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

Percentage Accurate: 69.2% → 99.8%

Time: 15.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-*l* 74.3%

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

   2. +-commutative 74.3%

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

   3. +-commutative 74.3%

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

   4. +-commutative 74.3%

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

   5. associate-*l* 74.3%

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

   6. associate-*l/ 83.7%

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

   7. *-commutative 83.7%

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

   8. *-commutative 83.7%

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

   9. distribute-rgt1-in 66.5%

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

   10. fma-def 83.7%

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

   11. +-commutative 83.7%

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

   12. +-commutative 83.7%

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

   13. cube-unmult 83.7%

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

   14. +-commutative 83.7%

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

3. Simplified 83.7%

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

   1. associate-*r/ 74.3%

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

   2. fma-udef 60.0%

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

   3. cube-mult 59.9%

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

   4. distribute-rgt1-in 74.3%

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

   5. associate-+r+ 74.3%

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

   6. *-commutative 74.3%

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

   7. frac-times 88.3%

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

   8. associate-/r* 99.8%

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

   9. associate-*l/ 99.9%

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

   10. +-commutative 99.9%

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

   11. +-commutative 99.9%

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

   12. associate-+l+ 99.9%

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

   13. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 64.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{y \cdot x}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.3e-89)
   (/ (/ y x) (+ x 1.0))
   (if (<= y 1.1e+95)
     (/ (* y x) (* (+ y 1.0) (* (+ y x) (+ y x))))
     (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-89) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.1e+95) {
		tmp = (y * x) / ((y + 1.0) * ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.3d-89) then
        tmp = (y / x) / (x + 1.0d0)
    else if (y <= 1.1d+95) then
        tmp = (y * x) / ((y + 1.0d0) * ((y + x) * (y + x)))
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.3e-89) {
		tmp = (y / x) / (x + 1.0);
	} else if (y <= 1.1e+95) {
		tmp = (y * x) / ((y + 1.0) * ((y + x) * (y + x)));
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.3e-89:
		tmp = (y / x) / (x + 1.0)
	elif y <= 1.1e+95:
		tmp = (y * x) / ((y + 1.0) * ((y + x) * (y + x)))
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.3e-89)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	elseif (y <= 1.1e+95)
		tmp = Float64(Float64(y * x) / Float64(Float64(y + 1.0) * Float64(Float64(y + x) * Float64(y + x))));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.3e-89)
		tmp = (y / x) / (x + 1.0);
	elseif (y <= 1.1e+95)
		tmp = (y * x) / ((y + 1.0) * ((y + x) * (y + x)));
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.3e-89], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+95], N[(N[(y * x), $MachinePrecision] / N[(N[(y + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.2999999999999999e-89

   1. Initial program 78.1%

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

      1. associate-*l* 78.1%

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

      2. +-commutative 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-*l* 78.1%

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

      6. *-commutative 78.1%

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

      7. times-frac 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around 0 54.7%

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

      1. associate-/r* 59.0%

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

      2. +-commutative 59.0%

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

   7. Simplified 59.0%

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

   if 1.2999999999999999e-89 < y < 1.0999999999999999e95

   1. Initial program 93.6%

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

   3. Taylor expanded in x around 0 90.8%

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

      1. +-commutative 90.8%

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

   5. Simplified 90.8%

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

   if 1.0999999999999999e95 < y

   1. Initial program 52.4%

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

      1. associate-+r+ 52.4%

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

      2. *-commutative 52.4%

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

      3. frac-times 81.3%

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

      4. associate-*l/ 81.3%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.9%

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

      2. +-commutative 99.9%

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

      3. frac-times 98.8%

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

      4. *-un-lft-identity 98.8%

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

      5. +-commutative 98.8%

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

      6. +-commutative 98.8%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 86.0%

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

      1. *-commutative 86.0%

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

   9. Simplified 86.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\frac{y \cdot x}{\left(y + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.5d0) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5

   1. Initial program 79.0%

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

      1. associate-*l* 79.0%

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

      2. +-commutative 79.0%

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

      3. +-commutative 79.0%

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

      4. +-commutative 79.0%

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

      5. associate-*l* 79.0%

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

      6. associate-*l/ 85.6%

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

      7. *-commutative 85.6%

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

      8. *-commutative 85.6%

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

      9. distribute-rgt1-in 63.9%

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

      10. fma-def 85.7%

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

      11. +-commutative 85.7%

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

      12. +-commutative 85.7%

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

      13. cube-unmult 85.7%

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

      14. +-commutative 85.7%

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

   3. Simplified 85.7%

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

      1. associate-*r/ 79.1%

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

      2. fma-udef 59.2%

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

      3. cube-mult 59.2%

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

      4. distribute-rgt1-in 79.0%

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

      5. associate-+r+ 79.0%

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

      6. *-commutative 79.0%

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

      7. frac-times 89.5%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 82.0%

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

      1. +-commutative 82.0%

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

   9. Simplified 82.0%

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

   if 1.5 < y

   1. Initial program 63.0%

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

      1. associate-+r+ 63.0%

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

      2. *-commutative 63.0%

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

      3. frac-times 85.4%

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

      4. associate-*l/ 85.4%

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

      5. times-frac 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-+l+ 99.9%

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

      9. +-commutative 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. clear-num 99.8%

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

      2. +-commutative 99.8%

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

      3. frac-times 99.0%

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

      4. *-un-lft-identity 99.0%

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

      5. +-commutative 99.0%

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

      6. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in y around inf 86.6%

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

      1. *-commutative 86.6%

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

   9. Simplified 86.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-42)
   (/ (* (/ y (+ y x)) (/ x (+ x 1.0))) (+ y x))
   (/ (/ x (+ y 1.0)) (* (+ y x) (/ (+ y x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-42) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.5d-42) then
        tmp = ((y / (y + x)) * (x / (x + 1.0d0))) / (y + x)
    else
        tmp = (x / (y + 1.0d0)) / ((y + x) * ((y + x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-42) {
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	} else {
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-42:
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x)
	else:
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-42)
		tmp = Float64(Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(x + 1.0))) / Float64(y + x));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(Float64(y + x) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-42)
		tmp = ((y / (y + x)) * (x / (x + 1.0))) / (y + x);
	else
		tmp = (x / (y + 1.0)) / ((y + x) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-42], N[(N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.50000000000000001e-42

   1. Initial program 78.6%

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

      1. associate-*l* 78.6%

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

      2. +-commutative 78.6%

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

      3. +-commutative 78.6%

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

      4. +-commutative 78.6%

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

      5. associate-*l* 78.6%

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

      6. associate-*l/ 85.5%

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

      7. *-commutative 85.5%

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

      8. *-commutative 85.5%

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

      9. distribute-rgt1-in 62.6%

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

      10. fma-def 85.5%

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

      11. +-commutative 85.5%

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

      12. +-commutative 85.5%

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

      13. cube-unmult 85.5%

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

      14. +-commutative 85.5%

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

   3. Simplified 85.5%

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

      1. associate-*r/ 78.7%

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

      2. fma-udef 57.8%

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

      3. cube-mult 57.7%

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

      4. distribute-rgt1-in 78.6%

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

      5. associate-+r+ 78.6%

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

      6. *-commutative 78.6%

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

      7. frac-times 89.0%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 81.4%

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

      1. +-commutative 81.4%

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

   9. Simplified 81.4%

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

   if 2.50000000000000001e-42 < y

   1. Initial program 65.5%

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

      1. associate-+r+ 65.5%

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

      2. *-commutative 65.5%

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

      3. frac-times 86.9%

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

      4. associate-*l/ 86.6%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. +-commutative 99.8%

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

      3. frac-times 99.0%

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

      4. *-un-lft-identity 99.0%

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

      5. +-commutative 99.0%

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

      6. +-commutative 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in x around 0 95.3%

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

      1. +-commutative 95.3%

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

   9. Simplified 95.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{y}{y + x} \cdot \frac{x}{x + 1}}{y + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{\left(y + x\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-+r+ 74.3%

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

   2. *-commutative 74.3%

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

   3. frac-times 88.3%

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

   4. associate-*l/ 84.5%

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

   5. times-frac 99.9%

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

   6. +-commutative 99.9%

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

   7. +-commutative 99.9%

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

   8. associate-+l+ 99.9%

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

   9. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

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

Alternative 6: 61.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-88)
   (/ (/ y x) (+ x 1.0))
   (/ (/ x (+ y (+ x 1.0))) (+ y (* x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-88) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + (x + 1.0d0))) / (y + (x * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-88:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-88)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + Float64(x + 1.0))) / Float64(y + Float64(x * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-88)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + (x + 1.0))) / (y + (x * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-88], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.9999999999999999e-88

   1. Initial program 78.1%

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

      1. associate-*l* 78.1%

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

      2. +-commutative 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-*l* 78.1%

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

      6. *-commutative 78.1%

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

      7. times-frac 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around 0 54.7%

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

      1. associate-/r* 59.0%

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

      2. +-commutative 59.0%

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

   7. Simplified 59.0%

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

   if 2.9999999999999999e-88 < y

   1. Initial program 67.7%

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

      1. associate-+r+ 67.7%

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

      2. *-commutative 67.7%

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

      3. frac-times 88.1%

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

      4. associate-*l/ 86.8%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. clear-num 99.8%

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

      2. +-commutative 99.8%

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

      3. frac-times 99.1%

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

      4. *-un-lft-identity 99.1%

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

      5. +-commutative 99.1%

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

      6. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in y around inf 79.6%

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

      1. *-commutative 79.6%

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

   9. Simplified 79.6%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + \left(x + 1\right)}}{y + x \cdot 2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y + x}}{-1 - \left(y + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.85e-88)
   (/ (/ y x) (+ x 1.0))
   (/ (/ (- x) (+ y x)) (- -1.0 (+ y x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (-x / (y + x)) / (-1.0 - (y + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.85d-88) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (-x / (y + x)) / ((-1.0d0) - (y + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.85e-88) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (-x / (y + x)) / (-1.0 - (y + x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.85e-88:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (-x / (y + x)) / (-1.0 - (y + x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.85e-88)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(-x) / Float64(y + x)) / Float64(-1.0 - Float64(y + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.85e-88)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (-x / (y + x)) / (-1.0 - (y + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.85e-88], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.8499999999999999e-88

   1. Initial program 78.1%

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

      1. associate-*l* 78.1%

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

      2. +-commutative 78.1%

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

      3. +-commutative 78.1%

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

      4. +-commutative 78.1%

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

      5. associate-*l* 78.1%

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

      6. *-commutative 78.1%

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

      7. times-frac 88.4%

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

      8. +-commutative 88.4%

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

      9. +-commutative 88.4%

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

      10. +-commutative 88.4%

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

      11. associate-+l+ 88.4%

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

   3. Simplified 88.4%

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

   5. Taylor expanded in y around 0 54.7%

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

      1. associate-/r* 59.0%

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

      2. +-commutative 59.0%

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

   7. Simplified 59.0%

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

   if 1.8499999999999999e-88 < y

   1. Initial program 67.7%

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

      1. associate-*l* 67.7%

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

      2. +-commutative 67.7%

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

      3. +-commutative 67.7%

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

      4. +-commutative 67.7%

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

      5. associate-*l* 67.7%

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

      6. associate-*l/ 82.0%

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

      7. *-commutative 82.0%

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

      8. *-commutative 82.0%

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

      9. distribute-rgt1-in 74.7%

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

      10. fma-def 82.0%

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

      11. +-commutative 82.0%

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

      12. +-commutative 82.0%

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

      13. cube-unmult 81.9%

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

      14. +-commutative 81.9%

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

   3. Simplified 81.9%

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

      1. associate-*r/ 67.7%

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

      2. fma-udef 64.5%

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

      3. cube-mult 64.6%

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

      4. distribute-rgt1-in 67.7%

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

      5. associate-+r+ 67.7%

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

      6. *-commutative 67.7%

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

      7. frac-times 88.1%

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

      8. associate-/r* 99.8%

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

      9. frac-2neg 99.8%

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

      10. frac-times 88.2%

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

      11. +-commutative 88.2%

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

      12. +-commutative 88.2%

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

      13. associate-+r+ 88.2%

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

      14. +-commutative 88.2%

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

      15. distribute-neg-in 88.2%

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

      16. metadata-eval 88.2%

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

      17. +-commutative 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. associate-/r* 99.9%

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

      2. associate-*l/ 74.6%

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

      3. distribute-rgt-neg-out 74.6%

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

      4. *-commutative 74.6%

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

      5. distribute-rgt-neg-in 74.6%

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

      6. +-commutative 74.6%

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

      7. +-commutative 74.6%

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

      8. unsub-neg 74.6%

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

      9. +-commutative 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in x around 0 79.1%

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

       1. neg-mul-1 79.1%

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

   11. Simplified 79.1%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-88}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{y + x}}{-1 - \left(y + x\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-84) (/ (/ y x) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ y x))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d-84) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-84:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-84)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e-84)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-84], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.0000000000000003e-84

   1. Initial program 78.2%

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

      1. associate-*l* 78.2%

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

      2. +-commutative 78.2%

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

      3. +-commutative 78.2%

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

      4. +-commutative 78.2%

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

      5. associate-*l* 78.2%

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

      6. *-commutative 78.2%

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

      7. times-frac 88.5%

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

      8. +-commutative 88.5%

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

      9. +-commutative 88.5%

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

      10. +-commutative 88.5%

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

      11. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in y around 0 54.4%

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

      1. associate-/r* 58.6%

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

      2. +-commutative 58.6%

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

   7. Simplified 58.6%

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

   if 8.0000000000000003e-84 < y

   1. Initial program 67.4%

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

      1. associate-*l* 67.4%

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

      2. +-commutative 67.4%

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

      3. +-commutative 67.4%

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

      4. +-commutative 67.4%

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

      5. associate-*l* 67.4%

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

      6. associate-*l/ 81.8%

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

      7. *-commutative 81.8%

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

      8. *-commutative 81.8%

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

      9. distribute-rgt1-in 74.4%

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

      10. fma-def 81.8%

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

      11. +-commutative 81.8%

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

      12. +-commutative 81.8%

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

      13. cube-unmult 81.7%

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

      14. +-commutative 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 67.4%

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

      2. fma-udef 64.2%

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

      3. cube-mult 64.2%

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

      4. distribute-rgt1-in 67.4%

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

      5. associate-+r+ 67.4%

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

      6. *-commutative 67.4%

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

      7. frac-times 88.0%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around 0 78.1%

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

      1. +-commutative 78.1%

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

   9. Simplified 78.1%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.76) (- (/ x y) x) (* (/ x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.76d0) then
        tmp = (x / y) - x
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.76) {
		tmp = (x / y) - x;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.76:
		tmp = (x / y) - x
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.76)
		tmp = Float64(Float64(x / y) - x);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.76)
		tmp = (x / y) - x;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.76], N[(N[(x / y), $MachinePrecision] - x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.76000000000000001

   1. Initial program 79.0%

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

      1. associate-*l* 79.0%

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

      2. +-commutative 79.0%

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

      3. +-commutative 79.0%

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

      4. +-commutative 79.0%

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

      5. associate-*l* 79.0%

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

      6. *-commutative 79.0%

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

      7. times-frac 89.5%

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

      8. +-commutative 89.5%

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

      9. +-commutative 89.5%

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

      10. +-commutative 89.5%

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

      11. associate-+l+ 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in x around 0 42.0%

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

      1. +-commutative 42.0%

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

   7. Simplified 42.0%

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

   8. Taylor expanded in y around 0 18.7%

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

      1. neg-mul-1 18.7%

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

      2. +-commutative 18.7%

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

      3. unsub-neg 18.7%

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

   10. Simplified 18.7%

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

   if 0.76000000000000001 < y

   1. Initial program 63.0%

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

      1. associate-*l* 62.9%

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

      2. +-commutative 62.9%

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

      3. +-commutative 62.9%

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

      4. +-commutative 62.9%

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

      5. associate-*l* 63.0%

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

      6. *-commutative 63.0%

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

      7. times-frac 85.4%

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

      8. +-commutative 85.4%

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

      9. +-commutative 85.4%

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

      10. +-commutative 85.4%

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

      11. associate-+l+ 85.4%

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y around inf 85.6%

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

   6. Taylor expanded in y around inf 82.9%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.76:\\ \;\;\;\;\frac{x}{y} - x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 50.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 30000000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 30000000000000.0) (/ x (* y (+ y 1.0))) (* (/ x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 30000000000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 30000000000000.0d0) then
        tmp = x / (y * (y + 1.0d0))
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 30000000000000.0) {
		tmp = x / (y * (y + 1.0));
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 30000000000000.0:
		tmp = x / (y * (y + 1.0))
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 30000000000000.0)
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 30000000000000.0)
		tmp = x / (y * (y + 1.0));
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 30000000000000.0], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3e13

   1. Initial program 79.7%

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

      1. associate-*l* 79.7%

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

      2. +-commutative 79.7%

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

      3. +-commutative 79.7%

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

      4. +-commutative 79.7%

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

      5. associate-*l* 79.7%

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

      6. *-commutative 79.7%

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

      7. times-frac 89.8%

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

      8. +-commutative 89.8%

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

      9. +-commutative 89.8%

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

      10. +-commutative 89.8%

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

      11. associate-+l+ 89.8%

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

   3. Simplified 89.8%

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

   5. Taylor expanded in x around 0 42.8%

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

      1. +-commutative 42.8%

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

   7. Simplified 42.8%

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

   if 3e13 < y

   1. Initial program 59.8%

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

      1. associate-*l* 59.8%

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

      2. +-commutative 59.8%

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

      3. +-commutative 59.8%

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

      4. +-commutative 59.8%

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

      5. associate-*l* 59.8%

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

      6. *-commutative 59.8%

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

      7. times-frac 84.2%

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

      8. +-commutative 84.2%

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

      9. +-commutative 84.2%

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

      10. +-commutative 84.2%

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

      11. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in y around inf 87.2%

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

   6. Taylor expanded in y around inf 86.9%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 30000000000000:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-84) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-84) {
		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 <= 8d-84) 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 <= 8e-84) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-84:
		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 <= 8e-84)
		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 (y <= 8e-84)
		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, 8e-84], 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 y < 8.0000000000000003e-84

   1. Initial program 78.2%

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

      1. associate-*l* 78.2%

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

      2. +-commutative 78.2%

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

      3. +-commutative 78.2%

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

      4. +-commutative 78.2%

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

      5. associate-*l* 78.2%

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

      6. *-commutative 78.2%

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

      7. times-frac 88.5%

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

      8. +-commutative 88.5%

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

      9. +-commutative 88.5%

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

      10. +-commutative 88.5%

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

      11. associate-+l+ 88.5%

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

   3. Simplified 88.5%

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

   5. Taylor expanded in y around 0 54.4%

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

      1. associate-/r* 58.6%

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

      2. +-commutative 58.6%

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

   7. Simplified 58.6%

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

   if 8.0000000000000003e-84 < y

   1. Initial program 67.4%

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

      1. associate-*l* 67.4%

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

      2. +-commutative 67.4%

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

      3. +-commutative 67.4%

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

      4. +-commutative 67.4%

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

      5. associate-*l* 67.4%

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

      6. associate-*l/ 81.8%

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

      7. *-commutative 81.8%

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

      8. *-commutative 81.8%

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

      9. distribute-rgt1-in 74.4%

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

      10. fma-def 81.8%

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

      11. +-commutative 81.8%

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

      12. +-commutative 81.8%

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

      13. cube-unmult 81.7%

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

      14. +-commutative 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 67.4%

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

      2. fma-udef 64.2%

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

      3. cube-mult 64.2%

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

      4. distribute-rgt1-in 67.4%

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

      5. associate-+r+ 67.4%

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

      6. *-commutative 67.4%

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

      7. frac-times 88.0%

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

      8. associate-/r* 99.8%

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

      9. associate-*l/ 99.9%

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

      10. +-commutative 99.9%

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

      11. +-commutative 99.9%

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

      12. associate-+l+ 99.9%

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

      13. +-commutative 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. add-cube-cbrt 99.7%

         \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\frac{y}{y + x}} \cdot \sqrt[3]{\frac{y}{y + x}}\right) \cdot \sqrt[3]{\frac{y}{y + x}}\right)} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

      2. pow3 99.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{y + x}}\right)}^{3}} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

   8. Applied egg-rr 99.7%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{y + x}}\right)}^{3}} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

   9. Taylor expanded in x around 0 71.7%

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

       1. associate-/r* 77.8%

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

   11. Simplified 77.8%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{x}{y}}{y + 1} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-*l* 74.3%

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

   2. +-commutative 74.3%

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

   3. +-commutative 74.3%

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

   4. +-commutative 74.3%

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

   5. associate-*l* 74.3%

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

   6. associate-*l/ 83.7%

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

   7. *-commutative 83.7%

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

   8. *-commutative 83.7%

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

   9. distribute-rgt1-in 66.5%

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

   10. fma-def 83.7%

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

   11. +-commutative 83.7%

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

   12. +-commutative 83.7%

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

   13. cube-unmult 83.7%

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

   14. +-commutative 83.7%

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

3. Simplified 83.7%

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

   1. associate-*r/ 74.3%

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

   2. fma-udef 60.0%

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

   3. cube-mult 59.9%

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

   4. distribute-rgt1-in 74.3%

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

   5. associate-+r+ 74.3%

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

   6. *-commutative 74.3%

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

   7. frac-times 88.3%

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

   8. associate-/r* 99.8%

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

   9. associate-*l/ 99.9%

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

   10. +-commutative 99.9%

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

   11. +-commutative 99.9%

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

   12. associate-+l+ 99.9%

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

   13. +-commutative 99.9%

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

6. Applied egg-rr 99.9%

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

   1. add-cube-cbrt 99.4%

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\frac{y}{y + x}} \cdot \sqrt[3]{\frac{y}{y + x}}\right) \cdot \sqrt[3]{\frac{y}{y + x}}\right)} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

   2. pow3 99.4%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{y + x}}\right)}^{3}} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

8. Applied egg-rr 99.4%

   \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{\frac{y}{y + x}}\right)}^{3}} \cdot \frac{x}{y + \left(1 + x\right)}}{y + x} \]

9. Taylor expanded in x around 0 52.7%

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

    1. associate-/r* 55.4%

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

11. Simplified 55.4%

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

12. Final simplification 55.4%

    \[\leadsto \frac{\frac{x}{y}}{y + 1} \]
13. Add Preprocessing

Alternative 13: 27.7% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -6.5) (/ 1.0 x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -6.5) {
		tmp = 1.0 / 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 <= (-6.5d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -6.5) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -6.5:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -6.5)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -6.5)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -6.5], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.5

   1. Initial program 67.3%

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

      1. associate-+r+ 67.3%

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

      2. *-commutative 67.3%

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

      3. frac-times 82.3%

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

      4. associate-*l/ 82.3%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

      8. associate-+l+ 99.8%

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

      9. +-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around inf 75.7%

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

   6. Taylor expanded in y around inf 6.2%

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

   if -6.5 < x

   1. Initial program 76.8%

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

      1. associate-*l* 76.8%

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

      2. +-commutative 76.8%

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

      3. +-commutative 76.8%

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

      4. +-commutative 76.8%

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

      5. associate-*l* 76.8%

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

      6. *-commutative 76.8%

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

      7. times-frac 90.4%

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

      8. +-commutative 90.4%

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

      9. +-commutative 90.4%

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

      10. +-commutative 90.4%

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

      11. associate-+l+ 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in x around 0 64.9%

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

      1. +-commutative 64.9%

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

   7. Simplified 64.9%

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

   8. Taylor expanded in y around 0 34.2%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 4.2% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. associate-+r+ 74.3%

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

   2. *-commutative 74.3%

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

   3. frac-times 88.3%

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

   4. associate-*l/ 84.5%

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

   5. times-frac 99.9%

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

   6. +-commutative 99.9%

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

   7. +-commutative 99.9%

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

   8. associate-+l+ 99.9%

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

   9. +-commutative 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in x around inf 36.2%

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

6. Taylor expanded in y around inf 4.1%

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

7. Final simplification 4.1%

   \[\leadsto \frac{1}{x} \]
8. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x))))

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