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

Percentage Accurate: 69.6% → 99.2%

Time: 14.6s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.6% 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: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* 71.6%

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

   2. times-frac 95.5%

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

   3. +-commutative 95.5%

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

   4. +-commutative 95.5%

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

   5. associate-+r+ 95.5%

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

   6. +-commutative 95.5%

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

   7. associate-+l+ 95.5%

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

4. Applied egg-rr 95.5%

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

   1. associate-/r* 99.8%

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

   2. +-commutative 99.8%

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

   3. associate-/r* 95.5%

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

   4. clear-num 95.5%

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

   5. associate-/r* 99.8%

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

   6. frac-times 99.7%

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

   7. metadata-eval 99.7%

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

   8. times-frac 99.7%

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

   9. *-un-lft-identity 99.7%

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

   10. *-un-lft-identity 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 76.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{\left(y + x\right) \cdot \left(y + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -4.2e+82)
   (/ (/ y (+ y x)) (+ y x))
   (if (<= x -1.75e-14)
     (* x (/ y (* (* (+ y x) (+ y x)) (+ x (+ y 1.0)))))
     (if (<= x 1.4e-14)
       (/ x (* (+ y x) (/ (* (+ y x) (+ y 1.0)) y)))
       (/ (/ x y) (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+82) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= -1.75e-14) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else if (x <= 1.4e-14) {
		tmp = x / ((y + x) * (((y + x) * (y + 1.0)) / y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.2d+82)) then
        tmp = (y / (y + x)) / (y + x)
    else if (x <= (-1.75d-14)) then
        tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0d0))))
    else if (x <= 1.4d-14) then
        tmp = x / ((y + x) * (((y + x) * (y + 1.0d0)) / y))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -4.2e+82) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= -1.75e-14) {
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	} else if (x <= 1.4e-14) {
		tmp = x / ((y + x) * (((y + x) * (y + 1.0)) / y));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -4.2e+82:
		tmp = (y / (y + x)) / (y + x)
	elif x <= -1.75e-14:
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))))
	elif x <= 1.4e-14:
		tmp = x / ((y + x) * (((y + x) * (y + 1.0)) / y))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -4.2e+82)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(y + x));
	elseif (x <= -1.75e-14)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(y + x) * Float64(y + x)) * Float64(x + Float64(y + 1.0)))));
	elseif (x <= 1.4e-14)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(Float64(Float64(y + x) * Float64(y + 1.0)) / y)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.2e+82)
		tmp = (y / (y + x)) / (y + x);
	elseif (x <= -1.75e-14)
		tmp = x * (y / (((y + x) * (y + x)) * (x + (y + 1.0))));
	elseif (x <= 1.4e-14)
		tmp = x / ((y + x) * (((y + x) * (y + 1.0)) / y));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -4.2e+82], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.75e-14], N[(x * N[(y / N[(N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-14], N[(x / N[(N[(y + x), $MachinePrecision] * N[(N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.2e82

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 58.8%

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

      1. associate-*l* 58.8%

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

      2. +-commutative 58.8%

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

      3. times-frac 87.4%

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

      4. +-commutative 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. associate-*r/ 87.4%

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

      2. associate-*l/ 64.9%

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

      3. associate-*r/ 87.4%

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

      4. *-commutative 87.4%

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

      5. associate-*r/ 87.4%

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

      6. associate-*l/ 87.4%

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

      7. associate-/l* 64.8%

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

      8. times-frac 85.5%

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

      9. *-inverses 85.5%

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

      10. *-rgt-identity 85.5%

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

   7. Simplified 85.5%

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

   if -4.2e82 < x < -1.7500000000000001e-14

   1. Initial program 88.8%

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

      1. associate-/l* 88.6%

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

      2. associate-+l+ 88.6%

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

   3. Simplified 88.6%

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

   if -1.7500000000000001e-14 < x < 1.4e-14

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 73.4%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 99.8%

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

      1. +-commutative 99.8%

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

   7. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.7%

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

   9. Applied egg-rr 99.8%

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

   if 1.4e-14 < x

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. times-frac 88.2%

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

      3. +-commutative 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. +-commutative 88.2%

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

      7. associate-+l+ 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 88.2%

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

      4. clear-num 88.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. associate-/r* 18.1%

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

      2. +-commutative 18.1%

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

   9. Simplified 18.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \frac{\left(y + x\right) \cdot \left(y + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e+82)
   (/ (/ y (+ y x)) (+ y x))
   (if (<= x 5.8e-49)
     (* x (/ (/ y (* (+ y x) (+ y (+ x 1.0)))) (+ y x)))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+82) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= 5.8e-49) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d+82)) then
        tmp = (y / (y + x)) / (y + x)
    else if (x <= 5.8d-49) then
        tmp = x * ((y / ((y + x) * (y + (x + 1.0d0)))) / (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e+82) {
		tmp = (y / (y + x)) / (y + x);
	} else if (x <= 5.8e-49) {
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.2e+82:
		tmp = (y / (y + x)) / (y + x)
	elif x <= 5.8e-49:
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.2e+82)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(y + x));
	elseif (x <= 5.8e-49)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e+82)
		tmp = (y / (y + x)) / (y + x);
	elseif (x <= 5.8e-49)
		tmp = x * ((y / ((y + x) * (y + (x + 1.0)))) / (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.2e+82], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-49], N[(x * N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.1999999999999997e82

   1. Initial program 58.8%

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

   3. Taylor expanded in x around inf 58.8%

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

      1. associate-*l* 58.8%

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

      2. +-commutative 58.8%

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

      3. times-frac 87.4%

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

      4. +-commutative 87.4%

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

   5. Applied egg-rr 87.4%

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

      1. associate-*r/ 87.4%

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

      2. associate-*l/ 64.9%

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

      3. associate-*r/ 87.4%

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

      4. *-commutative 87.4%

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

      5. associate-*r/ 87.4%

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

      6. associate-*l/ 87.4%

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

      7. associate-/l* 64.8%

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

      8. times-frac 85.5%

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

      9. *-inverses 85.5%

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

      10. *-rgt-identity 85.5%

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

   7. Simplified 85.5%

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

   if -5.1999999999999997e82 < x < 5.8e-49

   1. Initial program 74.9%

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

      1. associate-/l* 85.1%

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

      2. associate-+l+ 85.1%

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

   3. Simplified 85.1%

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

      1. *-un-lft-identity 85.1%

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

      2. associate-+r+ 85.1%

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

      3. associate-*l* 85.1%

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

      4. times-frac 99.6%

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

      5. +-commutative 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+r+ 99.6%

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

      8. +-commutative 99.6%

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

      9. associate-+l+ 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. +-commutative 99.6%

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

   8. Simplified 99.6%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

   10. Applied egg-rr 99.7%

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

   if 5.8e-49 < x

   1. Initial program 72.1%

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

      1. associate-*l* 72.1%

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

      2. times-frac 89.1%

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

      3. +-commutative 89.1%

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

      4. +-commutative 89.1%

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

      5. associate-+r+ 89.1%

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

      6. +-commutative 89.1%

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

      7. associate-+l+ 89.1%

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

   4. Applied egg-rr 89.1%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 89.1%

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

      4. clear-num 89.1%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 20.6%

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

      1. associate-/r* 19.9%

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

      2. +-commutative 19.9%

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

   9. Simplified 19.9%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{y + x}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{y + x}{x} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{y + x}}{y + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.0016)
   (/ (/ y x) (* (/ (+ y x) x) (+ x 1.0)))
   (if (<= x 1.4e-14)
     (* x (/ (/ (/ y (+ y x)) (+ y 1.0)) (+ y x)))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	} else if (x <= 1.4e-14) {
		tmp = x * (((y / (y + x)) / (y + 1.0)) / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.0016d0)) then
        tmp = (y / x) / (((y + x) / x) * (x + 1.0d0))
    else if (x <= 1.4d-14) then
        tmp = x * (((y / (y + x)) / (y + 1.0d0)) / (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.0016) {
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	} else if (x <= 1.4e-14) {
		tmp = x * (((y / (y + x)) / (y + 1.0)) / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.0016:
		tmp = (y / x) / (((y + x) / x) * (x + 1.0))
	elif x <= 1.4e-14:
		tmp = x * (((y / (y + x)) / (y + 1.0)) / (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.0016)
		tmp = Float64(Float64(y / x) / Float64(Float64(Float64(y + x) / x) * Float64(x + 1.0)));
	elseif (x <= 1.4e-14)
		tmp = Float64(x * Float64(Float64(Float64(y / Float64(y + x)) / Float64(y + 1.0)) / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.0016)
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	elseif (x <= 1.4e-14)
		tmp = x * (((y / (y + x)) / (y + 1.0)) / (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.0016], N[(N[(y / x), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-14], N[(x * N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00160000000000000008

   1. Initial program 67.1%

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

      1. associate-*l* 67.0%

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

      2. times-frac 93.0%

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

      3. +-commutative 93.0%

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

      4. +-commutative 93.0%

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

      5. associate-+r+ 93.0%

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

      6. +-commutative 93.0%

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

      7. associate-+l+ 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in y around 0 77.5%

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

      1. clear-num 77.5%

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

      2. associate-/r* 77.7%

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

      3. +-commutative 77.7%

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

      4. frac-times 77.7%

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

      5. *-un-lft-identity 77.7%

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

   7. Applied egg-rr 77.7%

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

   if -0.00160000000000000008 < x < 1.4e-14

   1. Initial program 73.6%

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

      1. associate-*l* 73.6%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   7. Simplified 99.7%

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

      1. associate-*l/ 99.7%

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

   9. Applied egg-rr 99.7%

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

       1. associate-/l* 99.6%

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

       2. associate-/r* 99.6%

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

   11. Simplified 99.6%

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

   if 1.4e-14 < x

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. times-frac 88.2%

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

      3. +-commutative 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. +-commutative 88.2%

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

      7. associate-+l+ 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 88.2%

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

      4. clear-num 88.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. associate-/r* 18.1%

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

      2. +-commutative 18.1%

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

   9. Simplified 18.1%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{y + x}{x} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{\frac{\frac{y}{y + x}}{y + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00145:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{y + x}{x} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -0.00145)
   (/ (/ y x) (* (/ (+ y x) x) (+ x 1.0)))
   (if (<= x 6.4e-105)
     (* (/ x (+ y x)) (/ y (* (+ y x) (+ y 1.0))))
     (/ (/ x y) (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -0.00145) {
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	} else if (x <= 6.4e-105) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-0.00145d0)) then
        tmp = (y / x) / (((y + x) / x) * (x + 1.0d0))
    else if (x <= 6.4d-105) then
        tmp = (x / (y + x)) * (y / ((y + x) * (y + 1.0d0)))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -0.00145) {
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	} else if (x <= 6.4e-105) {
		tmp = (x / (y + x)) * (y / ((y + x) * (y + 1.0)));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -0.00145:
		tmp = (y / x) / (((y + x) / x) * (x + 1.0))
	elif x <= 6.4e-105:
		tmp = (x / (y + x)) * (y / ((y + x) * (y + 1.0)))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -0.00145)
		tmp = Float64(Float64(y / x) / Float64(Float64(Float64(y + x) / x) * Float64(x + 1.0)));
	elseif (x <= 6.4e-105)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -0.00145)
		tmp = (y / x) / (((y + x) / x) * (x + 1.0));
	elseif (x <= 6.4e-105)
		tmp = (x / (y + x)) * (y / ((y + x) * (y + 1.0)));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -0.00145], N[(N[(y / x), $MachinePrecision] / N[(N[(N[(y + x), $MachinePrecision] / x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.4e-105], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.00145

   1. Initial program 67.1%

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

      1. associate-*l* 67.0%

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

      2. times-frac 93.0%

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

      3. +-commutative 93.0%

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

      4. +-commutative 93.0%

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

      5. associate-+r+ 93.0%

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

      6. +-commutative 93.0%

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

      7. associate-+l+ 93.0%

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

   4. Applied egg-rr 93.0%

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

   5. Taylor expanded in y around 0 77.5%

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

      1. clear-num 77.5%

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

      2. associate-/r* 77.7%

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

      3. +-commutative 77.7%

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

      4. frac-times 77.7%

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

      5. *-un-lft-identity 77.7%

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

   7. Applied egg-rr 77.7%

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

   if -0.00145 < x < 6.39999999999999962e-105

   1. Initial program 71.0%

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

      1. associate-*l* 71.0%

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

      2. times-frac 99.8%

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

      3. +-commutative 99.8%

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

      4. +-commutative 99.8%

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

      5. associate-+r+ 99.8%

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

      6. +-commutative 99.8%

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

      7. associate-+l+ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. +-commutative 99.7%

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

   7. Simplified 99.7%

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

   if 6.39999999999999962e-105 < x

   1. Initial program 75.8%

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

      1. associate-*l* 75.8%

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

      2. times-frac 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   4. Applied egg-rr 91.0%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 91.0%

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

      4. clear-num 90.9%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 29.4%

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

      1. associate-/r* 28.8%

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

      2. +-commutative 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00145:\\ \;\;\;\;\frac{\frac{y}{x}}{\frac{y + x}{x} \cdot \left(x + 1\right)}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.4e-105)
   (* (/ y (* (+ y x) (+ y (+ x 1.0)))) (/ x (+ y x)))
   (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.4e-105) {
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (x / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 6.4d-105) then
        tmp = (y / ((y + x) * (y + (x + 1.0d0)))) * (x / (y + x))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.4e-105) {
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (x / (y + x));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.4e-105:
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (x / (y + x))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.4e-105)
		tmp = Float64(Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0)))) * Float64(x / Float64(y + x)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.4e-105)
		tmp = (y / ((y + x) * (y + (x + 1.0)))) * (x / (y + x));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.4e-105], N[(N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.39999999999999962e-105

   1. Initial program 69.7%

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

      1. associate-*l* 69.7%

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

      2. times-frac 97.6%

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

      3. +-commutative 97.6%

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

      4. +-commutative 97.6%

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

      5. associate-+r+ 97.6%

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

      6. +-commutative 97.6%

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

      7. associate-+l+ 97.6%

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

   4. Applied egg-rr 97.6%

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

   if 6.39999999999999962e-105 < x

   1. Initial program 75.8%

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

      1. associate-*l* 75.8%

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

      2. times-frac 91.0%

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

      3. +-commutative 91.0%

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

      4. +-commutative 91.0%

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

      5. associate-+r+ 91.0%

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

      6. +-commutative 91.0%

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

      7. associate-+l+ 91.0%

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

   4. Applied egg-rr 91.0%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 91.0%

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

      4. clear-num 90.9%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 29.4%

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

      1. associate-/r* 28.8%

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

      2. +-commutative 28.8%

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

   9. Simplified 28.8%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.4 \cdot 10^{-105}:\\ \;\;\;\;\frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)} \cdot \frac{x}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.4e-14)
   (/ (* x (/ y (* (+ y x) (+ y (+ x 1.0))))) (+ y x))
   (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4e-14) {
		tmp = (x * (y / ((y + x) * (y + (x + 1.0))))) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.4d-14) then
        tmp = (x * (y / ((y + x) * (y + (x + 1.0d0))))) / (y + x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.4e-14) {
		tmp = (x * (y / ((y + x) * (y + (x + 1.0))))) / (y + x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4e-14:
		tmp = (x * (y / ((y + x) * (y + (x + 1.0))))) / (y + x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4e-14)
		tmp = Float64(Float64(x * Float64(y / Float64(Float64(y + x) * Float64(y + Float64(x + 1.0))))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.4e-14)
		tmp = (x * (y / ((y + x) * (y + (x + 1.0))))) / (y + x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4e-14], N[(N[(x * N[(y / N[(N[(y + x), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e-14

   1. Initial program 71.7%

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

      1. associate-*l* 71.6%

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

      2. times-frac 97.8%

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

      3. +-commutative 97.8%

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

      4. +-commutative 97.8%

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

      5. associate-+r+ 97.8%

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

      6. +-commutative 97.8%

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

      7. associate-+l+ 97.8%

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

   4. Applied egg-rr 97.8%

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

      1. associate-*l/ 97.9%

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

      2. associate-/r* 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-/r* 97.9%

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

   6. Applied egg-rr 97.9%

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

   if 1.4e-14 < x

   1. Initial program 71.3%

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

      1. associate-*l* 71.3%

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

      2. times-frac 88.2%

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

      3. +-commutative 88.2%

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

      4. +-commutative 88.2%

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

      5. associate-+r+ 88.2%

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

      6. +-commutative 88.2%

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

      7. associate-+l+ 88.2%

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

   4. Applied egg-rr 88.2%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 88.2%

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

      4. clear-num 88.2%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 18.8%

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

      1. associate-/r* 18.1%

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

      2. +-commutative 18.1%

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

   9. Simplified 18.1%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x \cdot \frac{y}{\left(y + x\right) \cdot \left(y + \left(x + 1\right)\right)}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ y x) x)
   (if (<= x -1.45e-111) (/ y x) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.45e-111) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-1.45d-111)) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -1.45e-111) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -1.45e-111:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -1.45e-111)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -1.45e-111)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -1.45e-111], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 65.9%

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

      1. associate-*l* 65.8%

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

      2. times-frac 92.8%

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

      3. +-commutative 92.8%

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

      4. +-commutative 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. associate-+l+ 92.8%

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in y around 0 76.8%

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

      1. clear-num 76.8%

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

      2. associate-/r* 76.8%

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

      3. +-commutative 76.8%

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

      4. frac-times 76.8%

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

      5. *-un-lft-identity 76.8%

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

   7. Applied egg-rr 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

   if -1 < x < -1.45000000000000001e-111

   1. Initial program 82.1%

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

      1. associate-*l* 81.9%

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

      2. times-frac 99.5%

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

      3. +-commutative 99.5%

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

      4. +-commutative 99.5%

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

      5. associate-+r+ 99.5%

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

      6. +-commutative 99.5%

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

      7. associate-+l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

   5. Taylor expanded in x around 0 93.3%

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

      1. +-commutative 93.3%

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

   7. Simplified 93.3%

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

      1. *-commutative 93.3%

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

      2. clear-num 93.2%

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

      3. frac-times 93.2%

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

      4. *-un-lft-identity 93.2%

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

   9. Applied egg-rr 93.2%

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

   10. Taylor expanded in y around 0 43.4%

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

   if -1.45000000000000001e-111 < x

   1. Initial program 71.8%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 56.4%

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

      1. +-commutative 56.4%

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

   7. Simplified 56.4%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.45e-111) (/ (/ y (+ y x)) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-111) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.45d-111)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.45e-111) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.45e-111:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.45e-111)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.45e-111)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.45e-111], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.45000000000000001e-111

   1. Initial program 71.1%

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

      1. associate-*l* 71.0%

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

      2. times-frac 95.0%

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

      3. +-commutative 95.0%

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

      4. +-commutative 95.0%

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

      5. associate-+r+ 95.0%

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

      6. +-commutative 95.0%

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

      7. associate-+l+ 95.0%

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

   4. Applied egg-rr 95.0%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/r* 95.0%

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

      4. clear-num 95.0%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 68.3%

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

      1. +-commutative 68.3%

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

   9. Simplified 68.3%

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

   if -1.45000000000000001e-111 < x

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. times-frac 95.8%

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

      3. +-commutative 95.8%

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

      4. +-commutative 95.8%

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

      5. associate-+r+ 95.8%

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

      6. +-commutative 95.8%

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

      7. associate-+l+ 95.8%

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

   4. Applied egg-rr 95.8%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 95.8%

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

      4. clear-num 95.7%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.6%

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

      7. metadata-eval 99.6%

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

      8. times-frac 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 56.4%

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

      1. associate-/r* 56.1%

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

      2. +-commutative 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.45 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0) (/ (/ y x) x) (if (<= x -2.85e-201) (/ y x) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -2.85e-201) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (y / x) / x
    else if (x <= (-2.85d-201)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = (y / x) / x;
	} else if (x <= -2.85e-201) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = (y / x) / x
	elif x <= -2.85e-201:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(y / x) / x);
	elseif (x <= -2.85e-201)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (y / x) / x;
	elseif (x <= -2.85e-201)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(N[(y / x), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, -2.85e-201], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 65.9%

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

      1. associate-*l* 65.8%

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

      2. times-frac 92.8%

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

      3. +-commutative 92.8%

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

      4. +-commutative 92.8%

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

      5. associate-+r+ 92.8%

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

      6. +-commutative 92.8%

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

      7. associate-+l+ 92.8%

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

   4. Applied egg-rr 92.8%

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

   5. Taylor expanded in y around 0 76.8%

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

      1. clear-num 76.8%

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

      2. associate-/r* 76.8%

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

      3. +-commutative 76.8%

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

      4. frac-times 76.8%

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

      5. *-un-lft-identity 76.8%

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

   7. Applied egg-rr 76.8%

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

   8. Taylor expanded in x around inf 72.9%

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

   if -1 < x < -2.85e-201

   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-*l* 76.2%

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

      2. times-frac 99.6%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. associate-+r+ 99.6%

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

      6. +-commutative 99.6%

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

      7. associate-+l+ 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in x around 0 96.3%

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

      1. +-commutative 96.3%

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

   7. Simplified 96.3%

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

      1. *-commutative 96.3%

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

      2. clear-num 96.2%

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

      3. frac-times 96.3%

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

      4. *-un-lft-identity 96.3%

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

   9. Applied egg-rr 96.3%

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

   10. Taylor expanded in y around 0 36.3%

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

   if -2.85e-201 < x

   1. Initial program 72.1%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. +-commutative 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in y around 0 35.6%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{y}{x}}{x}\\ \mathbf{elif}\;x \leq -2.85 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.2e-82) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.2d-82) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.2e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.2e-82:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.2e-82)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.2e-82)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.2e-82], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.19999999999999986e-82

   1. Initial program 69.9%

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

      1. associate-/l* 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around 0 60.4%

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

   if 2.19999999999999986e-82 < y

   1. Initial program 75.4%

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

      1. associate-/l* 84.6%

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

      2. associate-+l+ 84.6%

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

   3. Simplified 84.6%

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

   5. Taylor expanded in x around 0 67.5%

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

      1. +-commutative 67.5%

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

   7. Simplified 67.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.2 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.15e-82) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.15d-82) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.15e-82) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.15e-82:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.15e-82)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.15e-82)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.15e-82], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.14999999999999998e-82

   1. Initial program 69.9%

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

      1. associate-/l* 82.9%

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

      2. associate-+l+ 82.9%

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

   3. Simplified 82.9%

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

   5. Taylor expanded in y around 0 60.4%

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

   if 1.14999999999999998e-82 < y

   1. Initial program 75.4%

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

      1. associate-*l* 75.4%

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

      2. times-frac 91.6%

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

      3. +-commutative 91.6%

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

      4. +-commutative 91.6%

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

      5. associate-+r+ 91.6%

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

      6. +-commutative 91.6%

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

      7. associate-+l+ 91.6%

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

   4. Applied egg-rr 91.6%

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

      1. associate-/r* 99.7%

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

      2. +-commutative 99.7%

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

      3. associate-/r* 91.6%

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

      4. clear-num 91.5%

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

      5. associate-/r* 99.6%

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

      6. frac-times 99.8%

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

      7. metadata-eval 99.8%

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

      8. times-frac 99.8%

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

      9. *-un-lft-identity 99.8%

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

      10. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 67.5%

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

      1. associate-/r* 65.9%

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

      2. +-commutative 65.9%

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

   9. Simplified 65.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-82}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -8e-112) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -8e-112) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8d-112)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -8e-112) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -8e-112:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -8e-112)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8e-112)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -8e-112], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.9999999999999996e-112

   1. Initial program 71.1%

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

      1. associate-/l* 86.9%

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

      2. associate-+l+ 86.9%

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

   3. Simplified 86.9%

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

   5. Taylor expanded in y around 0 67.8%

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

      1. associate-/r* 67.9%

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

      2. +-commutative 67.9%

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

   7. Simplified 67.9%

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

   if -7.9999999999999996e-112 < x

   1. Initial program 71.8%

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

      1. associate-*l* 71.8%

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

      2. times-frac 95.8%

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

      3. +-commutative 95.8%

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

      4. +-commutative 95.8%

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

      5. associate-+r+ 95.8%

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

      6. +-commutative 95.8%

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

      7. associate-+l+ 95.8%

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

   4. Applied egg-rr 95.8%

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

      1. associate-/r* 99.8%

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

      2. +-commutative 99.8%

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

      3. associate-/r* 95.8%

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

      4. clear-num 95.7%

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

      5. associate-/r* 99.8%

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

      6. frac-times 99.6%

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

      7. metadata-eval 99.6%

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

      8. times-frac 99.6%

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

      9. *-un-lft-identity 99.6%

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

      10. *-un-lft-identity 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0 56.4%

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

      1. associate-/r* 56.1%

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

      2. +-commutative 56.1%

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

   9. Simplified 56.1%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -3.05e-201) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-201) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3.05d-201)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -3.05e-201) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -3.05e-201:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -3.05e-201)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3.05e-201)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -3.05e-201], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.05000000000000013e-201

   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-*l* 70.7%

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

      2. times-frac 96.0%

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

      3. +-commutative 96.0%

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

      4. +-commutative 96.0%

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

      5. associate-+r+ 96.0%

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

      6. +-commutative 96.0%

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

      7. associate-+l+ 96.0%

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in x around 0 77.9%

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

      1. +-commutative 77.9%

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

   7. Simplified 77.9%

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

      1. *-commutative 77.9%

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

      2. clear-num 77.9%

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

      3. frac-times 85.8%

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

      4. *-un-lft-identity 85.8%

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

   9. Applied egg-rr 85.8%

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

   10. Taylor expanded in y around 0 34.1%

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

   if -3.05000000000000013e-201 < x

   1. Initial program 72.1%

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

      1. associate-/l* 81.7%

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

      2. associate-+l+ 81.7%

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

   3. Simplified 81.7%

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

   5. Taylor expanded in x around 0 53.8%

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

      1. +-commutative 53.8%

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

   7. Simplified 53.8%

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

   8. Taylor expanded in y around 0 35.6%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{-201}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 4.1% 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 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* 71.6%

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

   2. times-frac 95.5%

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

   3. +-commutative 95.5%

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

   4. +-commutative 95.5%

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

   5. associate-+r+ 95.5%

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

   6. +-commutative 95.5%

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

   7. associate-+l+ 95.5%

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

4. Applied egg-rr 95.5%

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

5. Taylor expanded in y around inf 36.1%

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

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 16: 26.2% 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 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-/l* 83.5%

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

   2. associate-+l+ 83.5%

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

3. Simplified 83.5%

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

5. Taylor expanded in x around 0 49.7%

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

   1. +-commutative 49.7%

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

7. Simplified 49.7%

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

8. Taylor expanded in y around 0 26.0%

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

9. Final simplification 26.0%

   \[\leadsto \frac{x}{y} \]
10. Add Preprocessing

Alternative 17: 3.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l* 71.6%

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

   2. times-frac 95.5%

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

   3. +-commutative 95.5%

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

   4. +-commutative 95.5%

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

   5. associate-+r+ 95.5%

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

   6. +-commutative 95.5%

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

   7. associate-+l+ 95.5%

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

4. Applied egg-rr 95.5%

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

5. Taylor expanded in y around 0 51.3%

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

   1. clear-num 51.3%

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

   2. associate-/r* 52.4%

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

   3. +-commutative 52.4%

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

   4. frac-times 52.4%

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

   5. *-un-lft-identity 52.4%

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

7. Applied egg-rr 52.4%

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

8. Taylor expanded in x around 0 3.5%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 3.5%

   \[\leadsto 1 \]
10. 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 2024084 
(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))))