88.3%

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

      6. associate-*r/ 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

      10. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in x around 0 63.5%

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

   6. Taylor expanded in y around 0 41.1%

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

4. Final simplification 50.7%

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

Alternative 11: 57.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 67.5%

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

      1. associate-+r+ 67.5%

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

      2. *-un-lft-identity 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. +-commutative 75.9%

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

      8. +-commutative 75.9%

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

      9. associate-+l+ 75.9%

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

   4. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 81.5%

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

   8. Taylor expanded in y around 0 81.1%

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

   if -1 < x < -1.8000000000000001e-154

   1. Initial program 76.4%

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

      1. associate-+r+ 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. associate-*l* 76.4%

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

      4. times-frac 81.0%

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

      5. +-commutative 81.0%

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

      6. *-commutative 81.0%

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

      7. +-commutative 81.0%

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

      8. +-commutative 81.0%

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

      9. associate-+l+ 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. *-commutative 81.0%

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

      2. times-frac 99.4%

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

      3. +-commutative 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in x around 0 96.9%

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

      1. +-commutative 96.9%

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

   9. Simplified 96.9%

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

   10. Taylor expanded in y around 0 44.7%

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

   if -1.8000000000000001e-154 < x

   1. Initial program 72.8%

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

      1. associate-/r* 78.2%

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

      2. +-commutative 78.2%

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

      3. +-commutative 78.2%

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

      4. associate-/l* 88.3%

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

      5. +-commutative 88.3%

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

      6. associate-*r/ 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

      10. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in x around 0 63.5%

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

4. Final simplification 65.2%

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

Alternative 12: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.75e-56)
   (* (/ 1.0 (+ y x)) (/ y (+ 1.0 x)))
   (/ (/ x y) (+ y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-56) {
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	} else {
		tmp = (x / y) / (y + (1.0 + 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 <= 1.75d-56) then
        tmp = (1.0d0 / (y + x)) * (y / (1.0d0 + x))
    else
        tmp = (x / y) / (y + (1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.75e-56) {
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	} else {
		tmp = (x / y) / (y + (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.75e-56:
		tmp = (1.0 / (y + x)) * (y / (1.0 + x))
	else:
		tmp = (x / y) / (y + (1.0 + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.75e-56)
		tmp = Float64(Float64(1.0 / Float64(y + x)) * Float64(y / Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.75e-56)
		tmp = (1.0 / (y + x)) * (y / (1.0 + x));
	else
		tmp = (x / y) / (y + (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.75e-56], N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.7499999999999999e-56

   1. Initial program 72.2%

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

      1. associate-+r+ 72.2%

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

      2. *-un-lft-identity 72.2%

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

      3. associate-*l* 72.2%

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

      4. times-frac 76.8%

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

      5. +-commutative 76.8%

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

      6. *-commutative 76.8%

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

      7. +-commutative 76.8%

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

      8. +-commutative 76.8%

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

      9. associate-+l+ 76.8%

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

   4. Applied egg-rr 76.8%

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

      1. *-commutative 76.8%

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

      2. times-frac 99.7%

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

      3. +-commutative 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in y around 0 57.0%

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

      1. +-commutative 57.0%

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

   9. Simplified 57.0%

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

   if 1.7499999999999999e-56 < y

   1. Initial program 71.6%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-/l* 89.8%

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

      5. +-commutative 89.8%

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

      6. associate-*r/ 86.2%

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

      7. +-commutative 86.2%

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

      8. +-commutative 86.2%

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

      9. +-commutative 86.2%

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

      10. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around inf 67.8%

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

      1. add0 67.8%

         \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{y}}{x + \left(y + 1\right)} + 0} \]

      2. associate-/l/ 67.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      3. un-div-inv 67.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      4. associate-+r+ 67.8%

         \[\leadsto \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot y} + 0 \]

      5. +-commutative 67.8%

         \[\leadsto \frac{x}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot y} + 0 \]

      6. associate-+r+ 67.8%

         \[\leadsto \frac{x}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot y} + 0 \]

      7. +-commutative 67.8%

         \[\leadsto \frac{x}{\left(y + \color{blue}{\left(1 + x\right)}\right) \cdot y} + 0 \]

   7. Applied egg-rr 67.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(1 + x\right)\right) \cdot y} + 0} \]
   8. Step-by-step derivation

      1. associate-/l/ 70.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + \left(1 + x\right)}} + 0 \]

      2. add0 70.1%

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

      3. +-commutative 70.1%

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

   9. Simplified 70.1%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{y + x} \cdot \frac{y}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(1 + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-56) (/ y (* x (+ 1.0 x))) (/ (/ x y) (+ y (+ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y + (1.0 + 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 <= 3.1d-56) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / y) / (y + (1.0d0 + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (y + (1.0 + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-56:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (y + (1.0 + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-56)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + Float64(1.0 + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-56)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / y) / (y + (1.0 + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-56], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999987e-56

   1. Initial program 72.2%

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

      1. associate-/r* 76.5%

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

      2. +-commutative 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-/l* 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-*r/ 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

      10. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.2%

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

      1. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   if 3.09999999999999987e-56 < y

   1. Initial program 71.6%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-/l* 89.8%

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

      5. +-commutative 89.8%

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

      6. associate-*r/ 86.2%

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

      7. +-commutative 86.2%

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

      8. +-commutative 86.2%

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

      9. +-commutative 86.2%

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

      10. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in y around inf 67.8%

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

      1. add0 67.8%

         \[\leadsto \color{blue}{x \cdot \frac{\frac{1}{y}}{x + \left(y + 1\right)} + 0} \]

      2. associate-/l/ 67.8%

         \[\leadsto x \cdot \color{blue}{\frac{1}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      3. un-div-inv 67.8%

         \[\leadsto \color{blue}{\frac{x}{\left(x + \left(y + 1\right)\right) \cdot y}} + 0 \]

      4. associate-+r+ 67.8%

         \[\leadsto \frac{x}{\color{blue}{\left(\left(x + y\right) + 1\right)} \cdot y} + 0 \]

      5. +-commutative 67.8%

         \[\leadsto \frac{x}{\left(\color{blue}{\left(y + x\right)} + 1\right) \cdot y} + 0 \]

      6. associate-+r+ 67.8%

         \[\leadsto \frac{x}{\color{blue}{\left(y + \left(x + 1\right)\right)} \cdot y} + 0 \]

      7. +-commutative 67.8%

         \[\leadsto \frac{x}{\left(y + \color{blue}{\left(1 + x\right)}\right) \cdot y} + 0 \]

   7. Applied egg-rr 67.8%

      \[\leadsto \color{blue}{\frac{x}{\left(y + \left(1 + x\right)\right) \cdot y} + 0} \]
   8. Step-by-step derivation

      1. associate-/l/ 70.1%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + \left(1 + x\right)}} + 0 \]

      2. add0 70.1%

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

      3. +-commutative 70.1%

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

   9. Simplified 70.1%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + \left(1 + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 43.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 67.5%

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

      1. associate-+r+ 67.5%

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

      2. *-un-lft-identity 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 75.9%

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

      5. +-commutative 75.9%

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

      6. *-commutative 75.9%

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

      7. +-commutative 75.9%

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

      8. +-commutative 75.9%

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

      9. associate-+l+ 75.9%

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

   4. Applied egg-rr 75.9%

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

      1. *-commutative 75.9%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

   6. Simplified 99.8%

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

   7. Taylor expanded in x around inf 81.5%

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

   8. Taylor expanded in y around 0 81.1%

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

   if -1 < x

   1. Initial program 73.4%

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

      1. associate-/r* 78.7%

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

      2. +-commutative 78.7%

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

      3. +-commutative 78.7%

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

      4. associate-/l* 90.1%

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

      5. +-commutative 90.1%

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

      6. associate-*r/ 87.6%

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

      7. +-commutative 87.6%

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

      8. +-commutative 87.6%

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

      9. +-commutative 87.6%

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

      10. associate-+l+ 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in x around 0 62.1%

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

   6. Taylor expanded in y around 0 36.8%

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

4. Final simplification 47.0%

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

Alternative 15: 60.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-56) (/ y (* x (+ 1.0 x))) (/ x (* y (+ 1.0 y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / (y * (1.0 + 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 <= 2.5d-56) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = x / (y * (1.0d0 + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = x / (y * (1.0 + y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-56:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = x / (y * (1.0 + y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-56)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(x / Float64(y * Float64(1.0 + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-56)
		tmp = y / (x * (1.0 + x));
	else
		tmp = x / (y * (1.0 + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-56], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(1.0 + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.49999999999999999e-56

   1. Initial program 72.2%

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

      1. associate-/r* 76.5%

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

      2. +-commutative 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-/l* 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-*r/ 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

      10. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.2%

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

      1. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   if 2.49999999999999999e-56 < y

   1. Initial program 71.6%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-/l* 89.8%

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

      5. +-commutative 89.8%

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

      6. associate-*r/ 86.2%

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

      7. +-commutative 86.2%

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

      8. +-commutative 86.2%

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

      9. +-commutative 86.2%

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

      10. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in x around 0 66.2%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(1 + y\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 61.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.45e-56) (/ y (* x (+ 1.0 x))) (/ (/ x y) (+ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (1.0 + 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 <= 3.45d-56) then
        tmp = y / (x * (1.0d0 + x))
    else
        tmp = (x / y) / (1.0d0 + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-56) {
		tmp = y / (x * (1.0 + x));
	} else {
		tmp = (x / y) / (1.0 + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.45e-56:
		tmp = y / (x * (1.0 + x))
	else:
		tmp = (x / y) / (1.0 + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.45e-56)
		tmp = Float64(y / Float64(x * Float64(1.0 + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(1.0 + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.45e-56)
		tmp = y / (x * (1.0 + x));
	else
		tmp = (x / y) / (1.0 + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.45e-56], N[(y / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(1.0 + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4499999999999998e-56

   1. Initial program 72.2%

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

      1. associate-/r* 76.5%

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

      2. +-commutative 76.5%

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

      3. +-commutative 76.5%

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

      4. associate-/l* 88.6%

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

      5. +-commutative 88.6%

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

      6. associate-*r/ 85.3%

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

      7. +-commutative 85.3%

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

      8. +-commutative 85.3%

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

      9. +-commutative 85.3%

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

      10. associate-+l+ 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around 0 55.2%

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

      1. +-commutative 55.2%

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

   7. Simplified 55.2%

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

   if 3.4499999999999998e-56 < y

   1. Initial program 71.6%

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

      1. associate-/r* 81.6%

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

      2. +-commutative 81.6%

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

      3. +-commutative 81.6%

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

      4. associate-/l* 89.8%

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

      5. +-commutative 89.8%

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

      6. associate-*r/ 86.2%

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

      7. +-commutative 86.2%

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

      8. +-commutative 86.2%

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

      9. +-commutative 86.2%

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

      10. associate-+l+ 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in x around 0 66.2%

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

      1. associate-/r* 69.6%

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

      2. +-commutative 69.6%

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

   7. Simplified 69.6%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-56}:\\ \;\;\;\;\frac{y}{x \cdot \left(1 + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{1 + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.2% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. associate-+r+ 72.0%

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

   2. *-un-lft-identity 72.0%

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

   3. associate-*l* 72.0%

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

   4. times-frac 78.2%

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

   5. +-commutative 78.2%

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

   6. *-commutative 78.2%

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

   7. +-commutative 78.2%

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

   8. +-commutative 78.2%

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

   9. associate-+l+ 78.2%

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

4. Applied egg-rr 78.2%

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

   1. *-commutative 78.2%

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

   2. times-frac 99.7%

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

   3. +-commutative 99.7%

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

6. Simplified 99.7%

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

7. Taylor expanded in x around 0 52.5%

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

   1. +-commutative 52.5%

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

9. Simplified 52.5%

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

10. Taylor expanded in y around 0 29.2%

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

    1. add0 29.2%

       \[\leadsto \color{blue}{\frac{1}{y + x} \cdot x + 0} \]

    2. associate-*l/ 29.3%

       \[\leadsto \color{blue}{\frac{1 \cdot x}{y + x}} + 0 \]

    3. *-un-lft-identity 29.3%

       \[\leadsto \frac{\color{blue}{x}}{y + x} + 0 \]

12. Applied egg-rr 29.3%

    \[\leadsto \color{blue}{\frac{x}{y + x} + 0} \]
13. Step-by-step derivation

    1. add0 29.3%

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

14. Simplified 29.3%

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

15. Final simplification 29.3%

    \[\leadsto \frac{x}{y + x} \]
16. Add Preprocessing

Alternative 18: 4.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.0%

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

   1. associate-/r* 78.1%

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

   2. +-commutative 78.1%

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

   3. +-commutative 78.1%

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

   4. associate-/l* 89.0%

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

   5. +-commutative 89.0%

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

   6. associate-*r/ 85.6%

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

   7. +-commutative 85.6%

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

   8. +-commutative 85.6%

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

   9. +-commutative 85.6%

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

   10. associate-+l+ 85.6%

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

3. Simplified 85.6%

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

5. Taylor expanded in y around inf 52.1%

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

6. Taylor expanded in x around inf 4.0%

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

7. Final simplification 4.0%

   \[\leadsto \frac{1}{y} \]
8. Add Preprocessing

Alternative 19: 25.9% 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 72.0%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. associate-/r* 78.1%

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{\left(x + y\right) \cdot \left(x + y\right)}}{\left(x + y\right) + 1}} \]

   2. +-commutative 78.1%

      \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)}}{\left(x + y\right) + 1} \]

   3. +-commutative 78.1%

      \[\leadsto \frac{\frac{x \cdot y}{\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}}}{\left(x + y\right) + 1} \]

   4. associate-/l* 89.0%

      \[\leadsto \frac{\color{blue}{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}}{\left(x + y\right) + 1} \]

   5. +-commutative 89.0%

      \[\leadsto \frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{\color{blue}{\left(y + x\right)} + 1} \]

   6. associate-*r/ 85.6%

      \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + x\right)}}{\left(y + x\right) + 1}} \]

   7. +-commutative 85.6%

      \[\leadsto x \cdot \frac{\frac{y}{\color{blue}{\left(x + y\right)} \cdot \left(y + x\right)}}{\left(y + x\right) + 1} \]

   8. +-commutative 85.6%

      \[\leadsto x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \color{blue}{\left(x + y\right)}}}{\left(y + x\right) + 1} \]

   9. +-commutative 85.6%

      \[\leadsto x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{\left(x + y\right)} + 1} \]

   10. associate-+l+ 85.6%

       \[\leadsto x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}}{\color{blue}{x + \left(y + 1\right)}} \]

3. Simplified 85.6%

   \[\leadsto \color{blue}{x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(x + y\right)}}{x + \left(y + 1\right)}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 51.5%

   \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

6. Taylor expanded in y around 0 28.9%

   \[\leadsto \frac{x}{\color{blue}{y}} \]

7. Final simplification 28.9%

   \[\leadsto \frac{x}{y} \]
8. 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 2024046 
(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))))