Octave 3.8, jcobi/4

Percentage Accurate: 15.5% → 84.6%

Time: 13.5s

Alternatives: 8

Speedup: 53.0×

Specification

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 1\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 15.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := i \cdot \left(\left(\alpha + \beta\right) + i\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := t\_1 \cdot t\_1\\ \frac{\frac{t\_0 \cdot \left(\beta \cdot \alpha + t\_0\right)}{t\_2}}{t\_2 - 1} \end{array} \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* i (+ (+ alpha beta) i)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (* t_1 t_1)))
   (/ (/ (* t_0 (+ (* beta alpha) t_0)) t_2) (- t_2 1.0))))

C

double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Fortran

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = i * ((alpha + beta) + i)
    t_1 = (alpha + beta) + (2.0d0 * i)
    t_2 = t_1 * t_1
    code = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0d0)
end function

Java

public static double code(double alpha, double beta, double i) {
	double t_0 = i * ((alpha + beta) + i);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = t_1 * t_1;
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
}

Python

def code(alpha, beta, i):
	t_0 = i * ((alpha + beta) + i)
	t_1 = (alpha + beta) + (2.0 * i)
	t_2 = t_1 * t_1
	return ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0)

Julia

function code(alpha, beta, i)
	t_0 = Float64(i * Float64(Float64(alpha + beta) + i))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(t_1 * t_1)
	return Float64(Float64(Float64(t_0 * Float64(Float64(beta * alpha) + t_0)) / t_2) / Float64(t_2 - 1.0))
end

MATLAB

function tmp = code(alpha, beta, i)
	t_0 = i * ((alpha + beta) + i);
	t_1 = (alpha + beta) + (2.0 * i);
	t_2 = t_1 * t_1;
	tmp = ((t_0 * ((beta * alpha) + t_0)) / t_2) / (t_2 - 1.0);
end

Wolfram

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * t$95$1), $MachinePrecision]}, N[(N[(N[(t$95$0 * N[(N[(beta * alpha), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(t$95$2 - 1.0), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 84.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{i}{\alpha + \left(t\_0 + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\right)}{\alpha + \left(t\_0 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (if (<= beta 1.5e+186)
     (*
      (/ i (+ alpha (+ t_0 1.0)))
      (/
       (* i (+ 0.25 (/ (* 0.25 (- (* 2.0 (+ beta alpha)) (+ beta alpha))) i)))
       (+ alpha (+ t_0 -1.0))))
     (/ (/ (+ i alpha) (/ beta i)) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 1.5e+186) {
		tmp = (i / (alpha + (t_0 + 1.0))) * ((i * (0.25 + ((0.25 * ((2.0 * (beta + alpha)) - (beta + alpha))) / i))) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    if (beta <= 1.5d+186) then
        tmp = (i / (alpha + (t_0 + 1.0d0))) * ((i * (0.25d0 + ((0.25d0 * ((2.0d0 * (beta + alpha)) - (beta + alpha))) / i))) / (alpha + (t_0 + (-1.0d0))))
    else
        tmp = ((i + alpha) / (beta / i)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 1.5e+186) {
		tmp = (i / (alpha + (t_0 + 1.0))) * ((i * (0.25 + ((0.25 * ((2.0 * (beta + alpha)) - (beta + alpha))) / i))) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	tmp = 0
	if beta <= 1.5e+186:
		tmp = (i / (alpha + (t_0 + 1.0))) * ((i * (0.25 + ((0.25 * ((2.0 * (beta + alpha)) - (beta + alpha))) / i))) / (alpha + (t_0 + -1.0)))
	else:
		tmp = ((i + alpha) / (beta / i)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 1.5e+186)
		tmp = Float64(Float64(i / Float64(alpha + Float64(t_0 + 1.0))) * Float64(Float64(i * Float64(0.25 + Float64(Float64(0.25 * Float64(Float64(2.0 * Float64(beta + alpha)) - Float64(beta + alpha))) / i))) / Float64(alpha + Float64(t_0 + -1.0))));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	tmp = 0.0;
	if (beta <= 1.5e+186)
		tmp = (i / (alpha + (t_0 + 1.0))) * ((i * (0.25 + ((0.25 * ((2.0 * (beta + alpha)) - (beta + alpha))) / i))) / (alpha + (t_0 + -1.0)));
	else
		tmp = ((i + alpha) / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 1.5e+186], N[(N[(i / N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * N[(0.25 + N[(N[(0.25 * N[(N[(2.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision] - N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 1.49999999999999991e186

   1. Initial program 17.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({i}^{2} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, i\right)\right)\right), 1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{i}\right)\right)\right), 1\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)\right), i\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

   5. Simplified 36.8%

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot \left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - 1} \]

      2. difference-of-sqr-1 N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right) \cdot \left(\left(\left(\alpha + \color{blue}{\beta}\right) + 2 \cdot i\right) - 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot i\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \color{blue}{\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{\left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) \cdot 0.25}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}} \]

   if 1.49999999999999991e186 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 86.2%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + i}{\frac{\beta}{i}}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.5 \cdot 10^{+186}:\\ \;\;\;\;\frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right)}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+185}:\\ \;\;\;\;\frac{i}{\alpha + \left(t\_0 + 1\right)} \cdot \frac{0.25 \cdot \left(\beta + i\right)}{\alpha + \left(t\_0 + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ beta (* i 2.0))))
   (if (<= beta 4.8e+185)
     (*
      (/ i (+ alpha (+ t_0 1.0)))
      (/ (* 0.25 (+ beta i)) (+ alpha (+ t_0 -1.0))))
     (/ (/ (+ i alpha) (/ beta i)) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 4.8e+185) {
		tmp = (i / (alpha + (t_0 + 1.0))) * ((0.25 * (beta + i)) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = beta + (i * 2.0d0)
    if (beta <= 4.8d+185) then
        tmp = (i / (alpha + (t_0 + 1.0d0))) * ((0.25d0 * (beta + i)) / (alpha + (t_0 + (-1.0d0))))
    else
        tmp = ((i + alpha) / (beta / i)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double t_0 = beta + (i * 2.0);
	double tmp;
	if (beta <= 4.8e+185) {
		tmp = (i / (alpha + (t_0 + 1.0))) * ((0.25 * (beta + i)) / (alpha + (t_0 + -1.0)));
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	t_0 = beta + (i * 2.0)
	tmp = 0
	if beta <= 4.8e+185:
		tmp = (i / (alpha + (t_0 + 1.0))) * ((0.25 * (beta + i)) / (alpha + (t_0 + -1.0)))
	else:
		tmp = ((i + alpha) / (beta / i)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	t_0 = Float64(beta + Float64(i * 2.0))
	tmp = 0.0
	if (beta <= 4.8e+185)
		tmp = Float64(Float64(i / Float64(alpha + Float64(t_0 + 1.0))) * Float64(Float64(0.25 * Float64(beta + i)) / Float64(alpha + Float64(t_0 + -1.0))));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	t_0 = beta + (i * 2.0);
	tmp = 0.0;
	if (beta <= 4.8e+185)
		tmp = (i / (alpha + (t_0 + 1.0))) * ((0.25 * (beta + i)) / (alpha + (t_0 + -1.0)));
	else
		tmp = ((i + alpha) / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.8e+185], N[(N[(i / N[(alpha + N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(0.25 * N[(beta + i), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 4.79999999999999978e185

   1. Initial program 17.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({i}^{2} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, i\right)\right)\right), 1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{i}\right)\right)\right), 1\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)\right), i\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

   5. Simplified 36.8%

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot \left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - 1} \]

      2. difference-of-sqr-1 N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right) \cdot \left(\left(\left(\alpha + \color{blue}{\beta}\right) + 2 \cdot i\right) - 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot i\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \color{blue}{\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{\left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) \cdot 0.25}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\beta}, \frac{1}{4}\right), i\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 86.3%

       \[\leadsto \frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{\color{blue}{\beta} \cdot 0.25}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]

   11. Taylor expanded in i around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot i\right)}, \mathsf{+.f64}\left(\alpha, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]
   12. Step-by-step derivation

       1. distribute-lft-out N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(\beta + i\right)\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \left(\beta + i\right)\right), \mathsf{+.f64}\left(\color{blue}{\alpha}, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]

       3. +-lowering-+.f64 86.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{4}, \mathsf{+.f64}\left(\beta, i\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]

   13. Simplified 86.3%

       \[\leadsto \frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{\color{blue}{0.25 \cdot \left(\beta + i\right)}}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]

   if 4.79999999999999978e185 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 86.2%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + i}{\frac{\beta}{i}}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.8 \cdot 10^{+185}:\\ \;\;\;\;\frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{0.25 \cdot \left(\beta + i\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + \frac{\beta}{i} \cdot 0.0625\right) + \frac{2 \cdot \left(\left(1 + \left(\beta + \alpha\right)\right) + \left(\alpha + \left(\beta + -1\right)\right)\right)}{i} \cdot -0.015625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.4e+185)
   (+
    (+ 0.0625 (* (/ beta i) 0.0625))
    (*
     (/ (* 2.0 (+ (+ 1.0 (+ beta alpha)) (+ alpha (+ beta -1.0)))) i)
     -0.015625))
   (/ (/ (+ i alpha) (/ beta i)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+185) {
		tmp = (0.0625 + ((beta / i) * 0.0625)) + (((2.0 * ((1.0 + (beta + alpha)) + (alpha + (beta + -1.0)))) / i) * -0.015625);
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.4d+185) then
        tmp = (0.0625d0 + ((beta / i) * 0.0625d0)) + (((2.0d0 * ((1.0d0 + (beta + alpha)) + (alpha + (beta + (-1.0d0))))) / i) * (-0.015625d0))
    else
        tmp = ((i + alpha) / (beta / i)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.4e+185) {
		tmp = (0.0625 + ((beta / i) * 0.0625)) + (((2.0 * ((1.0 + (beta + alpha)) + (alpha + (beta + -1.0)))) / i) * -0.015625);
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.4e+185:
		tmp = (0.0625 + ((beta / i) * 0.0625)) + (((2.0 * ((1.0 + (beta + alpha)) + (alpha + (beta + -1.0)))) / i) * -0.015625)
	else:
		tmp = ((i + alpha) / (beta / i)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.4e+185)
		tmp = Float64(Float64(0.0625 + Float64(Float64(beta / i) * 0.0625)) + Float64(Float64(Float64(2.0 * Float64(Float64(1.0 + Float64(beta + alpha)) + Float64(alpha + Float64(beta + -1.0)))) / i) * -0.015625));
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.4e+185)
		tmp = (0.0625 + ((beta / i) * 0.0625)) + (((2.0 * ((1.0 + (beta + alpha)) + (alpha + (beta + -1.0)))) / i) * -0.015625);
	else
		tmp = ((i + alpha) / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.4e+185], N[(N[(0.0625 + N[(N[(beta / i), $MachinePrecision] * 0.0625), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.0 * N[(N[(1.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] + N[(alpha + N[(beta + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision] * -0.015625), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.40000000000000017e185

   1. Initial program 17.1%

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left({i}^{2} \cdot \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)}, \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({i}^{2}\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right)}, 1\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(i \cdot i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\left(\frac{1}{4} + \frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i}\right) - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)\right), 1\right)\right) \]

      4. associate--l+ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{1}{4} \cdot \frac{2 \cdot \alpha + 2 \cdot \beta}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{1}{4} \cdot \frac{\alpha + \beta}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(\color{blue}{2}, i\right)\right)\right), 1\right)\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)}{i} - \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, \color{blue}{i}\right)\right)\right), 1\right)\right) \]

      7. div-sub N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \left(\frac{\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)}{i}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \color{blue}{\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right)}\right), 1\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(i, i\right), \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\left(\frac{1}{4} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - \frac{1}{4} \cdot \left(\alpha + \beta\right)\right), i\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \mathsf{*.f64}\left(2, i\right)\right), \mathsf{+.f64}\left(\mathsf{+.f64}\left(\alpha, \beta\right), \color{blue}{\mathsf{*.f64}\left(2, i\right)}\right)\right), 1\right)\right) \]

   5. Simplified 36.8%

      \[\leadsto \frac{\color{blue}{\left(i \cdot i\right) \cdot \left(0.25 + \frac{0.25 \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\color{blue}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - 1} \]

      2. difference-of-sqr-1 N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]

      3. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 1\right) \cdot \left(\left(\left(\alpha + \color{blue}{\beta}\right) + 2 \cdot i\right) - 1\right)} \]

      4. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\color{blue}{\left(\alpha + \beta\right)} + 2 \cdot i\right) - 1\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + i \cdot 2\right) - 1\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \frac{i \cdot \left(i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)\right)}{\left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1\right) \cdot \left(\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1\right)} \]

      7. times-frac N/A

         \[\leadsto \frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1} \cdot \color{blue}{\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}} \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{i}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) + 1}\right), \color{blue}{\left(\frac{i \cdot \left(\frac{1}{4} + \frac{\frac{1}{4} \cdot \left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right)}{i}\right)}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}\right)}\right) \]

   7. Applied egg-rr 83.2%

      \[\leadsto \color{blue}{\frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{\left(2 \cdot \left(\alpha + \beta\right) - \left(\alpha + \beta\right)\right) \cdot 0.25}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)}} \]

   8. Taylor expanded in beta around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \mathsf{+.f64}\left(\alpha, \mathsf{+.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\frac{1}{4}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\color{blue}{\beta}, \frac{1}{4}\right), i\right)\right)\right), \mathsf{+.f64}\left(\alpha, \mathsf{\_.f64}\left(\mathsf{+.f64}\left(\beta, \mathsf{*.f64}\left(i, 2\right)\right), 1\right)\right)\right)\right) \]
   9. Step-by-step derivation

   10. Simplified 86.3%

       \[\leadsto \frac{i}{\alpha + \left(\left(\beta + i \cdot 2\right) + 1\right)} \cdot \frac{i \cdot \left(0.25 + \frac{\color{blue}{\beta} \cdot 0.25}{i}\right)}{\alpha + \left(\left(\beta + i \cdot 2\right) - 1\right)} \]

   11. Taylor expanded in i around inf

       \[\leadsto \color{blue}{\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\beta}{i}\right) - \frac{1}{64} \cdot \frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}} \]
   12. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\beta}{i}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{64} \cdot \frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)\right)} \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{16} + \frac{1}{16} \cdot \frac{\beta}{i}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{64} \cdot \frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right)\right)}\right) \]

       3. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{1}{16} \cdot \frac{\beta}{i}\right)\right), \left(\mathsf{neg}\left(\color{blue}{\frac{1}{64} \cdot \frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right)\right)\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \left(\frac{\beta}{i} \cdot \frac{1}{16}\right)\right), \left(\mathsf{neg}\left(\frac{1}{64} \cdot \color{blue}{\frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(\left(\frac{\beta}{i}\right), \frac{1}{16}\right)\right), \left(\mathsf{neg}\left(\frac{1}{64} \cdot \color{blue}{\frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}}\right)\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, i\right), \frac{1}{16}\right)\right), \left(\mathsf{neg}\left(\frac{1}{64} \cdot \frac{\color{blue}{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}}{i}\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, i\right), \frac{1}{16}\right)\right), \left(\mathsf{neg}\left(\frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i} \cdot \frac{1}{64}\right)\right)\right) \]

       8. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, i\right), \frac{1}{16}\right)\right), \left(\frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{64}\right)\right)}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{+.f64}\left(\frac{1}{16}, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\beta, i\right), \frac{1}{16}\right)\right), \mathsf{*.f64}\left(\left(\frac{2 \cdot \left(1 + \left(\alpha + \beta\right)\right) + 2 \cdot \left(\left(\alpha + \beta\right) - 1\right)}{i}\right), \color{blue}{\left(\mathsf{neg}\left(\frac{1}{64}\right)\right)}\right)\right) \]

   13. Simplified 81.8%

       \[\leadsto \color{blue}{\left(0.0625 + \frac{\beta}{i} \cdot 0.0625\right) + \frac{2 \cdot \left(\left(1 + \left(\beta + \alpha\right)\right) + \left(\alpha + \left(\beta + -1\right)\right)\right)}{i} \cdot -0.015625} \]

   if 3.40000000000000017e185 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 86.2%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + i}{\frac{\beta}{i}}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+185}:\\ \;\;\;\;\left(0.0625 + \frac{\beta}{i} \cdot 0.0625\right) + \frac{2 \cdot \left(\left(1 + \left(\beta + \alpha\right)\right) + \left(\alpha + \left(\beta + -1\right)\right)\right)}{i} \cdot -0.015625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.0% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8e+185) 0.0625 (/ (/ (+ i alpha) (/ beta i)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+185) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 8d+185) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / (beta / i)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8e+185) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / (beta / i)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8e+185:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / (beta / i)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8e+185)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / Float64(beta / i)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8e+185)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / (beta / i)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8e+185], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.9999999999999998e185

   1. Initial program 17.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 82.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 7.9999999999999998e185 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{i}{\beta}\right), \beta\right) \]

       2. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\left(i + \alpha\right) \cdot \frac{1}{\frac{\beta}{i}}\right), \beta\right) \]

       3. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{i + \alpha}{\frac{\beta}{i}}\right), \beta\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       5. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \left(\frac{\beta}{i}\right)\right), \beta\right) \]

       7. /-lowering-/.f64 86.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\beta, i\right)\right), \beta\right) \]

   11. Applied egg-rr 86.2%

       \[\leadsto \frac{\color{blue}{\frac{\alpha + i}{\frac{\beta}{i}}}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\frac{\beta}{i}}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 3.8 \cdot 10^{+185}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 3.8e+185) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+185) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 3.8d+185) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 3.8e+185) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 3.8e+185:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 3.8e+185)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 3.8e+185)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 3.8e+185], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 3.7999999999999998e185

   1. Initial program 17.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 82.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 3.7999999999999998e185 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(\alpha + i\right) \cdot i}{\color{blue}{\beta} \cdot \beta} \]

      2. times-frac N/A

         \[\leadsto \frac{\alpha + i}{\beta} \cdot \color{blue}{\frac{i}{\beta}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\alpha + i}{\beta}\right), \color{blue}{\left(\frac{i}{\beta}\right)}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\alpha + i\right), \beta\right), \left(\frac{\color{blue}{i}}{\beta}\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(i + \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \left(\frac{i}{\beta}\right)\right) \]

      7. /-lowering-/.f64 86.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(i, \alpha\right), \beta\right), \mathsf{/.f64}\left(i, \color{blue}{\beta}\right)\right) \]

   9. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+186}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 6.5e+186) 0.0625 (/ (* i (/ i beta)) beta)))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+186) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 6.5d+186) then
        tmp = 0.0625d0
    else
        tmp = (i * (i / beta)) / beta
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 6.5e+186) {
		tmp = 0.0625;
	} else {
		tmp = (i * (i / beta)) / beta;
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 6.5e+186:
		tmp = 0.0625
	else:
		tmp = (i * (i / beta)) / beta
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 6.5e+186)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i * Float64(i / beta)) / beta);
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 6.5e+186)
		tmp = 0.0625;
	else
		tmp = (i * (i / beta)) / beta;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 6.5e+186], 0.0625, N[(N[(i * N[(i / beta), $MachinePrecision]), $MachinePrecision] / beta), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 6.4999999999999997e186

   1. Initial program 17.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 82.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 6.4999999999999997e186 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]

   10. Taylor expanded in i around inf

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \color{blue}{i}\right), \beta\right) \]
   11. Step-by-step derivation

   12. Simplified 89.3%

       \[\leadsto \frac{\frac{i}{\beta} \cdot \color{blue}{i}}{\beta} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+186}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+187}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;i \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.05e+187) 0.0625 (* i (/ (/ i beta) beta))))

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.05e+187) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / beta) / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: tmp
    if (beta <= 2.05d+187) then
        tmp = 0.0625d0
    else
        tmp = i * ((i / beta) / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.05e+187) {
		tmp = 0.0625;
	} else {
		tmp = i * ((i / beta) / beta);
	}
	return tmp;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.05e+187:
		tmp = 0.0625
	else:
		tmp = i * ((i / beta) / beta)
	return tmp

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.05e+187)
		tmp = 0.0625;
	else
		tmp = Float64(i * Float64(Float64(i / beta) / beta));
	end
	return tmp
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.05e+187)
		tmp = 0.0625;
	else
		tmp = i * ((i / beta) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.05e+187], 0.0625, N[(i * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.05e187

   1. Initial program 17.1%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{1}{16}} \]
   6. Step-by-step derivation

   7. Simplified 82.6%

      \[\leadsto \color{blue}{0.0625} \]

   if 2.05e187 < beta

   1. Initial program 0.0%

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

   3. Simplified 23.3%

      \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
   4. Add Preprocessing

   5. Taylor expanded in beta around inf

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{{\beta}^{2}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(i \cdot \left(\alpha + i\right)\right), \color{blue}{\left({\beta}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \left(\alpha + i\right)\right), \left({\color{blue}{\beta}}^{2}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left({\beta}^{2}\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \left(\beta \cdot \color{blue}{\beta}\right)\right) \]

      5. *-lowering-*.f64 42.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(i, \mathsf{+.f64}\left(\alpha, i\right)\right), \mathsf{*.f64}\left(\beta, \color{blue}{\beta}\right)\right) \]

   7. Simplified 42.7%

      \[\leadsto \color{blue}{\frac{i \cdot \left(\alpha + i\right)}{\beta \cdot \beta}} \]
   8. Step-by-step derivation

      1. times-frac N/A

         \[\leadsto \frac{i}{\beta} \cdot \color{blue}{\frac{\alpha + i}{\beta}} \]

      2. associate-*r/ N/A

         \[\leadsto \frac{\frac{i}{\beta} \cdot \left(\alpha + i\right)}{\color{blue}{\beta}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{i}{\beta} \cdot \left(\alpha + i\right)\right), \color{blue}{\beta}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{i}{\beta}\right), \left(\alpha + i\right)\right), \beta\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(\alpha + i\right)\right), \beta\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \left(i + \alpha\right)\right), \beta\right) \]

      7. +-lowering-+.f64 86.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \mathsf{+.f64}\left(i, \alpha\right)\right), \beta\right) \]

   9. Applied egg-rr 86.1%

      \[\leadsto \color{blue}{\frac{\frac{i}{\beta} \cdot \left(i + \alpha\right)}{\beta}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\left(i + \alpha\right) \cdot \frac{i}{\beta}}{\beta} \]

       2. associate-/l* N/A

          \[\leadsto \left(i + \alpha\right) \cdot \color{blue}{\frac{\frac{i}{\beta}}{\beta}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(i + \alpha\right), \color{blue}{\left(\frac{\frac{i}{\beta}}{\beta}\right)}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\alpha + i\right), \left(\frac{\color{blue}{\frac{i}{\beta}}}{\beta}\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \left(\frac{\color{blue}{\frac{i}{\beta}}}{\beta}\right)\right) \]

       6. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\left(\frac{i}{\beta}\right), \color{blue}{\beta}\right)\right) \]

       7. /-lowering-/.f64 66.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\alpha, i\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \beta\right)\right) \]

   11. Applied egg-rr 66.4%

       \[\leadsto \color{blue}{\left(\alpha + i\right) \cdot \frac{\frac{i}{\beta}}{\beta}} \]

   12. Taylor expanded in alpha around 0

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{i}, \mathsf{/.f64}\left(\mathsf{/.f64}\left(i, \beta\right), \beta\right)\right) \]
   13. Step-by-step derivation

   14. Simplified 66.4%

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

Alternative 8: 69.9% accurate, 53.0× speedup

Math

\[\begin{array}{l} [alpha, beta, i] = \mathsf{sort}([alpha, beta, i])\\ \\ 0.0625 \end{array} \]

FPCore

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
(FPCore (alpha beta i) :precision binary64 0.0625)

C

assert(alpha < beta && beta < i);
double code(double alpha, double beta, double i) {
	return 0.0625;
}

Fortran

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    code = 0.0625d0
end function

Java

assert alpha < beta && beta < i;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

[alpha, beta, i] = sort([alpha, beta, i])
def code(alpha, beta, i):
	return 0.0625

Julia

alpha, beta, i = sort([alpha, beta, i])
function code(alpha, beta, i)
	return 0.0625
end

MATLAB

alpha, beta, i = num2cell(sort([alpha, beta, i])){:}
function tmp = code(alpha, beta, i)
	tmp = 0.0625;
end

Wolfram

NOTE: alpha, beta, and i should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := 0.0625

Derivation

1. Initial program 15.1%

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

   1. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right), \color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}\right) \]

3. Simplified 40.0%

   \[\leadsto \color{blue}{\frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right) + \alpha \cdot \beta\right) \cdot \frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{\alpha + \left(\beta + i \cdot 2\right)}}{\alpha + \left(\beta + i \cdot 2\right)}}{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) + -1}} \]
4. Add Preprocessing

5. Taylor expanded in i around inf

   \[\leadsto \color{blue}{\frac{1}{16}} \]
6. Step-by-step derivation

7. Simplified 74.4%

   \[\leadsto \color{blue}{0.0625} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))