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

Percentage Accurate: 69.3% → 99.3%

Time: 12.5s

Alternatives: 15

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ x y)) (/ (+ x y) (/ y (+ y (+ x 1.0))))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (x + y)) / ((x + y) / (y / (y + (x + 1.0d0))))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (x + y)) / ((x + y) / (y / (y + (x + 1.0))))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) / Float64(y / Float64(y + Float64(x + 1.0)))))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (x + y)) / ((x + y) / (y / (y + (x + 1.0))));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] / N[(y / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

   1. associate-*l* 70.4%

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

   2. times-frac 93.1%

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

   3. +-commutative 93.1%

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

   4. +-commutative 93.1%

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

   5. associate-+r+ 93.1%

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

   6. +-commutative 93.1%

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

   7. associate-+l+ 93.1%

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

4. Applied egg-rr 93.1%

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

   1. clear-num 92.7%

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

   2. inv-pow 92.7%

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

   3. +-commutative 92.7%

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

6. Applied egg-rr 92.7%

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

   1. unpow-1 92.7%

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

   2. associate-/l* 99.0%

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

8. Simplified 99.0%

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

   1. un-div-inv 99.1%

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

   2. +-commutative 99.1%

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

10. Applied egg-rr 99.1%

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

    1. clear-num 99.1%

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

    2. div-inv 99.2%

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

12. Applied egg-rr 99.2%

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

13. Final simplification 99.2%

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

Alternative 2: 95.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \left(x + y\right) \cdot \frac{t\_0}{y}\\ \mathbf{if}\;y \leq -1.36 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{t\_1}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (* (+ x y) (/ t_0 y))))
   (if (<= y -1.36e-117)
     (/ 1.0 t_1)
     (if (<= y 3.55e+112)
       (* x (/ (/ y (+ x y)) (* (+ x y) t_0)))
       (/ (/ x y) t_1)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (x + y) * (t_0 / y);
	double tmp;
	if (y <= -1.36e-117) {
		tmp = 1.0 / t_1;
	} else if (y <= 3.55e+112) {
		tmp = x * ((y / (x + y)) / ((x + y) * t_0));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (x + y) * (t_0 / y)
    if (y <= (-1.36d-117)) then
        tmp = 1.0d0 / t_1
    else if (y <= 3.55d+112) then
        tmp = x * ((y / (x + y)) / ((x + y) * t_0))
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (x + y) * (t_0 / y);
	double tmp;
	if (y <= -1.36e-117) {
		tmp = 1.0 / t_1;
	} else if (y <= 3.55e+112) {
		tmp = x * ((y / (x + y)) / ((x + y) * t_0));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (x + y) * (t_0 / y)
	tmp = 0
	if y <= -1.36e-117:
		tmp = 1.0 / t_1
	elif y <= 3.55e+112:
		tmp = x * ((y / (x + y)) / ((x + y) * t_0))
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(x + y) * Float64(t_0 / y))
	tmp = 0.0
	if (y <= -1.36e-117)
		tmp = Float64(1.0 / t_1);
	elseif (y <= 3.55e+112)
		tmp = Float64(x * Float64(Float64(y / Float64(x + y)) / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (x + y) * (t_0 / y);
	tmp = 0.0;
	if (y <= -1.36e-117)
		tmp = 1.0 / t_1;
	elseif (y <= 3.55e+112)
		tmp = x * ((y / (x + y)) / ((x + y) * t_0));
	else
		tmp = (x / y) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.36e-117], N[(1.0 / t$95$1), $MachinePrecision], If[LessEqual[y, 3.55e+112], N[(x * N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35999999999999996e-117

   1. Initial program 79.7%

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

      1. associate-*l* 79.8%

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

      2. times-frac 95.4%

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

      3. +-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. associate-+r+ 95.4%

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

      6. +-commutative 95.4%

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

      7. associate-+l+ 95.4%

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

   4. Applied egg-rr 95.4%

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

      1. clear-num 95.4%

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

      2. inv-pow 95.4%

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

      3. +-commutative 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. unpow-1 95.4%

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

      2. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

      1. un-div-inv 99.8%

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

      2. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around inf 34.4%

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

   if -1.35999999999999996e-117 < y < 3.55e112

   1. Initial program 73.7%

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

      1. associate-/l* 83.9%

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

      2. associate-+l+ 83.9%

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

   3. Simplified 83.9%

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

      1. add-sqr-sqrt 55.9%

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

      2. associate-+r+ 55.9%

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

      3. associate-*l* 55.9%

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

      4. times-frac 64.1%

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

      5. +-commutative 64.1%

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

      6. +-commutative 64.1%

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

      7. associate-+r+ 64.1%

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

      8. +-commutative 64.1%

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

      9. associate-+l+ 64.1%

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

   6. Applied egg-rr 64.1%

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

      1. associate-*r/ 64.2%

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

      2. times-frac 63.4%

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

      3. +-commutative 63.4%

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

   8. Simplified 63.4%

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

      1. frac-times 64.2%

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

   10. Applied egg-rr 64.2%

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

       1. associate-*l/ 64.1%

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

       2. rem-square-sqrt 96.5%

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

   12. Simplified 96.5%

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

   if 3.55e112 < y

   1. Initial program 40.2%

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

      1. associate-*l* 40.2%

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

      2. times-frac 71.1%

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

      3. +-commutative 71.1%

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

      4. +-commutative 71.1%

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

      5. associate-+r+ 71.1%

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

      6. +-commutative 71.1%

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

      7. associate-+l+ 71.1%

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

   4. Applied egg-rr 71.1%

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

      1. clear-num 71.1%

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

      2. inv-pow 71.1%

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

      3. +-commutative 71.1%

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

   6. Applied egg-rr 71.1%

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

      1. unpow-1 71.1%

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

      2. associate-/l* 99.6%

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

   8. Simplified 99.6%

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

      1. un-div-inv 99.7%

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

      2. +-commutative 99.7%

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

   10. Applied egg-rr 99.7%

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

   11. Taylor expanded in x around 0 90.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.36 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;y \leq 3.55 \cdot 10^{+112}:\\ \;\;\;\;x \cdot \frac{\frac{y}{x + y}}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \left(x + y\right) \cdot \frac{t\_0}{y}\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{t\_1}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (* (+ x y) (/ t_0 y))))
   (if (<= y -1.35e-117)
     (/ 1.0 t_1)
     (if (<= y 6.9e+72)
       (* x (/ (/ y (* (+ x y) t_0)) (+ x y)))
       (/ (/ x y) t_1)))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (x + y) * (t_0 / y);
	double tmp;
	if (y <= -1.35e-117) {
		tmp = 1.0 / t_1;
	} else if (y <= 6.9e+72) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (x + y) * (t_0 / y)
    if (y <= (-1.35d-117)) then
        tmp = 1.0d0 / t_1
    else if (y <= 6.9d+72) then
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    else
        tmp = (x / y) / t_1
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (x + y) * (t_0 / y);
	double tmp;
	if (y <= -1.35e-117) {
		tmp = 1.0 / t_1;
	} else if (y <= 6.9e+72) {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	} else {
		tmp = (x / y) / t_1;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (x + y) * (t_0 / y)
	tmp = 0
	if y <= -1.35e-117:
		tmp = 1.0 / t_1
	elif y <= 6.9e+72:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	else:
		tmp = (x / y) / t_1
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(x + y) * Float64(t_0 / y))
	tmp = 0.0
	if (y <= -1.35e-117)
		tmp = Float64(1.0 / t_1);
	elseif (y <= 6.9e+72)
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	else
		tmp = Float64(Float64(x / y) / t_1);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (x + y) * (t_0 / y);
	tmp = 0.0;
	if (y <= -1.35e-117)
		tmp = 1.0 / t_1;
	elseif (y <= 6.9e+72)
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	else
		tmp = (x / y) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.35e-117], N[(1.0 / t$95$1), $MachinePrecision], If[LessEqual[y, 6.9e+72], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.35000000000000001e-117

   1. Initial program 79.7%

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

      1. associate-*l* 79.8%

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

      2. times-frac 95.4%

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

      3. +-commutative 95.4%

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

      4. +-commutative 95.4%

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

      5. associate-+r+ 95.4%

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

      6. +-commutative 95.4%

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

      7. associate-+l+ 95.4%

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

   4. Applied egg-rr 95.4%

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

      1. clear-num 95.4%

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

      2. inv-pow 95.4%

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

      3. +-commutative 95.4%

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

   6. Applied egg-rr 95.4%

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

      1. unpow-1 95.4%

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

      2. associate-/l* 99.7%

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

   8. Simplified 99.7%

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

      1. un-div-inv 99.8%

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

      2. +-commutative 99.8%

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

   10. Applied egg-rr 99.8%

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

   11. Taylor expanded in x around inf 34.4%

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

   if -1.35000000000000001e-117 < y < 6.90000000000000034e72

   1. Initial program 73.5%

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

      1. associate-/l* 84.2%

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

      2. associate-+l+ 84.2%

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

   3. Simplified 84.2%

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

      1. *-un-lft-identity 84.2%

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

      2. associate-+r+ 84.2%

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

      3. associate-*l* 84.2%

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

      4. times-frac 96.7%

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

      5. +-commutative 96.7%

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

      6. +-commutative 96.7%

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

      7. associate-+r+ 96.7%

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

      8. +-commutative 96.7%

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

      9. associate-+l+ 96.7%

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

   6. Applied egg-rr 96.7%

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

      1. associate-*l/ 96.8%

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

      2. *-lft-identity 96.8%

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

      3. +-commutative 96.8%

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

   8. Simplified 96.8%

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

   if 6.90000000000000034e72 < y

   1. Initial program 45.7%

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

      1. associate-*l* 45.7%

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

      2. times-frac 74.1%

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

      3. +-commutative 74.1%

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

      4. +-commutative 74.1%

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

      5. associate-+r+ 74.1%

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

      6. +-commutative 74.1%

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

      7. associate-+l+ 74.1%

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

   4. Applied egg-rr 74.1%

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

      1. clear-num 74.2%

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

      2. inv-pow 74.2%

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

      3. +-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. unpow-1 74.2%

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

      2. associate-/l* 98.5%

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

   8. Simplified 98.5%

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

      1. un-div-inv 98.6%

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

      2. +-commutative 98.6%

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

   10. Applied egg-rr 98.6%

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

   11. Taylor expanded in x around 0 88.7%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+72}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{t\_0}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= y 2.1e+79)
     (* (/ x (+ x y)) (/ y (* (+ x y) t_0)))
     (/ (/ x y) (* (+ x y) (/ t_0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 2.1e+79) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 <= 2.1d+79) then
        tmp = (x / (x + y)) * (y / ((x + y) * t_0))
    else
        tmp = (x / y) / ((x + y) * (t_0 / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (y <= 2.1e+79) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if y <= 2.1e+79:
		tmp = (x / (x + y)) * (y / ((x + y) * t_0))
	else:
		tmp = (x / y) / ((x + y) * (t_0 / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (y <= 2.1e+79)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + y) * Float64(t_0 / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (y <= 2.1e+79)
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	else
		tmp = (x / y) / ((x + y) * (t_0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.1e+79], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000008e79

   1. Initial program 76.1%

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

      1. associate-*l* 76.1%

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

      2. times-frac 97.5%

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

      3. +-commutative 97.5%

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

      4. +-commutative 97.5%

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

      5. associate-+r+ 97.5%

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

      6. +-commutative 97.5%

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

      7. associate-+l+ 97.5%

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

   4. Applied egg-rr 97.5%

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

   if 2.10000000000000008e79 < y

   1. Initial program 45.7%

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

      1. associate-*l* 45.7%

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

      2. times-frac 74.1%

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

      3. +-commutative 74.1%

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

      4. +-commutative 74.1%

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

      5. associate-+r+ 74.1%

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

      6. +-commutative 74.1%

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

      7. associate-+l+ 74.1%

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

   4. Applied egg-rr 74.1%

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

      1. clear-num 74.2%

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

      2. inv-pow 74.2%

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

      3. +-commutative 74.2%

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

   6. Applied egg-rr 74.2%

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

      1. unpow-1 74.2%

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

      2. associate-/l* 98.5%

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

   8. Simplified 98.5%

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

      1. un-div-inv 98.6%

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

      2. +-commutative 98.6%

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

   10. Applied egg-rr 98.6%

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

   11. Taylor expanded in x around 0 88.7%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -8.2e-5)
   (/ y (* (+ x y) (+ y (+ x 1.0))))
   (/ (/ x (+ x y)) (* (+ x y) (/ (+ y 1.0) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -8.2e-5) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-8.2d-5)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) / ((x + y) * ((y + 1.0d0) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -8.2e-5) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -8.2e-5:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -8.2e-5)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + 1.0) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -8.2e-5)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) / ((x + y) * ((y + 1.0) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -8.2e-5], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + 1.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.20000000000000009e-5

   1. Initial program 69.7%

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

      1. associate-*l* 69.7%

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

      2. times-frac 86.6%

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

      3. +-commutative 86.6%

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

      4. +-commutative 86.6%

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

      5. associate-+r+ 86.6%

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

      6. +-commutative 86.6%

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

      7. associate-+l+ 86.6%

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

   4. Applied egg-rr 86.6%

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

   5. Taylor expanded in x around inf 80.9%

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

   if -8.20000000000000009e-5 < x

   1. Initial program 70.7%

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

      1. associate-*l* 70.7%

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

      2. times-frac 95.3%

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

      3. +-commutative 95.3%

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

      4. +-commutative 95.3%

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

      5. associate-+r+ 95.3%

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

      6. +-commutative 95.3%

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

      7. associate-+l+ 95.3%

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

   4. Applied egg-rr 95.3%

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

      1. clear-num 94.8%

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

      2. inv-pow 94.8%

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

      3. +-commutative 94.8%

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

   6. Applied egg-rr 94.8%

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

      1. unpow-1 94.8%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. un-div-inv 99.3%

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

      2. +-commutative 99.3%

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

   10. Applied egg-rr 99.3%

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

   11. Taylor expanded in x around 0 86.1%

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

       1. +-commutative 86.1%

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

   13. Simplified 86.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{x + y}}{\left(x + y\right) \cdot \frac{y + 1}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -2.35 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{t\_0}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -2.35e-189)
     (/ y (* (+ x y) t_0))
     (/ (/ x y) (* (+ x y) (/ t_0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2.35e-189) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
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 (x <= (-2.35d-189)) then
        tmp = y / ((x + y) * t_0)
    else
        tmp = (x / y) / ((x + y) * (t_0 / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -2.35e-189) {
		tmp = y / ((x + y) * t_0);
	} else {
		tmp = (x / y) / ((x + y) * (t_0 / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -2.35e-189:
		tmp = y / ((x + y) * t_0)
	else:
		tmp = (x / y) / ((x + y) * (t_0 / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -2.35e-189)
		tmp = Float64(y / Float64(Float64(x + y) * t_0));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + y) * Float64(t_0 / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -2.35e-189)
		tmp = y / ((x + y) * t_0);
	else
		tmp = (x / y) / ((x + y) * (t_0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.35e-189], N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(t$95$0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3499999999999998e-189

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

      2. times-frac 91.7%

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

      3. +-commutative 91.7%

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

      4. +-commutative 91.7%

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

      5. associate-+r+ 91.7%

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

      6. +-commutative 91.7%

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

      7. associate-+l+ 91.7%

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -2.3499999999999998e-189 < x

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 66.7%

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

      2. times-frac 94.2%

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

      3. +-commutative 94.2%

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

      4. +-commutative 94.2%

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

      5. associate-+r+ 94.2%

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

      6. +-commutative 94.2%

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

      7. associate-+l+ 94.2%

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

   4. Applied egg-rr 94.2%

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

      1. clear-num 93.6%

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

      2. inv-pow 93.6%

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

      3. +-commutative 93.6%

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

   6. Applied egg-rr 93.6%

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

      1. unpow-1 93.6%

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

      2. associate-/l* 99.2%

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

   8. Simplified 99.2%

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

      1. un-div-inv 99.2%

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

      2. +-commutative 99.2%

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

   10. Applied egg-rr 99.2%

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

   11. Taylor expanded in x around 0 62.9%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + y\right) \cdot \frac{y + \left(x + 1\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.3% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (/ x (+ x y)) (* (+ x y) (/ (+ y (+ x 1.0)) y))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x / (x + y)) / ((x + y) * ((y + (x + 1.0d0)) / y))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return (x / (x + y)) / ((x + y) * ((y + (x + 1.0)) / y))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(x / Float64(x + y)) / Float64(Float64(x + y) * Float64(Float64(y + Float64(x + 1.0)) / y)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (x / (x + y)) / ((x + y) * ((y + (x + 1.0)) / y));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x + y), $MachinePrecision] * N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

   1. associate-*l* 70.4%

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

   2. times-frac 93.1%

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

   3. +-commutative 93.1%

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

   4. +-commutative 93.1%

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

   5. associate-+r+ 93.1%

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

   6. +-commutative 93.1%

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

   7. associate-+l+ 93.1%

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

4. Applied egg-rr 93.1%

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

   1. clear-num 92.7%

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

   2. inv-pow 92.7%

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

   3. +-commutative 92.7%

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

6. Applied egg-rr 92.7%

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

   1. unpow-1 92.7%

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

   2. associate-/l* 99.0%

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

8. Simplified 99.0%

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

   1. un-div-inv 99.1%

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

   2. +-commutative 99.1%

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

10. Applied egg-rr 99.1%

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

11. Final simplification 99.1%

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

Alternative 8: 85.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2.35e-189)
   (/ y (* (+ x y) (+ y (+ x 1.0))))
   (* (/ x (+ x y)) (/ 1.0 (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2.35e-189) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2.35d-189)) then
        tmp = y / ((x + y) * (y + (x + 1.0d0)))
    else
        tmp = (x / (x + y)) * (1.0d0 / (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2.35e-189) {
		tmp = y / ((x + y) * (y + (x + 1.0)));
	} else {
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2.35e-189:
		tmp = y / ((x + y) * (y + (x + 1.0)))
	else:
		tmp = (x / (x + y)) * (1.0 / (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2.35e-189)
		tmp = Float64(y / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(1.0 / Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2.35e-189)
		tmp = y / ((x + y) * (y + (x + 1.0)));
	else
		tmp = (x / (x + y)) * (1.0 / (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2.35e-189], N[(y / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.3499999999999998e-189

   1. Initial program 75.9%

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

      1. associate-*l* 75.9%

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

      2. times-frac 91.7%

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

      3. +-commutative 91.7%

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

      4. +-commutative 91.7%

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

      5. associate-+r+ 91.7%

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

      6. +-commutative 91.7%

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

      7. associate-+l+ 91.7%

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in x around inf 72.5%

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

   if -2.3499999999999998e-189 < x

   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. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l* 66.7%

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

      2. times-frac 94.2%

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

      3. +-commutative 94.2%

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

      4. +-commutative 94.2%

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

      5. associate-+r+ 94.2%

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

      6. +-commutative 94.2%

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

      7. associate-+l+ 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in x around 0 56.1%

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

      1. +-commutative 56.1%

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

   7. Simplified 56.1%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.35 \cdot 10^{-189}:\\ \;\;\;\;\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{1}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{y + \left(x + 1\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-128)
   (/ (/ 1.0 x) (/ (+ y (+ x 1.0)) y))
   (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-128) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d-128)) then
        tmp = (1.0d0 / x) / ((y + (x + 1.0d0)) / y)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-128) {
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.2e-128:
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-128)
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(y + Float64(x + 1.0)) / y));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e-128)
		tmp = (1.0 / x) / ((y + (x + 1.0)) / y);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.2e-128], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999961e-128

   1. Initial program 76.1%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

      1. clear-num 82.4%

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

      2. associate-+r+ 82.4%

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

      3. *-commutative 82.4%

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

      4. distribute-rgt1-in 49.8%

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

      5. cube-mult 49.8%

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

      6. un-div-inv 49.9%

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

      7. cube-mult 49.9%

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

      8. distribute-rgt1-in 82.5%

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

      9. *-commutative 82.5%

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

      10. associate-/l* 84.6%

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

      11. pow2 84.6%

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

      12. +-commutative 84.6%

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

   6. Applied egg-rr 84.6%

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

      1. associate-/r* 90.1%

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

      2. +-commutative 90.1%

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

   8. Simplified 90.1%

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

   9. Taylor expanded in x around inf 64.4%

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

   if -5.19999999999999961e-128 < x

   1. Initial program 67.5%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 55.2%

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

      2. +-commutative 55.2%

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

   7. Simplified 55.2%

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

      1. associate-/l/ 55.2%

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

      2. *-commutative 55.2%

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

      3. div-inv 55.2%

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

      4. associate-/r* 57.1%

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

   9. Applied egg-rr 57.1%

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

Alternative 10: 82.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-128}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.5e-128) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-128) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.5d-128)) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.5e-128) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.5e-128:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.5e-128)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.5e-128)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.5e-128], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999999e-128

   1. Initial program 76.1%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around 0 63.0%

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

      1. associate-/r* 63.9%

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

      2. +-commutative 63.9%

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

   7. Simplified 63.9%

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

   if -4.4999999999999999e-128 < x

   1. Initial program 67.5%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 55.2%

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

      2. +-commutative 55.2%

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

   7. Simplified 55.2%

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

      1. associate-/l/ 55.2%

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

      2. *-commutative 55.2%

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

      3. div-inv 55.2%

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

      4. associate-/r* 57.1%

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

   9. Applied egg-rr 57.1%

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

Alternative 11: 80.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -5.2e-128) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-128) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-5.2d-128)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -5.2e-128) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -5.2e-128:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -5.2e-128)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.2e-128)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -5.2e-128], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.19999999999999961e-128

   1. Initial program 76.1%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around 0 63.0%

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

   if -5.19999999999999961e-128 < x

   1. Initial program 67.5%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. associate-/r* 55.2%

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

      2. +-commutative 55.2%

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

   7. Simplified 55.2%

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

      1. associate-/l/ 55.2%

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

      2. *-commutative 55.2%

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

      3. div-inv 55.2%

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

      4. associate-/r* 57.1%

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

   9. Applied egg-rr 57.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -4.1e-128) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-128) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-4.1d-128)) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -4.1e-128) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -4.1e-128:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -4.1e-128)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -4.1e-128)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -4.1e-128], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.1e-128

   1. Initial program 76.1%

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

      1. associate-/l* 84.3%

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

      2. associate-+l+ 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around 0 63.0%

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

   if -4.1e-128 < x

   1. Initial program 67.5%

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

      1. associate-/l* 80.3%

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

      2. associate-+l+ 80.3%

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

   3. Simplified 80.3%

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

   5. Taylor expanded in x around 0 55.2%

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

      1. +-commutative 55.2%

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

   7. Simplified 55.2%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{-128}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 49.3% accurate, 2.4× speedup

Math

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

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x (* y (+ y 1.0))))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (y * (y + 1.0d0))
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / (y * (y + 1.0))

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / Float64(y * Float64(y + 1.0)))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / (y * (y + 1.0));
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

   1. associate-/l* 81.6%

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

   2. associate-+l+ 81.6%

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

3. Simplified 81.6%

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

5. Taylor expanded in x around 0 46.2%

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

   1. +-commutative 46.2%

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

7. Simplified 46.2%

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

Alternative 14: 26.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ x y))

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / y
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return x / y

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(x / y)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = x / y;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 70.4%

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

   1. associate-/l* 81.6%

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

   2. associate-+l+ 81.6%

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

3. Simplified 81.6%

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

5. Taylor expanded in x around 0 46.2%

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

   1. +-commutative 46.2%

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

7. Simplified 46.2%

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

8. Taylor expanded in y around 0 25.2%

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

Alternative 15: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

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

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 70.4%

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

   1. associate-*l* 70.4%

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

   2. times-frac 93.1%

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

   3. +-commutative 93.1%

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

   4. +-commutative 93.1%

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

   5. associate-+r+ 93.1%

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

   6. +-commutative 93.1%

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

   7. associate-+l+ 93.1%

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

4. Applied egg-rr 93.1%

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

5. Taylor expanded in y around 0 50.1%

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

6. Taylor expanded in x around 0 3.6%

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

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024102 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

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

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