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

Percentage Accurate: 69.9% → 99.8%

Time: 13.7s

Alternatives: 18

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: 69.9% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

   1. +-commutative 65.4%

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

   2. +-commutative 65.4%

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

   3. +-commutative 65.4%

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

   4. *-commutative 65.4%

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

   5. distribute-rgt1-in 53.8%

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

   6. fma-define 65.4%

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

   7. +-commutative 65.4%

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

   8. +-commutative 65.4%

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

   9. cube-unmult 65.4%

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

   10. +-commutative 65.4%

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

3. Simplified 65.4%

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

   1. *-commutative 65.4%

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

   2. fma-define 53.8%

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

   3. cube-mult 53.8%

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

   4. distribute-rgt1-in 65.4%

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

   5. *-commutative 65.4%

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

   6. associate-*l* 65.4%

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

   7. times-frac 93.0%

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

   8. associate-+r+ 93.0%

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

6. Applied egg-rr 93.0%

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

   1. associate-*r/ 93.0%

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

   2. +-commutative 93.0%

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

   3. +-commutative 93.0%

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

   4. +-commutative 93.0%

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

   5. associate-+l+ 93.0%

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

   6. +-commutative 93.0%

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

8. Applied egg-rr 93.0%

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

   1. *-un-lft-identity 93.0%

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

   2. times-frac 99.7%

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

   3. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 99.9%

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

    2. *-lft-identity 99.9%

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

    3. associate-/l* 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 96.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+50} \lor \neg \left(y \leq 1.35 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x \cdot \frac{t\_0}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (or (<= y -4e+50) (not (<= y 1.35e-24)))
     (/ (* x (/ t_0 (+ x y))) (+ x y))
     (* t_0 (/ x (* (+ x y) (+ x 1.0)))))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if ((y <= -4e+50) || !(y <= 1.35e-24)) {
		tmp = (x * (t_0 / (x + y))) / (x + y);
	} else {
		tmp = t_0 * (x / ((x + y) * (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if ((y <= (-4d+50)) .or. (.not. (y <= 1.35d-24))) then
        tmp = (x * (t_0 / (x + y))) / (x + y)
    else
        tmp = t_0 * (x / ((x + y) * (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if ((y <= -4e+50) || !(y <= 1.35e-24)) {
		tmp = (x * (t_0 / (x + y))) / (x + y);
	} else {
		tmp = t_0 * (x / ((x + y) * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if (y <= -4e+50) or not (y <= 1.35e-24):
		tmp = (x * (t_0 / (x + y))) / (x + y)
	else:
		tmp = t_0 * (x / ((x + y) * (x + 1.0)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if ((y <= -4e+50) || !(y <= 1.35e-24))
		tmp = Float64(Float64(x * Float64(t_0 / Float64(x + y))) / Float64(x + y));
	else
		tmp = Float64(t_0 * Float64(x / Float64(Float64(x + y) * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if ((y <= -4e+50) || ~((y <= 1.35e-24)))
		tmp = (x * (t_0 / (x + y))) / (x + y);
	else
		tmp = t_0 * (x / ((x + y) * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y, -4e+50], N[Not[LessEqual[y, 1.35e-24]], $MachinePrecision]], N[(N[(x * N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.0000000000000003e50 or 1.35000000000000003e-24 < y

   1. Initial program 57.7%

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

      1. +-commutative 57.7%

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

      2. +-commutative 57.7%

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

      3. +-commutative 57.7%

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

      4. *-commutative 57.7%

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

      5. distribute-rgt1-in 43.8%

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

      6. fma-define 57.7%

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

      7. +-commutative 57.7%

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

      8. +-commutative 57.7%

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

      9. cube-unmult 57.7%

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

      10. +-commutative 57.7%

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

   3. Simplified 57.7%

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

      1. *-commutative 57.7%

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

      2. fma-define 43.8%

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

      3. cube-mult 43.8%

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

      4. distribute-rgt1-in 57.7%

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

      5. *-commutative 57.7%

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

      6. associate-*l* 57.7%

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

      7. times-frac 86.0%

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

      8. associate-+r+ 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. associate-*r/ 86.0%

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

      2. +-commutative 86.0%

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

      3. +-commutative 86.0%

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

      4. +-commutative 86.0%

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

      5. associate-+l+ 86.0%

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

      6. +-commutative 86.0%

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

   8. Applied egg-rr 86.0%

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

      1. *-un-lft-identity 86.0%

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

      2. times-frac 99.8%

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

      3. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 99.8%

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

   if -4.0000000000000003e50 < y < 1.35000000000000003e-24

   1. Initial program 73.0%

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

   3. Taylor expanded in x around inf 72.7%

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

      1. *-commutative 72.7%

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

      2. associate-*l* 72.7%

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

      3. times-frac 99.6%

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

      4. +-commutative 99.6%

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

      5. +-commutative 99.6%

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

   5. Applied egg-rr 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+50} \lor \neg \left(y \leq 1.35 \cdot 10^{-24}\right):\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{x + y}}{x + y}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 0.014:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 0.014)
     (* (/ y (+ x y)) (/ x (* (+ x y) (+ x 1.0))))
     (if (<= y 1.22e+149)
       (/ x (* (+ x y) t_0))
       (* (/ 1.0 t_0) (/ x (+ x y)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 0.014) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.22e+149) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 0.014d0) then
        tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0d0)))
    else if (y <= 1.22d+149) then
        tmp = x / ((x + y) * t_0)
    else
        tmp = (1.0d0 / t_0) * (x / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 0.014) {
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	} else if (y <= 1.22e+149) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 0.014:
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)))
	elif y <= 1.22e+149:
		tmp = x / ((x + y) * t_0)
	else:
		tmp = (1.0 / t_0) * (x / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 0.014)
		tmp = Float64(Float64(y / Float64(x + y)) * Float64(x / Float64(Float64(x + y) * Float64(x + 1.0))));
	elseif (y <= 1.22e+149)
		tmp = Float64(x / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 0.014)
		tmp = (y / (x + y)) * (x / ((x + y) * (x + 1.0)));
	elseif (y <= 1.22e+149)
		tmp = x / ((x + y) * t_0);
	else
		tmp = (1.0 / t_0) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.014], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(x + y), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.22e+149], N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0140000000000000003

   1. Initial program 64.7%

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

   3. Taylor expanded in x around inf 62.1%

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

      1. *-commutative 62.1%

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

      2. associate-*l* 62.1%

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

      3. times-frac 88.8%

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

      4. +-commutative 88.8%

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

      5. +-commutative 88.8%

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

   5. Applied egg-rr 88.8%

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

   if 0.0140000000000000003 < y < 1.22e149

   1. Initial program 68.4%

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

      1. +-commutative 68.4%

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

      2. +-commutative 68.4%

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

      3. +-commutative 68.4%

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

      4. *-commutative 68.4%

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

      5. distribute-rgt1-in 60.0%

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

      6. fma-define 68.4%

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

      7. +-commutative 68.4%

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

      8. +-commutative 68.4%

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

      9. cube-unmult 68.5%

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

      10. +-commutative 68.5%

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

   3. Simplified 68.5%

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

      1. *-commutative 68.5%

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

      2. fma-define 60.1%

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

      3. cube-mult 60.0%

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

      4. distribute-rgt1-in 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-*l* 68.4%

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

      7. times-frac 87.1%

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

      8. associate-+r+ 87.1%

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

   6. Applied egg-rr 87.1%

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

      1. associate-*r/ 87.1%

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

      2. +-commutative 87.1%

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

      3. +-commutative 87.1%

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

      4. +-commutative 87.1%

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

      5. associate-+l+ 87.1%

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

      6. +-commutative 87.1%

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

   8. Applied egg-rr 87.1%

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

   9. Taylor expanded in y around inf 84.1%

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

   if 1.22e149 < y

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. +-commutative 66.7%

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

      3. +-commutative 66.7%

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

      4. *-commutative 66.7%

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

      5. distribute-rgt1-in 66.7%

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

      6. fma-define 66.7%

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

      7. +-commutative 66.7%

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

      8. +-commutative 66.7%

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

      9. cube-unmult 66.7%

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

      10. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. fma-define 66.7%

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

      3. cube-mult 66.7%

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

      4. distribute-rgt1-in 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-*l* 66.7%

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

      7. times-frac 84.5%

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

      8. associate-+r+ 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. *-commutative 84.5%

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

      3. +-commutative 84.5%

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

      4. associate-+l+ 84.5%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 93.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.014:\\ \;\;\;\;\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \frac{x}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 3.8e-85)
     (/ (/ y (+ x y)) (+ x 1.0))
     (if (<= y 5e+146) (/ x (* (+ x y) t_0)) (* (/ 1.0 t_0) (/ x (+ x y)))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.8e-85) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (y <= 5e+146) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (y <= 3.8d-85) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else if (y <= 5d+146) then
        tmp = x / ((x + y) * t_0)
    else
        tmp = (1.0d0 / t_0) * (x / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 3.8e-85) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (y <= 5e+146) {
		tmp = x / ((x + y) * t_0);
	} else {
		tmp = (1.0 / t_0) * (x / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 3.8e-85:
		tmp = (y / (x + y)) / (x + 1.0)
	elif y <= 5e+146:
		tmp = x / ((x + y) * t_0)
	else:
		tmp = (1.0 / t_0) * (x / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 3.8e-85)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	elseif (y <= 5e+146)
		tmp = Float64(x / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(1.0 / t_0) * Float64(x / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 3.8e-85)
		tmp = (y / (x + y)) / (x + 1.0);
	elseif (y <= 5e+146)
		tmp = x / ((x + y) * t_0);
	else
		tmp = (1.0 / t_0) * (x / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 3.8e-85], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5e+146], N[(x / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.7999999999999999e-85

   1. Initial program 64.0%

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

   3. Taylor expanded in x around inf 61.1%

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

   4. Taylor expanded in x around inf 44.1%

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

      1. times-frac 60.6%

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

      2. +-commutative 60.6%

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

      3. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. associate-/r* 63.4%

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

      3. *-inverses 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-*r/ 63.5%

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

      6. associate-*l/ 63.5%

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

      7. *-rgt-identity 63.5%

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

   8. Simplified 63.5%

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

   if 3.7999999999999999e-85 < y < 4.9999999999999999e146

   1. Initial program 69.2%

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

      1. +-commutative 69.2%

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

      2. +-commutative 69.2%

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

      3. +-commutative 69.2%

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

      4. *-commutative 69.2%

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

      5. distribute-rgt1-in 62.3%

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

      6. fma-define 69.2%

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

      7. +-commutative 69.2%

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

      8. +-commutative 69.2%

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

      9. cube-unmult 69.3%

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

      10. +-commutative 69.3%

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

   3. Simplified 69.3%

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

      1. *-commutative 69.3%

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

      2. fma-define 62.4%

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

      3. cube-mult 62.3%

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

      4. distribute-rgt1-in 69.2%

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

      5. *-commutative 69.2%

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

      6. associate-*l* 69.2%

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

      7. times-frac 92.2%

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

      8. associate-+r+ 92.2%

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

   6. Applied egg-rr 92.2%

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

      1. associate-*r/ 92.3%

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

      2. +-commutative 92.3%

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

      3. +-commutative 92.3%

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

      4. +-commutative 92.3%

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

      5. associate-+l+ 92.3%

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

      6. +-commutative 92.3%

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

   8. Applied egg-rr 92.3%

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

   9. Taylor expanded in y around inf 71.0%

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

   if 4.9999999999999999e146 < y

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. +-commutative 66.7%

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

      3. +-commutative 66.7%

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

      4. *-commutative 66.7%

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

      5. distribute-rgt1-in 66.7%

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

      6. fma-define 66.7%

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

      7. +-commutative 66.7%

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

      8. +-commutative 66.7%

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

      9. cube-unmult 66.7%

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

      10. +-commutative 66.7%

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

   3. Simplified 66.7%

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

      1. *-commutative 66.7%

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

      2. fma-define 66.7%

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

      3. cube-mult 66.7%

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

      4. distribute-rgt1-in 66.7%

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

      5. *-commutative 66.7%

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

      6. associate-*l* 66.7%

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

      7. times-frac 84.5%

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

      8. associate-+r+ 84.5%

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

   6. Applied egg-rr 84.5%

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

      1. *-un-lft-identity 84.5%

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

      2. *-commutative 84.5%

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

      3. +-commutative 84.5%

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

      4. associate-+l+ 84.5%

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

      5. times-frac 99.8%

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

      6. +-commutative 99.8%

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

      7. +-commutative 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around inf 93.4%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y + \left(x + 1\right)} \cdot \frac{x}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5e-85)
   (/ (/ y (+ x y)) (+ x 1.0))
   (if (<= y 7.2e+143) (/ x (* (+ x y) (+ y (+ x 1.0)))) (/ (/ x y) (+ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5e-85) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (y <= 7.2e+143) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (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 <= 5d-85) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else if (y <= 7.2d+143) then
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / y) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-85) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else if (y <= 7.2e+143) {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / y) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5e-85:
		tmp = (y / (x + y)) / (x + 1.0)
	elif y <= 7.2e+143:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / y) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5e-85)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	elseif (y <= 7.2e+143)
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / y) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-85)
		tmp = (y / (x + y)) / (x + 1.0);
	elseif (y <= 7.2e+143)
		tmp = x / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / y) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5e-85], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+143], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 5.0000000000000002e-85

   1. Initial program 64.0%

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

   3. Taylor expanded in x around inf 61.1%

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

   4. Taylor expanded in x around inf 44.1%

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

      1. times-frac 60.6%

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

      2. +-commutative 60.6%

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

      3. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. associate-/r* 63.4%

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

      3. *-inverses 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-*r/ 63.5%

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

      6. associate-*l/ 63.5%

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

      7. *-rgt-identity 63.5%

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

   8. Simplified 63.5%

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

   if 5.0000000000000002e-85 < y < 7.1999999999999998e143

   1. Initial program 70.5%

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

      1. +-commutative 70.5%

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

      2. +-commutative 70.5%

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

      3. +-commutative 70.5%

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

      4. *-commutative 70.5%

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

      5. distribute-rgt1-in 63.5%

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

      6. fma-define 70.5%

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

      7. +-commutative 70.5%

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

      8. +-commutative 70.5%

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

      9. cube-unmult 70.6%

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

      10. +-commutative 70.6%

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

   3. Simplified 70.6%

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

      1. *-commutative 70.6%

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

      2. fma-define 63.6%

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

      3. cube-mult 63.5%

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

      4. distribute-rgt1-in 70.5%

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

      5. *-commutative 70.5%

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

      6. associate-*l* 70.5%

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

      7. times-frac 93.8%

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

      8. associate-+r+ 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. associate-*r/ 93.9%

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

      2. +-commutative 93.9%

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

      3. +-commutative 93.9%

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

      4. +-commutative 93.9%

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

      5. associate-+l+ 93.9%

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

      6. +-commutative 93.9%

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

   8. Applied egg-rr 93.9%

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

   9. Taylor expanded in y around inf 72.2%

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

   if 7.1999999999999998e143 < y

   1. Initial program 64.5%

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

      1. +-commutative 64.5%

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

      2. +-commutative 64.5%

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

      3. +-commutative 64.5%

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

      4. *-commutative 64.5%

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

      5. distribute-rgt1-in 64.5%

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

      6. fma-define 64.5%

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

      7. +-commutative 64.5%

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

      8. +-commutative 64.5%

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

      9. cube-unmult 64.5%

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

      10. +-commutative 64.5%

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

   3. Simplified 64.5%

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

      1. *-commutative 64.5%

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

      2. fma-define 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 64.5%

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

      4. distribute-rgt1-in 64.5%

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

      5. *-commutative 64.5%

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

      6. associate-*l* 64.5%

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

      7. times-frac 82.0%

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

      8. associate-+r+ 82.0%

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

   6. Applied egg-rr 82.0%

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

      1. associate-*r/ 82.0%

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

      2. +-commutative 82.0%

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

      3. +-commutative 82.0%

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

      4. +-commutative 82.0%

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

      5. associate-+l+ 82.0%

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

      6. +-commutative 82.0%

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

   8. Applied egg-rr 82.0%

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

      1. *-un-lft-identity 82.0%

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

      2. times-frac 99.7%

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

      3. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in y around inf 90.7%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-143}:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y -2.8e-143)
   (/ y (* x x))
   (if (<= y 5.5e-118) (/ y x) (if (<= y 1.0) (/ x y) (/ x (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-143) {
		tmp = y / (x * x);
	} else if (y <= 5.5e-118) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * 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 <= (-2.8d-143)) then
        tmp = y / (x * x)
    else if (y <= 5.5d-118) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= -2.8e-143) {
		tmp = y / (x * x);
	} else if (y <= 5.5e-118) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= -2.8e-143:
		tmp = y / (x * x)
	elif y <= 5.5e-118:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= -2.8e-143)
		tmp = Float64(y / Float64(x * x));
	elseif (y <= 5.5e-118)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= -2.8e-143)
		tmp = y / (x * x);
	elseif (y <= 5.5e-118)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, -2.8e-143], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.5e-118], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.7999999999999999e-143

   1. Initial program 61.9%

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

      1. associate-/l* 82.8%

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

      2. associate-+l+ 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around 0 40.8%

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

      1. +-commutative 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in x around inf 38.0%

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

   if -2.7999999999999999e-143 < y < 5.5000000000000003e-118

   1. Initial program 63.7%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 84.5%

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

      1. +-commutative 84.5%

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

   7. Simplified 84.5%

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

   8. Taylor expanded in x around 0 70.2%

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

   if 5.5000000000000003e-118 < y < 1

   1. Initial program 75.0%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 34.0%

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

   6. Taylor expanded in y around 0 34.0%

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

   if 1 < y

   1. Initial program 67.6%

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

      1. associate-/l* 79.7%

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

      2. associate-+l+ 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in x around 0 75.1%

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

   6. Taylor expanded in y around inf 75.0%

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

Alternative 7: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-115}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (* y y))))
   (if (<= y -4e+50)
     t_0
     (if (<= y 4.2e-115) (/ y x) (if (<= y 1.0) (/ x y) t_0)))))

C

double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -4e+50) {
		tmp = t_0;
	} else if (y <= 4.2e-115) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / (y * y)
    if (y <= (-4d+50)) then
        tmp = t_0
    else if (y <= 4.2d-115) then
        tmp = y / x
    else if (y <= 1.0d0) then
        tmp = x / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = x / (y * y);
	double tmp;
	if (y <= -4e+50) {
		tmp = t_0;
	} else if (y <= 4.2e-115) {
		tmp = y / x;
	} else if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = x / (y * y)
	tmp = 0
	if y <= -4e+50:
		tmp = t_0
	elif y <= 4.2e-115:
		tmp = y / x
	elif y <= 1.0:
		tmp = x / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(x / Float64(y * y))
	tmp = 0.0
	if (y <= -4e+50)
		tmp = t_0;
	elseif (y <= 4.2e-115)
		tmp = Float64(y / x);
	elseif (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = x / (y * y);
	tmp = 0.0;
	if (y <= -4e+50)
		tmp = t_0;
	elseif (y <= 4.2e-115)
		tmp = y / x;
	elseif (y <= 1.0)
		tmp = x / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+50], t$95$0, If[LessEqual[y, 4.2e-115], N[(y / x), $MachinePrecision], If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.0000000000000003e50 or 1 < y

   1. Initial program 57.5%

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

      1. associate-/l* 78.9%

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

      2. associate-+l+ 78.9%

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

   3. Simplified 78.9%

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

   5. Taylor expanded in x around 0 71.8%

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

   6. Taylor expanded in y around inf 71.8%

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

   if -4.0000000000000003e50 < y < 4.20000000000000003e-115

   1. Initial program 71.6%

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

      1. associate-/l* 84.1%

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

      2. associate-+l+ 84.1%

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

   3. Simplified 84.1%

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

   5. Taylor expanded in y around 0 79.9%

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

      1. +-commutative 79.9%

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

   7. Simplified 79.9%

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

   8. Taylor expanded in x around 0 58.6%

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

   if 4.20000000000000003e-115 < y < 1

   1. Initial program 75.0%

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

      1. associate-/l* 81.9%

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

      2. associate-+l+ 81.9%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in x around 0 34.0%

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

   6. Taylor expanded in y around 0 34.0%

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

Alternative 8: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{x + y}\\ \mathbf{if}\;y \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{x \cdot t\_0}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{t\_0}{x + y}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ y (+ x y))))
   (if (<= y 1.35e-24)
     (/ (/ (* x t_0) (+ x y)) (+ x 1.0))
     (/ (* x (/ t_0 (+ x y))) (+ x y)))))

C

double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= 1.35e-24) {
		tmp = ((x * t_0) / (x + y)) / (x + 1.0);
	} else {
		tmp = (x * (t_0 / (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) :: t_0
    real(8) :: tmp
    t_0 = y / (x + y)
    if (y <= 1.35d-24) then
        tmp = ((x * t_0) / (x + y)) / (x + 1.0d0)
    else
        tmp = (x * (t_0 / (x + y))) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y / (x + y);
	double tmp;
	if (y <= 1.35e-24) {
		tmp = ((x * t_0) / (x + y)) / (x + 1.0);
	} else {
		tmp = (x * (t_0 / (x + y))) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y / (x + y)
	tmp = 0
	if y <= 1.35e-24:
		tmp = ((x * t_0) / (x + y)) / (x + 1.0)
	else:
		tmp = (x * (t_0 / (x + y))) / (x + y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y / Float64(x + y))
	tmp = 0.0
	if (y <= 1.35e-24)
		tmp = Float64(Float64(Float64(x * t_0) / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x * Float64(t_0 / Float64(x + y))) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y / (x + y);
	tmp = 0.0;
	if (y <= 1.35e-24)
		tmp = ((x * t_0) / (x + y)) / (x + 1.0);
	else
		tmp = (x * (t_0 / (x + y))) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.35e-24], N[(N[(N[(x * t$95$0), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(t$95$0 / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.35000000000000003e-24

   1. Initial program 64.9%

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

   3. Taylor expanded in x around inf 62.2%

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

      1. *-commutative 62.2%

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

      2. associate-*l* 62.2%

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

      3. times-frac 88.4%

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

      4. +-commutative 88.4%

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

      5. +-commutative 88.4%

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

   5. Applied egg-rr 88.4%

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

      1. associate-*r/ 88.4%

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

      2. associate-/r* 87.1%

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

      3. *-commutative 87.1%

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

      4. div-inv 87.0%

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

      5. div-inv 87.1%

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

      6. +-commutative 87.1%

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

      7. +-commutative 87.1%

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

   7. Applied egg-rr 87.1%

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

   if 1.35000000000000003e-24 < y

   1. Initial program 66.9%

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

      1. +-commutative 66.9%

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

      2. +-commutative 66.9%

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

      3. +-commutative 66.9%

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

      4. *-commutative 66.9%

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

      5. distribute-rgt1-in 63.0%

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

      6. fma-define 66.9%

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

      7. +-commutative 66.9%

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

      8. +-commutative 66.9%

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

      9. cube-unmult 66.9%

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

      10. +-commutative 66.9%

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

   3. Simplified 66.9%

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

      1. *-commutative 66.9%

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

      2. fma-define 63.0%

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

      3. cube-mult 63.0%

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

      4. distribute-rgt1-in 66.9%

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

      5. *-commutative 66.9%

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

      6. associate-*l* 66.9%

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

      7. times-frac 87.2%

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

      8. associate-+r+ 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. associate-*r/ 87.3%

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

      2. +-commutative 87.3%

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

      3. +-commutative 87.3%

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

      4. +-commutative 87.3%

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

      5. associate-+l+ 87.3%

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

      6. +-commutative 87.3%

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

   8. Applied egg-rr 87.3%

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

      1. *-un-lft-identity 87.3%

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

      2. times-frac 99.8%

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

      3. *-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around inf 99.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{x \cdot \frac{y}{x + y}}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{\frac{y}{x + y}}{x + y}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.0)
   (/ y (* x x))
   (if (<= x -1.04e-147) (/ y x) (/ x (* y (+ y 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = y / (x * x)
    else if (x <= (-1.04d-147)) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.0) {
		tmp = y / (x * x);
	} else if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.0:
		tmp = y / (x * x)
	elif x <= -1.04e-147:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(y / Float64(x * x));
	elseif (x <= -1.04e-147)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = y / (x * x);
	elseif (x <= -1.04e-147)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.0], N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.04e-147], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 64.6%

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

      1. associate-/l* 76.2%

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

      2. associate-+l+ 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in y around 0 70.1%

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

      1. +-commutative 70.1%

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

   7. Simplified 70.1%

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

   8. Taylor expanded in x around inf 68.2%

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

   if -1 < x < -1.04000000000000003e-147

   1. Initial program 78.3%

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

      1. associate-/l* 99.5%

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

      2. associate-+l+ 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around 0 49.4%

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

      1. +-commutative 49.4%

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

   7. Simplified 49.4%

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

   8. Taylor expanded in x around 0 49.4%

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

   if -1.04000000000000003e-147 < x

   1. Initial program 63.2%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 56.4%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{y}{x \cdot x}\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

   1. +-commutative 65.4%

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

   2. +-commutative 65.4%

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

   3. +-commutative 65.4%

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

   4. *-commutative 65.4%

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

   5. distribute-rgt1-in 53.8%

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

   6. fma-define 65.4%

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

   7. +-commutative 65.4%

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

   8. +-commutative 65.4%

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

   9. cube-unmult 65.4%

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

   10. +-commutative 65.4%

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

3. Simplified 65.4%

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

   1. *-commutative 65.4%

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

   2. fma-define 53.8%

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

   3. cube-mult 53.8%

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

   4. distribute-rgt1-in 65.4%

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

   5. *-commutative 65.4%

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

   6. associate-*l* 65.4%

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

   7. times-frac 93.0%

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

   8. associate-+r+ 93.0%

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

6. Applied egg-rr 93.0%

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

   1. associate-*r/ 93.0%

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

   2. +-commutative 93.0%

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

   3. +-commutative 93.0%

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

   4. +-commutative 93.0%

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

   5. associate-+l+ 93.0%

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

   6. +-commutative 93.0%

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

8. Applied egg-rr 93.0%

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

   1. times-frac 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 11: 62.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.52e-83) (/ (/ y (+ x y)) (+ x 1.0)) (/ (/ x (+ y 1.0)) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.52e-83) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.52d-83) then
        tmp = (y / (x + y)) / (x + 1.0d0)
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.52e-83) {
		tmp = (y / (x + y)) / (x + 1.0);
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.52e-83:
		tmp = (y / (x + y)) / (x + 1.0)
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.52e-83)
		tmp = Float64(Float64(y / Float64(x + y)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.52e-83)
		tmp = (y / (x + y)) / (x + 1.0);
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.52e-83], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.51999999999999994e-83

   1. Initial program 64.0%

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

   3. Taylor expanded in x around inf 61.1%

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

   4. Taylor expanded in x around inf 44.1%

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

      1. times-frac 60.6%

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

      2. +-commutative 60.6%

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

      3. *-commutative 60.6%

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

   6. Applied egg-rr 60.6%

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

      1. associate-*r/ 60.6%

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

      2. associate-/r* 63.4%

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

      3. *-inverses 63.4%

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

      4. *-commutative 63.4%

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

      5. associate-*r/ 63.5%

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

      6. associate-*l/ 63.5%

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

      7. *-rgt-identity 63.5%

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

   8. Simplified 63.5%

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

   if 1.51999999999999994e-83 < y

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. +-commutative 68.3%

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

      3. +-commutative 68.3%

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

      4. *-commutative 68.3%

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

      5. distribute-rgt1-in 63.9%

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

      6. fma-define 68.3%

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

      7. +-commutative 68.3%

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

      8. +-commutative 68.3%

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

      9. cube-unmult 68.3%

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

      10. +-commutative 68.3%

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

   3. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. fma-define 63.9%

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

      3. cube-mult 63.9%

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

      4. distribute-rgt1-in 68.3%

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

      5. *-commutative 68.3%

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

      6. associate-*l* 68.3%

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

      7. times-frac 89.4%

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

      8. associate-+r+ 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

      3. +-commutative 89.5%

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

      4. +-commutative 89.5%

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

      5. associate-+l+ 89.5%

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

      6. +-commutative 89.5%

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

   8. Applied egg-rr 89.5%

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

      1. *-un-lft-identity 89.5%

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

      2. times-frac 99.7%

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

      3. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 67.9%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e-85) (/ y (* x (+ x 1.0))) (/ (/ x (+ y 1.0)) (+ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-85) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.8d-85) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / (y + 1.0d0)) / (x + y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.8e-85) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / (y + 1.0)) / (x + y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e-85:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / (y + 1.0)) / (x + y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e-85)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(x + y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e-85)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / (y + 1.0)) / (x + y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e-85], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.79999999999999977e-85

   1. Initial program 64.0%

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

      1. associate-/l* 82.3%

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

      2. associate-+l+ 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in y around 0 62.8%

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

      1. +-commutative 62.8%

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

   7. Simplified 62.8%

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

   if 7.79999999999999977e-85 < y

   1. Initial program 68.3%

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

      1. +-commutative 68.3%

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

      2. +-commutative 68.3%

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

      3. +-commutative 68.3%

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

      4. *-commutative 68.3%

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

      5. distribute-rgt1-in 63.9%

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

      6. fma-define 68.3%

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

      7. +-commutative 68.3%

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

      8. +-commutative 68.3%

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

      9. cube-unmult 68.3%

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

      10. +-commutative 68.3%

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

   3. Simplified 68.3%

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

      1. *-commutative 68.3%

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

      2. fma-define 63.9%

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

      3. cube-mult 63.9%

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

      4. distribute-rgt1-in 68.3%

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

      5. *-commutative 68.3%

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

      6. associate-*l* 68.3%

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

      7. times-frac 89.4%

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

      8. associate-+r+ 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. associate-*r/ 89.5%

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

      2. +-commutative 89.5%

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

      3. +-commutative 89.5%

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

      4. +-commutative 89.5%

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

      5. associate-+l+ 89.5%

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

      6. +-commutative 89.5%

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

   8. Applied egg-rr 89.5%

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

      1. *-un-lft-identity 89.5%

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

      2. times-frac 99.7%

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

      3. *-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. associate-*l/ 99.8%

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

       2. *-lft-identity 99.8%

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

       3. associate-/l* 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in x around 0 67.9%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 61.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-83) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-83) {
		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 <= 4.2d-83) 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 <= 4.2e-83) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e-83:
		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 <= 4.2e-83)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-83)
		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, 4.2e-83], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999998e-83

   1. Initial program 64.0%

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

      1. associate-/l* 82.3%

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

      2. associate-+l+ 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in y around 0 62.8%

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

      1. +-commutative 62.8%

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

   7. Simplified 62.8%

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

   if 4.1999999999999998e-83 < y

   1. Initial program 68.3%

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

      1. associate-/l* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 64.2%

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

      1. associate-/r* 67.4%

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

      2. +-commutative 67.4%

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

   7. Simplified 67.4%

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

Alternative 14: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.2e-83) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.2e-83) {
		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 <= 4.2d-83) 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 <= 4.2e-83) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.2e-83:
		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 <= 4.2e-83)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.2e-83)
		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, 4.2e-83], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.1999999999999998e-83

   1. Initial program 64.0%

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

      1. associate-/l* 82.3%

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

      2. associate-+l+ 82.3%

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

   3. Simplified 82.3%

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

   5. Taylor expanded in y around 0 62.8%

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

      1. +-commutative 62.8%

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

   7. Simplified 62.8%

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

   if 4.1999999999999998e-83 < y

   1. Initial program 68.3%

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

      1. associate-/l* 79.5%

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

      2. associate-+l+ 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in x around 0 64.2%

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

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.04e-147) (/ y x) (/ 1.0 (/ y x))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.04d-147)) then
        tmp = y / x
    else
        tmp = 1.0d0 / (y / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = 1.0 / (y / x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.04e-147:
		tmp = y / x
	else:
		tmp = 1.0 / (y / x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-147)
		tmp = Float64(y / x);
	else
		tmp = Float64(1.0 / Float64(y / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.04e-147)
		tmp = y / x;
	else
		tmp = 1.0 / (y / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.04e-147], N[(y / x), $MachinePrecision], N[(1.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04000000000000003e-147

   1. Initial program 68.8%

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

      1. associate-/l* 83.3%

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

      2. associate-+l+ 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in y around 0 63.8%

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

      1. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in x around 0 35.3%

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

   if -1.04000000000000003e-147 < x

   1. Initial program 63.2%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 56.4%

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

   6. Taylor expanded in y around 0 37.3%

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

      1. clear-num 38.7%

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

      2. inv-pow 38.7%

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

   8. Applied egg-rr 38.7%

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

      1. unpow-1 38.7%

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

   10. Simplified 38.7%

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

Alternative 16: 33.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.04 \cdot 10^{-147}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x -1.04e-147) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-1.04d-147)) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -1.04e-147) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.04e-147:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.04e-147)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.04e-147)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.04e-147], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.04000000000000003e-147

   1. Initial program 68.8%

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

      1. associate-/l* 83.3%

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

      2. associate-+l+ 83.3%

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

   3. Simplified 83.3%

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

   5. Taylor expanded in y around 0 63.8%

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

      1. +-commutative 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in x around 0 35.3%

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

   if -1.04000000000000003e-147 < x

   1. Initial program 63.2%

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

      1. associate-/l* 80.2%

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

      2. associate-+l+ 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in x around 0 56.4%

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

   6. Taylor expanded in y around 0 37.3%

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

Alternative 17: 25.6% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.4%

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

   1. associate-/l* 81.4%

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

   2. associate-+l+ 81.4%

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

3. Simplified 81.4%

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

5. Taylor expanded in x around 0 46.7%

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

6. Taylor expanded in y around 0 25.7%

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

Alternative 18: 3.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 65.4%

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

3. Taylor expanded in x around inf 57.9%

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

4. Taylor expanded in x around inf 36.1%

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

5. Taylor expanded in x around 0 3.4%

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

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024151 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

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

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