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

Percentage Accurate: 68.3% → 99.8%

Time: 16.2s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-*l* 67.8%

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

   2. times-frac 92.1%

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

   3. +-commutative 92.1%

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

   4. +-commutative 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l+ 92.1%

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

4. Applied egg-rr 92.1%

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

   1. *-un-lft-identity 92.1%

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

   2. +-commutative 92.1%

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

6. Applied egg-rr 92.1%

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

   1. *-lft-identity 92.1%

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

   2. associate-/r* 99.8%

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

8. Simplified 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 69.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{x + y}}{y + \left(x + 1\right)}\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{-199}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ y (+ x y)) (+ y (+ x 1.0)))))
   (if (<= y 2.8e-199)
     t_0
     (if (<= y 4.6e+98)
       (* x (/ y (* (* (+ x y) (+ x y)) (+ x (+ y 1.0)))))
       (* t_0 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = (y / (x + y)) / (y + (x + 1.0));
	double tmp;
	if (y <= 2.8e-199) {
		tmp = t_0;
	} else if (y <= 4.6e+98) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y / (x + y)) / (y + (x + 1.0d0))
    if (y <= 2.8d-199) then
        tmp = t_0
    else if (y <= 4.6d+98) then
        tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0d0))))
    else
        tmp = t_0 * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y / (x + y)) / (y + (x + 1.0));
	double tmp;
	if (y <= 2.8e-199) {
		tmp = t_0;
	} else if (y <= 4.6e+98) {
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	} else {
		tmp = t_0 * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y / (x + y)) / (y + (x + 1.0))
	tmp = 0
	if y <= 2.8e-199:
		tmp = t_0
	elif y <= 4.6e+98:
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))))
	else:
		tmp = t_0 * (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y / Float64(x + y)) / Float64(y + Float64(x + 1.0)))
	tmp = 0.0
	if (y <= 2.8e-199)
		tmp = t_0;
	elseif (y <= 4.6e+98)
		tmp = Float64(x * Float64(y / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(x + Float64(y + 1.0)))));
	else
		tmp = Float64(t_0 * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y / (x + y)) / (y + (x + 1.0));
	tmp = 0.0;
	if (y <= 2.8e-199)
		tmp = t_0;
	elseif (y <= 4.6e+98)
		tmp = x * (y / (((x + y) * (x + y)) * (x + (y + 1.0))));
	else
		tmp = t_0 * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.8e-199], t$95$0, If[LessEqual[y, 4.6e+98], N[(x * N[(y / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 2.80000000000000018e-199

   1. Initial program 66.4%

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

      1. associate-*l* 66.5%

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

      2. times-frac 92.7%

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

      3. +-commutative 92.7%

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

      4. +-commutative 92.7%

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

      5. associate-+r+ 92.7%

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

      6. +-commutative 92.7%

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

      7. associate-+l+ 92.7%

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

   4. Applied egg-rr 92.7%

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

      1. *-un-lft-identity 92.7%

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

      2. +-commutative 92.7%

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

   6. Applied egg-rr 92.7%

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

      1. *-lft-identity 92.7%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 56.7%

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

   if 2.80000000000000018e-199 < y < 4.60000000000000026e98

   1. Initial program 88.9%

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

      1. associate-/l* 93.0%

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

      2. associate-+l+ 93.0%

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

   3. Simplified 93.0%

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

   if 4.60000000000000026e98 < y

   1. Initial program 47.3%

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

      1. associate-*l* 47.3%

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

      2. times-frac 83.3%

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

      3. +-commutative 83.3%

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

      4. +-commutative 83.3%

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

      5. associate-+r+ 83.3%

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

      6. +-commutative 83.3%

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

      7. associate-+l+ 83.3%

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

   4. Applied egg-rr 83.3%

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

      1. *-un-lft-identity 83.3%

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

      2. +-commutative 83.3%

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

   6. Applied egg-rr 83.3%

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

      1. *-lft-identity 83.3%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 86.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-199}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+98}:\\ \;\;\;\;x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ t_1 := \frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+88}:\\ \;\;\;\;t\_1 \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))) (t_1 (/ (/ y (+ x y)) t_0)))
   (if (<= y -4e+88)
     (* t_1 (- 1.0 (/ y x)))
     (if (<= y 1.4e+124)
       (* (/ x (+ x y)) (/ y (* (+ x y) t_0)))
       (* t_1 (/ x y))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / (x + y)) / t_0;
	double tmp;
	if (y <= -4e+88) {
		tmp = t_1 * (1.0 - (y / x));
	} else if (y <= 1.4e+124) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = t_1 * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    t_1 = (y / (x + y)) / t_0
    if (y <= (-4d+88)) then
        tmp = t_1 * (1.0d0 - (y / x))
    else if (y <= 1.4d+124) then
        tmp = (x / (x + y)) * (y / ((x + y) * t_0))
    else
        tmp = t_1 * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double t_1 = (y / (x + y)) / t_0;
	double tmp;
	if (y <= -4e+88) {
		tmp = t_1 * (1.0 - (y / x));
	} else if (y <= 1.4e+124) {
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	} else {
		tmp = t_1 * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	t_1 = (y / (x + y)) / t_0
	tmp = 0
	if y <= -4e+88:
		tmp = t_1 * (1.0 - (y / x))
	elif y <= 1.4e+124:
		tmp = (x / (x + y)) * (y / ((x + y) * t_0))
	else:
		tmp = t_1 * (x / y)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	t_1 = Float64(Float64(y / Float64(x + y)) / t_0)
	tmp = 0.0
	if (y <= -4e+88)
		tmp = Float64(t_1 * Float64(1.0 - Float64(y / x)));
	elseif (y <= 1.4e+124)
		tmp = Float64(Float64(x / Float64(x + y)) * Float64(y / Float64(Float64(x + y) * t_0)));
	else
		tmp = Float64(t_1 * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	t_1 = (y / (x + y)) / t_0;
	tmp = 0.0;
	if (y <= -4e+88)
		tmp = t_1 * (1.0 - (y / x));
	elseif (y <= 1.4e+124)
		tmp = (x / (x + y)) * (y / ((x + y) * t_0));
	else
		tmp = t_1 * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y, -4e+88], N[(t$95$1 * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.4e+124], N[(N[(x / N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.99999999999999984e88

   1. Initial program 53.5%

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

      1. associate-*l* 53.5%

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

      2. times-frac 79.7%

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

      3. +-commutative 79.7%

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

      4. +-commutative 79.7%

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

      5. associate-+r+ 79.7%

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

      6. +-commutative 79.7%

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

      7. associate-+l+ 79.7%

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

   4. Applied egg-rr 79.7%

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

      1. *-un-lft-identity 79.7%

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

      2. +-commutative 79.7%

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

   6. Applied egg-rr 79.7%

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

      1. *-lft-identity 79.7%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around inf 16.9%

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

       1. mul-1-neg 16.9%

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

       2. unsub-neg 16.9%

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

   11. Simplified 16.9%

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

   if -3.99999999999999984e88 < y < 1.4e124

   1. Initial program 77.0%

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

      1. associate-*l* 77.1%

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

      2. times-frac 98.1%

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

      3. +-commutative 98.1%

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

      4. +-commutative 98.1%

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

      5. associate-+r+ 98.1%

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

      6. +-commutative 98.1%

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

      7. associate-+l+ 98.1%

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

   4. Applied egg-rr 98.1%

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

   if 1.4e124 < y

   1. Initial program 48.9%

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

      1. associate-*l* 48.9%

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

      2. times-frac 83.1%

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

      3. +-commutative 83.1%

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

      4. +-commutative 83.1%

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

      5. associate-+r+ 83.1%

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

      6. +-commutative 83.1%

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

      7. associate-+l+ 83.1%

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

   4. Applied egg-rr 83.1%

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

      1. *-un-lft-identity 83.1%

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

      2. +-commutative 83.1%

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

   6. Applied egg-rr 83.1%

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

      1. *-lft-identity 83.1%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 89.3%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+88}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+124}:\\ \;\;\;\;\frac{x}{x + y} \cdot \frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -6.5e+86)
     (/ (/ y (+ x y)) t_0)
     (* x (/ (/ y (* (+ x y) t_0)) (+ x y))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -6.5e+86) {
		tmp = (y / (x + y)) / t_0;
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-6.5d+86)) then
        tmp = (y / (x + y)) / t_0
    else
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -6.5e+86) {
		tmp = (y / (x + y)) / t_0;
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -6.5e+86:
		tmp = (y / (x + y)) / t_0
	else:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -6.5e+86)
		tmp = Float64(Float64(y / Float64(x + y)) / t_0);
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -6.5e+86)
		tmp = (y / (x + y)) / t_0;
	else
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -6.5e+86], N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.49999999999999996e86

   1. Initial program 46.2%

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

      1. associate-*l* 46.2%

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

      2. times-frac 75.0%

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

      3. +-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. associate-+r+ 75.0%

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

      6. +-commutative 75.0%

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

      7. associate-+l+ 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. +-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. *-lft-identity 75.0%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 81.0%

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

   if -6.49999999999999996e86 < x

   1. Initial program 71.9%

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

      1. associate-/l* 82.5%

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

      2. associate-+l+ 82.5%

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

   3. Simplified 82.5%

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

      1. *-un-lft-identity 82.5%

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

      2. associate-+r+ 82.5%

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

      3. associate-*l* 82.5%

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

      4. times-frac 93.9%

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

      5. +-commutative 93.9%

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

      6. +-commutative 93.9%

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

      7. associate-+r+ 93.9%

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

      8. +-commutative 93.9%

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

      9. associate-+l+ 93.9%

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

   6. Applied egg-rr 93.9%

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

      1. associate-*l/ 93.9%

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

      2. *-lft-identity 93.9%

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

      3. +-commutative 93.9%

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

   8. Simplified 93.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y + \left(x + 1\right)\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{t\_0} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot t\_0}}{x + y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ y (+ x 1.0))))
   (if (<= x -1.7e+87)
     (* (/ (/ y (+ x y)) t_0) (- 1.0 (/ y x)))
     (* x (/ (/ y (* (+ x y) t_0)) (+ x y))))))

C

double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.7e+87) {
		tmp = ((y / (x + y)) / t_0) * (1.0 - (y / x));
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y + (x + 1.0d0)
    if (x <= (-1.7d+87)) then
        tmp = ((y / (x + y)) / t_0) * (1.0d0 - (y / x))
    else
        tmp = x * ((y / ((x + y) * t_0)) / (x + y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y + (x + 1.0);
	double tmp;
	if (x <= -1.7e+87) {
		tmp = ((y / (x + y)) / t_0) * (1.0 - (y / x));
	} else {
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y + (x + 1.0)
	tmp = 0
	if x <= -1.7e+87:
		tmp = ((y / (x + y)) / t_0) * (1.0 - (y / x))
	else:
		tmp = x * ((y / ((x + y) * t_0)) / (x + y))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y + Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.7e+87)
		tmp = Float64(Float64(Float64(y / Float64(x + y)) / t_0) * Float64(1.0 - Float64(y / x)));
	else
		tmp = Float64(x * Float64(Float64(y / Float64(Float64(x + y) * t_0)) / Float64(x + y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y + (x + 1.0);
	tmp = 0.0;
	if (x <= -1.7e+87)
		tmp = ((y / (x + y)) / t_0) * (1.0 - (y / x));
	else
		tmp = x * ((y / ((x + y) * t_0)) / (x + y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e+87], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / N[(N[(x + y), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7000000000000001e87

   1. Initial program 46.2%

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

      1. associate-*l* 46.2%

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

      2. times-frac 75.0%

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

      3. +-commutative 75.0%

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

      4. +-commutative 75.0%

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

      5. associate-+r+ 75.0%

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

      6. +-commutative 75.0%

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

      7. associate-+l+ 75.0%

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

   4. Applied egg-rr 75.0%

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

      1. *-un-lft-identity 75.0%

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

      2. +-commutative 75.0%

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

   6. Applied egg-rr 75.0%

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

      1. *-lft-identity 75.0%

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

      2. associate-/r* 99.7%

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

   8. Simplified 99.7%

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

   9. Taylor expanded in x around inf 82.0%

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

       1. mul-1-neg 82.0%

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

       2. unsub-neg 82.0%

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

   11. Simplified 82.0%

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

   if -1.7000000000000001e87 < x

   1. Initial program 71.9%

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

      1. associate-/l* 82.5%

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

      2. associate-+l+ 82.5%

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

   3. Simplified 82.5%

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

      1. *-un-lft-identity 82.5%

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

      2. associate-+r+ 82.5%

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

      3. associate-*l* 82.5%

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

      4. times-frac 93.9%

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

      5. +-commutative 93.9%

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

      6. +-commutative 93.9%

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

      7. associate-+r+ 93.9%

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

      8. +-commutative 93.9%

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

      9. associate-+l+ 93.9%

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

   6. Applied egg-rr 93.9%

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

      1. associate-*l/ 93.9%

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

      2. *-lft-identity 93.9%

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

      3. +-commutative 93.9%

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

   8. Simplified 93.9%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{y}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}}{x + y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.1e-110)
   (/ (/ y x) (+ x 1.0))
   (* (/ (/ y (+ x y)) (+ y (+ x 1.0))) (/ x y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-110) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = ((y / (x + y)) / (y + (x + 1.0))) * (x / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.1d-110) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = ((y / (x + y)) / (y + (x + 1.0d0))) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.1e-110) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = ((y / (x + y)) / (y + (x + 1.0))) * (x / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.1e-110:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = ((y / (x + y)) / (y + (x + 1.0))) * (x / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.1e-110)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(y / Float64(x + y)) / Float64(y + Float64(x + 1.0))) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.1e-110)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = ((y / (x + y)) / (y + (x + 1.0))) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.1e-110], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.10000000000000007e-110

   1. Initial program 69.2%

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

      1. associate-/l* 80.9%

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

      2. associate-+l+ 80.9%

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

   3. Simplified 80.9%

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

   5. Taylor expanded in y around 0 57.3%

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

      1. associate-/r* 59.0%

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

      2. +-commutative 59.0%

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

   7. Simplified 59.0%

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

   if 3.10000000000000007e-110 < y

   1. Initial program 65.0%

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

      1. associate-*l* 65.0%

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

      2. times-frac 89.4%

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

      3. +-commutative 89.4%

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

      4. +-commutative 89.4%

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

      5. associate-+r+ 89.4%

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

      6. +-commutative 89.4%

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

      7. associate-+l+ 89.4%

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

   4. Applied egg-rr 89.4%

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

      1. *-un-lft-identity 89.4%

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

      2. +-commutative 89.4%

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

   6. Applied egg-rr 89.4%

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

      1. *-lft-identity 89.4%

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

      2. associate-/r* 99.8%

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

   8. Simplified 99.8%

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

   9. Taylor expanded in x around 0 82.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x + y}}{y + \left(x + 1\right)} \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.3e-107) (/ (/ y x) (+ x 1.0)) (/ x (* (+ x y) (+ y (+ x 1.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.3d-107) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = x / ((x + y) * (y + (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.3e-107) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = x / ((x + y) * (y + (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.3e-107:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = x / ((x + y) * (y + (x + 1.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.3e-107)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(x / Float64(Float64(x + y) * Float64(y + Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.3e-107)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = x / ((x + y) * (y + (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.3e-107], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(x + y), $MachinePrecision] * N[(y + N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.30000000000000004e-107

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   if 3.30000000000000004e-107 < y

   1. Initial program 64.6%

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

      1. associate-*l* 64.6%

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

      2. times-frac 89.2%

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

      3. +-commutative 89.2%

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

      4. +-commutative 89.2%

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

      5. associate-+r+ 89.2%

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

      6. +-commutative 89.2%

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

      7. associate-+l+ 89.2%

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

   4. Applied egg-rr 89.2%

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

      1. associate-*r/ 89.2%

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

      2. +-commutative 89.2%

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in x around 0 79.5%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.3 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(x + y\right) \cdot \left(y + \left(x + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3e-108) (/ y x) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3e-108) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3d-108) then
        tmp = y / x
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3e-108) {
		tmp = y / x;
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3e-108:
		tmp = y / x
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3e-108)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3e-108)
		tmp = y / x;
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3e-108], N[(y / x), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.99999999999999993e-108

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

      1. div-inv 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in x around 0 39.5%

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

   if 2.99999999999999993e-108 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-108}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e-107) (/ y (* x (+ x 1.0))) (/ x (* y (+ y 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e-107) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d-107) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e-107) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e-107:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e-107)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e-107)
		tmp = y / (x * (x + 1.0));
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e-107], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8e-107

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

   if 8e-107 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-106) (/ y (* x (+ x 1.0))) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-106) then
        tmp = y / (x * (x + 1.0d0))
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = y / (x * (x + 1.0));
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-106:
		tmp = y / (x * (x + 1.0))
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-106)
		tmp = Float64(y / Float64(x * Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-106)
		tmp = y / (x * (x + 1.0));
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-106], N[(y / N[(x * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-106

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

   if 1.9e-106 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. div-inv 69.2%

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

      3. associate-/r* 65.7%

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

   7. Applied egg-rr 65.7%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{y}{x \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-106) (/ (/ y x) (+ x 1.0)) (/ (/ x y) (+ y 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.9d-106) then
        tmp = (y / x) / (x + 1.0d0)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-106) {
		tmp = (y / x) / (x + 1.0);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.9e-106:
		tmp = (y / x) / (x + 1.0)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-106)
		tmp = Float64(Float64(y / x) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-106)
		tmp = (y / x) / (x + 1.0);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.9e-106], N[(N[(y / x), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.9e-106

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

   if 1.9e-106 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

      2. div-inv 69.2%

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

      3. associate-/r* 65.7%

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

   7. Applied egg-rr 65.7%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{y}{x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 9.8e-107) (/ y x) (/ x y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-107) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9.8d-107) then
        tmp = y / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9.8e-107) {
		tmp = y / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9.8e-107:
		tmp = y / x
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9.8e-107)
		tmp = Float64(y / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9.8e-107)
		tmp = y / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9.8e-107], N[(y / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.79999999999999959e-107

   1. Initial program 69.4%

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

      1. associate-/l* 81.1%

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

      2. associate-+l+ 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y around 0 57.5%

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

      1. associate-/r* 59.3%

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

      2. +-commutative 59.3%

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

   7. Simplified 59.3%

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

      1. div-inv 59.2%

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

   9. Applied egg-rr 59.2%

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

   10. Taylor expanded in x around 0 39.5%

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

   if 9.79999999999999959e-107 < y

   1. Initial program 64.6%

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

      1. associate-/l* 79.2%

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

      2. associate-+l+ 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in x around 0 69.2%

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

      1. +-commutative 69.2%

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

   7. Simplified 69.2%

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

   8. Taylor expanded in y around 0 29.5%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.8% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 80.4%

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

   2. associate-+l+ 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in x around 0 50.0%

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

   1. +-commutative 50.0%

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

7. Simplified 50.0%

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

8. Taylor expanded in y around 0 25.0%

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

9. Final simplification 25.0%

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

Alternative 14: 3.6% accurate, 8.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := (-x)

Derivation

1. Initial program 67.8%

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

   1. associate-/l* 80.4%

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

   2. associate-+l+ 80.4%

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

3. Simplified 80.4%

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

5. Taylor expanded in x around 0 50.0%

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

6. Taylor expanded in y around 0 15.0%

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

   1. neg-mul-1 15.0%

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

   2. sub-neg 15.0%

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

8. Simplified 15.0%

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

9. Taylor expanded in y around inf 3.7%

   \[\leadsto \color{blue}{-1 \cdot x} \]
10. Step-by-step derivation

    1. neg-mul-1 3.7%

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

11. Simplified 3.7%

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

12. Final simplification 3.7%

    \[\leadsto -x \]
13. Add Preprocessing

Alternative 15: 3.5% accurate, 17.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 1.0)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 67.8%

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

   1. associate-*l* 67.8%

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

   2. times-frac 92.1%

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

   3. +-commutative 92.1%

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

   4. +-commutative 92.1%

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

   5. associate-+r+ 92.1%

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

   6. +-commutative 92.1%

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

   7. associate-+l+ 92.1%

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

4. Applied egg-rr 92.1%

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

5. Taylor expanded in y around 0 48.2%

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

   1. +-commutative 48.2%

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

7. Simplified 48.2%

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

8. Taylor expanded in x around 0 3.3%

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

9. Final simplification 3.3%

   \[\leadsto 1 \]
10. Add Preprocessing

Developer target: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024055 
(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))))