Octave 3.8, jcobi/4

Percentage Accurate: 15.9% → 83.2%

Time: 20.0s

Alternatives: 11

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: 15.9% 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: 83.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+101}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, i + \alpha, \frac{i + \alpha}{\frac{\beta}{i \cdot i}}\right) + \frac{{\left(i + \alpha\right)}^{2}}{\frac{\beta}{i}}\right) - \frac{i}{\frac{\beta}{\left(i + \alpha\right) \cdot \mathsf{fma}\left(4, i, \alpha \cdot 2\right)}}}{t_0 \cdot t_0 + -1}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+200}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (* i 2.0) (+ beta alpha))))
   (if (<= beta 4.7e+101)
     0.0625
     (if (<= beta 9.2e+153)
       (/
        (-
         (+
          (fma i (+ i alpha) (/ (+ i alpha) (/ beta (* i i))))
          (/ (pow (+ i alpha) 2.0) (/ beta i)))
         (/ i (/ beta (* (+ i alpha) (fma 4.0 i (* alpha 2.0))))))
        (+ (* t_0 t_0) -1.0))
       (if (<= beta 1.5e+200)
         (+
          (+ 0.0625 (* 0.0625 (/ beta i)))
          (*
           0.00390625
           (/
            (-
             (*
              2.0
              (-
               (* 4.0 (- (- alpha (* beta -2.0)) beta))
               (* 8.0 (+ beta alpha))))
             (* beta 8.0))
            i)))
         (/ (/ (+ i alpha) beta) (/ beta i)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = (i * 2.0) + (beta + alpha);
	double tmp;
	if (beta <= 4.7e+101) {
		tmp = 0.0625;
	} else if (beta <= 9.2e+153) {
		tmp = ((fma(i, (i + alpha), ((i + alpha) / (beta / (i * i)))) + (pow((i + alpha), 2.0) / (beta / i))) - (i / (beta / ((i + alpha) * fma(4.0, i, (alpha * 2.0)))))) / ((t_0 * t_0) + -1.0);
	} else if (beta <= 1.5e+200) {
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.7e+101)
		tmp = 0.0625;
	elseif (beta <= 9.2e+153)
		tmp = Float64(Float64(Float64(fma(i, Float64(i + alpha), Float64(Float64(i + alpha) / Float64(beta / Float64(i * i)))) + Float64((Float64(i + alpha) ^ 2.0) / Float64(beta / i))) - Float64(i / Float64(beta / Float64(Float64(i + alpha) * fma(4.0, i, Float64(alpha * 2.0)))))) / Float64(Float64(t_0 * t_0) + -1.0));
	elseif (beta <= 1.5e+200)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(beta / i))) + Float64(0.00390625 * Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(alpha - Float64(beta * -2.0)) - beta)) - Float64(8.0 * Float64(beta + alpha)))) - Float64(beta * 8.0)) / i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.7e+101], 0.0625, If[LessEqual[beta, 9.2e+153], N[(N[(N[(N[(i * N[(i + alpha), $MachinePrecision] + N[(N[(i + alpha), $MachinePrecision] / N[(beta / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(i + alpha), $MachinePrecision], 2.0], $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(i / N[(beta / N[(N[(i + alpha), $MachinePrecision] * N[(4.0 * i + N[(alpha * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$0 * t$95$0), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+200], N[(N[(0.0625 + N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.00390625 * N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(alpha - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - N[(8.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(beta * 8.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.69999999999999971e101

   1. Initial program 24.4%

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

      1. associate-/l/ 22.4%

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

      2. associate-*l* 22.3%

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

      3. times-frac 28.3%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in i around inf 79.2%

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

   if 4.69999999999999971e101 < beta < 9.2000000000000005e153

   1. Initial program 18.8%

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

   2. Taylor expanded in beta around inf 36.2%

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

      1. +-commutative 36.2%

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

      2. fma-def 36.2%

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

      3. +-commutative 36.2%

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

      4. associate-/l* 36.2%

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

      5. +-commutative 36.2%

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

      6. unpow2 36.2%

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

      7. associate-/l* 36.4%

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

      8. +-commutative 36.4%

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

      9. associate-/l* 54.3%

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

      10. +-commutative 54.3%

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

      11. fma-def 54.3%

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

      12. *-commutative 54.3%

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

   4. Simplified 54.3%

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

   if 9.2000000000000005e153 < beta < 1.49999999999999995e200

   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. times-frac 11.2%

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

      2. +-commutative 11.2%

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

      3. +-commutative 11.2%

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

      4. *-commutative 11.2%

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

      5. fma-def 11.2%

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

      6. +-commutative 11.2%

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

      7. +-commutative 11.2%

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

      8. *-commutative 11.2%

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

      9. fma-udef 11.2%

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

      10. +-commutative 11.2%

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

      11. *-commutative 11.2%

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

      12. fma-def 11.2%

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

   3. Applied egg-rr 11.2%

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

      1. *-commutative 11.2%

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

      2. +-commutative 11.2%

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

      3. +-commutative 11.2%

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

      4. *-commutative 11.2%

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

      5. +-commutative 11.2%

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

      6. associate-/l* 11.2%

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

      7. +-commutative 11.2%

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

      8. +-commutative 11.2%

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

      9. +-commutative 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in alpha around 0 11.2%

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

   7. Taylor expanded in alpha around 0 11.2%

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

   8. Taylor expanded in i around inf 67.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) - 0.00390625 \cdot \frac{2 \cdot \left(8 \cdot \left(\beta + \alpha\right) + 4 \cdot \left(\beta + \left(-2 \cdot \beta + -1 \cdot \alpha\right)\right)\right) + 8 \cdot \beta}{i}} \]

   if 1.49999999999999995e200 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in beta around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 36.7%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 36.7%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 36.7%

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

   6. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 36.7%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 51.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 51.1%

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

      2. associate-/r/ 51.2%

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

   10. Simplified 51.2%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 51.1%

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

       2. add-cube-cbrt 51.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 50.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 50.3%

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

       5. associate-/r/ 50.3%

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

       6. +-commutative 50.3%

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

       7. *-commutative 50.3%

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

   12. Applied egg-rr 50.3%

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

       1. add-exp-log 51.2%

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

       2. associate-/r* 82.1%

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

   14. Applied egg-rr 82.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+101}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(\mathsf{fma}\left(i, i + \alpha, \frac{i + \alpha}{\frac{\beta}{i \cdot i}}\right) + \frac{{\left(i + \alpha\right)}^{2}}{\frac{\beta}{i}}\right) - \frac{i}{\frac{\beta}{\left(i + \alpha\right) \cdot \mathsf{fma}\left(4, i, \alpha \cdot 2\right)}}}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+200}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 2: 84.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := i \cdot 2 + \left(\beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+101}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot t_0\right)}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ i (+ beta alpha))) (t_1 (+ (* i 2.0) (+ beta alpha))))
   (if (<= beta 4.5e+101)
     0.0625
     (if (<= beta 3e+153)
       (/
        (*
         (/ (fma i t_0 (* beta alpha)) (fma i 2.0 (+ beta alpha)))
         (* (/ i (+ alpha (fma i 2.0 beta))) t_0))
        (+ (* t_1 t_1) -1.0))
       (if (<= beta 1.32e+200)
         (+
          (+ 0.0625 (* 0.0625 (/ beta i)))
          (*
           0.00390625
           (/
            (-
             (*
              2.0
              (-
               (* 4.0 (- (- alpha (* beta -2.0)) beta))
               (* 8.0 (+ beta alpha))))
             (* beta 8.0))
            i)))
         (/ (/ (+ i alpha) beta) (/ beta i)))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double t_0 = i + (beta + alpha);
	double t_1 = (i * 2.0) + (beta + alpha);
	double tmp;
	if (beta <= 4.5e+101) {
		tmp = 0.0625;
	} else if (beta <= 3e+153) {
		tmp = ((fma(i, t_0, (beta * alpha)) / fma(i, 2.0, (beta + alpha))) * ((i / (alpha + fma(i, 2.0, beta))) * t_0)) / ((t_1 * t_1) + -1.0);
	} else if (beta <= 1.32e+200) {
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	t_0 = Float64(i + Float64(beta + alpha))
	t_1 = Float64(Float64(i * 2.0) + Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 4.5e+101)
		tmp = 0.0625;
	elseif (beta <= 3e+153)
		tmp = Float64(Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / fma(i, 2.0, Float64(beta + alpha))) * Float64(Float64(i / Float64(alpha + fma(i, 2.0, beta))) * t_0)) / Float64(Float64(t_1 * t_1) + -1.0));
	elseif (beta <= 1.32e+200)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(beta / i))) + Float64(0.00390625 * Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(alpha - Float64(beta * -2.0)) - beta)) - Float64(8.0 * Float64(beta + alpha)))) - Float64(beta * 8.0)) / i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := Block[{t$95$0 = N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(i * 2.0), $MachinePrecision] + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 4.5e+101], 0.0625, If[LessEqual[beta, 3e+153], N[(N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.32e+200], N[(N[(0.0625 + N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.00390625 * N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(alpha - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - N[(8.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(beta * 8.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 4.5000000000000002e101

   1. Initial program 24.4%

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

      1. associate-/l/ 22.4%

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

      2. associate-*l* 22.3%

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

      3. times-frac 28.3%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in i around inf 79.2%

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

   if 4.5000000000000002e101 < beta < 3.00000000000000019e153

   1. Initial program 20.7%

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

      1. times-frac 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. fma-def 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. *-commutative 59.4%

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

      9. fma-udef 59.4%

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

      10. +-commutative 59.4%

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

      11. *-commutative 59.4%

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

      12. fma-def 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. *-commutative 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. +-commutative 59.4%

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

      6. associate-/l* 59.4%

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

      7. +-commutative 59.4%

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

      8. +-commutative 59.4%

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

      9. +-commutative 59.4%

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

   5. Simplified 59.4%

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

      1. associate-/r/ 59.5%

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

      2. fma-udef 59.5%

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

      3. *-commutative 59.5%

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

      4. +-commutative 59.5%

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

      5. +-commutative 59.5%

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

      6. associate-+l+ 59.5%

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

      7. *-commutative 59.5%

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

      8. +-commutative 59.5%

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

      9. fma-udef 59.5%

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

      10. +-commutative 59.5%

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

      11. +-commutative 59.5%

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

      12. +-commutative 59.5%

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

   7. Applied egg-rr 59.5%

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

   if 3.00000000000000019e153 < beta < 1.3199999999999999e200

   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. times-frac 10.7%

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

      2. +-commutative 10.7%

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

      3. +-commutative 10.7%

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

      4. *-commutative 10.7%

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

      5. fma-def 10.7%

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

      6. +-commutative 10.7%

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

      7. +-commutative 10.7%

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

      8. *-commutative 10.7%

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

      9. fma-udef 10.7%

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

      10. +-commutative 10.7%

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

      11. *-commutative 10.7%

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

      12. fma-def 10.7%

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

   3. Applied egg-rr 10.7%

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

      1. *-commutative 10.7%

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

      2. +-commutative 10.7%

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

      3. +-commutative 10.7%

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

      4. *-commutative 10.7%

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

      5. +-commutative 10.7%

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

      6. associate-/l* 10.7%

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

      7. +-commutative 10.7%

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

      8. +-commutative 10.7%

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

      9. +-commutative 10.7%

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

   5. Simplified 10.7%

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

   6. Taylor expanded in alpha around 0 10.7%

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

   7. Taylor expanded in alpha around 0 10.7%

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

   8. Taylor expanded in i around inf 63.9%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) - 0.00390625 \cdot \frac{2 \cdot \left(8 \cdot \left(\beta + \alpha\right) + 4 \cdot \left(\beta + \left(-2 \cdot \beta + -1 \cdot \alpha\right)\right)\right) + 8 \cdot \beta}{i}} \]

   if 1.3199999999999999e200 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in beta around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 36.7%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 36.7%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 36.7%

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

   6. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 36.7%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 51.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 51.1%

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

      2. associate-/r/ 51.2%

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

   10. Simplified 51.2%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 51.1%

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

       2. add-cube-cbrt 51.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 50.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 50.3%

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

       5. associate-/r/ 50.3%

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

       6. +-commutative 50.3%

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

       7. *-commutative 50.3%

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

   12. Applied egg-rr 50.3%

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

       1. add-exp-log 51.2%

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

       2. associate-/r* 82.1%

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

   14. Applied egg-rr 82.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+101}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(i, i + \left(\beta + \alpha\right), \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right)}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) \cdot \left(i \cdot 2 + \left(\beta + \alpha\right)\right) + -1}\\ \mathbf{elif}\;\beta \leq 1.32 \cdot 10^{+200}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 3: 83.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+201}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 1.8e+103)
   0.0625
   (if (<= beta 2.45e+154)
     (/ 1.0 (/ (* beta (/ beta i)) (+ i alpha)))
     (if (<= beta 1.5e+201)
       (+
        (+ 0.0625 (* 0.0625 (/ beta i)))
        (*
         0.00390625
         (/
          (-
           (*
            2.0
            (-
             (* 4.0 (- (- alpha (* beta -2.0)) beta))
             (* 8.0 (+ beta alpha))))
           (* beta 8.0))
          i)))
       (/ (/ (+ i alpha) beta) (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.8e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.45e+154) {
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	} else if (beta <= 1.5e+201) {
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 1.8d+103) then
        tmp = 0.0625d0
    else if (beta <= 2.45d+154) then
        tmp = 1.0d0 / ((beta * (beta / i)) / (i + alpha))
    else if (beta <= 1.5d+201) then
        tmp = (0.0625d0 + (0.0625d0 * (beta / i))) + (0.00390625d0 * (((2.0d0 * ((4.0d0 * ((alpha - (beta * (-2.0d0))) - beta)) - (8.0d0 * (beta + alpha)))) - (beta * 8.0d0)) / i))
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 1.8e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.45e+154) {
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	} else if (beta <= 1.5e+201) {
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i));
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 1.8e+103:
		tmp = 0.0625
	elif beta <= 2.45e+154:
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha))
	elif beta <= 1.5e+201:
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i))
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 1.8e+103)
		tmp = 0.0625;
	elseif (beta <= 2.45e+154)
		tmp = Float64(1.0 / Float64(Float64(beta * Float64(beta / i)) / Float64(i + alpha)));
	elseif (beta <= 1.5e+201)
		tmp = Float64(Float64(0.0625 + Float64(0.0625 * Float64(beta / i))) + Float64(0.00390625 * Float64(Float64(Float64(2.0 * Float64(Float64(4.0 * Float64(Float64(alpha - Float64(beta * -2.0)) - beta)) - Float64(8.0 * Float64(beta + alpha)))) - Float64(beta * 8.0)) / i)));
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 1.8e+103)
		tmp = 0.0625;
	elseif (beta <= 2.45e+154)
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	elseif (beta <= 1.5e+201)
		tmp = (0.0625 + (0.0625 * (beta / i))) + (0.00390625 * (((2.0 * ((4.0 * ((alpha - (beta * -2.0)) - beta)) - (8.0 * (beta + alpha)))) - (beta * 8.0)) / i));
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 1.8e+103], 0.0625, If[LessEqual[beta, 2.45e+154], N[(1.0 / N[(N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.5e+201], N[(N[(0.0625 + N[(0.0625 * N[(beta / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.00390625 * N[(N[(N[(2.0 * N[(N[(4.0 * N[(N[(alpha - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision] - N[(8.0 * N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(beta * 8.0), $MachinePrecision]), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 1.80000000000000008e103

   1. Initial program 24.4%

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

      1. associate-/l/ 22.4%

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

      2. associate-*l* 22.3%

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

      3. times-frac 28.3%

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

   3. Simplified 46.1%

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

   4. Taylor expanded in i around inf 79.2%

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

   if 1.80000000000000008e103 < beta < 2.4500000000000001e154

   1. Initial program 18.8%

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

      1. associate-/l/ 1.0%

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

      2. associate-*l* 1.0%

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

      3. times-frac 18.9%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in beta around inf 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-/l* 53.6%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 53.6%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 53.6%

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

   6. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 53.6%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 53.6%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 53.6%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 53.6%

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

      2. associate-/r/ 53.7%

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

   10. Simplified 53.7%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 53.6%

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

       2. add-cube-cbrt 52.9%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 49.6%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 49.6%

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

       5. associate-/r/ 49.6%

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

       6. +-commutative 49.6%

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

       7. *-commutative 49.6%

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

   12. Applied egg-rr 49.6%

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

       1. add-exp-log 53.7%

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

       2. clear-num 53.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}} \]

   14. Applied egg-rr 53.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}} \]

   if 2.4500000000000001e154 < beta < 1.50000000000000012e201

   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. times-frac 11.2%

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

      2. +-commutative 11.2%

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

      3. +-commutative 11.2%

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

      4. *-commutative 11.2%

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

      5. fma-def 11.2%

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

      6. +-commutative 11.2%

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

      7. +-commutative 11.2%

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

      8. *-commutative 11.2%

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

      9. fma-udef 11.2%

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

      10. +-commutative 11.2%

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

      11. *-commutative 11.2%

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

      12. fma-def 11.2%

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

   3. Applied egg-rr 11.2%

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

      1. *-commutative 11.2%

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

      2. +-commutative 11.2%

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

      3. +-commutative 11.2%

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

      4. *-commutative 11.2%

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

      5. +-commutative 11.2%

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

      6. associate-/l* 11.2%

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

      7. +-commutative 11.2%

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

      8. +-commutative 11.2%

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

      9. +-commutative 11.2%

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

   5. Simplified 11.2%

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

   6. Taylor expanded in alpha around 0 11.2%

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

   7. Taylor expanded in alpha around 0 11.2%

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

   8. Taylor expanded in i around inf 67.0%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) - 0.00390625 \cdot \frac{2 \cdot \left(8 \cdot \left(\beta + \alpha\right) + 4 \cdot \left(\beta + \left(-2 \cdot \beta + -1 \cdot \alpha\right)\right)\right) + 8 \cdot \beta}{i}} \]

   if 1.50000000000000012e201 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 18.5%

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

   4. Taylor expanded in beta around inf 35.0%

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

      1. *-commutative 35.0%

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

      2. associate-/l* 36.7%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 36.7%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 36.7%

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

   6. Simplified 36.7%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 36.7%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 51.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 51.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 51.1%

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

      2. associate-/r/ 51.2%

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

   10. Simplified 51.2%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 51.1%

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

       2. add-cube-cbrt 51.0%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 50.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 50.3%

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

       5. associate-/r/ 50.3%

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

       6. +-commutative 50.3%

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

       7. *-commutative 50.3%

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

   12. Applied egg-rr 50.3%

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

       1. add-exp-log 51.2%

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

       2. associate-/r* 82.1%

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

   14. Applied egg-rr 82.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 1.8 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.45 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.5 \cdot 10^{+201}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + 0.00390625 \cdot \frac{2 \cdot \left(4 \cdot \left(\left(\alpha - \beta \cdot -2\right) - \beta\right) - 8 \cdot \left(\beta + \alpha\right)\right) - \beta \cdot 8}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 4: 83.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+190}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.8e+102)
   0.0625
   (if (<= beta 4.6e+153)
     (* (+ i alpha) (/ (/ i beta) beta))
     (if (<= beta 6e+190) 0.0625 (* (/ (+ i alpha) beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 4.6e+153) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 6e+190) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 8.8d+102) then
        tmp = 0.0625d0
    else if (beta <= 4.6d+153) then
        tmp = (i + alpha) * ((i / beta) / beta)
    else if (beta <= 6d+190) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 4.6e+153) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 6e+190) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.8e+102:
		tmp = 0.0625
	elif beta <= 4.6e+153:
		tmp = (i + alpha) * ((i / beta) / beta)
	elif beta <= 6e+190:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 4.6e+153)
		tmp = Float64(Float64(i + alpha) * Float64(Float64(i / beta) / beta));
	elseif (beta <= 6e+190)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 4.6e+153)
		tmp = (i + alpha) * ((i / beta) / beta);
	elseif (beta <= 6e+190)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.8e+102], 0.0625, If[LessEqual[beta, 4.6e+153], N[(N[(i + alpha), $MachinePrecision] * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6e+190], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.8000000000000003e102 or 4.6000000000000003e153 < beta < 5.99999999999999964e190

   1. Initial program 22.9%

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

      1. associate-/l/ 21.0%

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

      2. associate-*l* 20.9%

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

      3. times-frac 26.5%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in i around inf 77.8%

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

   if 8.8000000000000003e102 < beta < 4.6000000000000003e153

   1. Initial program 20.7%

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

      1. times-frac 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. fma-def 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. *-commutative 59.4%

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

      9. fma-udef 59.4%

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

      10. +-commutative 59.4%

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

      11. *-commutative 59.4%

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

      12. fma-def 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. *-commutative 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. +-commutative 59.4%

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

      6. associate-/l* 59.4%

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

      7. +-commutative 59.4%

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

      8. +-commutative 59.4%

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

      9. +-commutative 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in beta around inf 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*r/ 58.6%

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

      3. unpow2 58.6%

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

      4. associate-/r* 58.8%

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

   8. Simplified 58.8%

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

   if 5.99999999999999964e190 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in beta around inf 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/l* 34.9%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 34.9%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 34.9%

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

   6. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 34.9%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 49.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 49.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 49.1%

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

      2. associate-/r/ 49.2%

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

   10. Simplified 49.2%

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

   11. Taylor expanded in beta around 0 33.3%

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

       1. unpow2 33.3%

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

       2. times-frac 76.4%

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

   13. Simplified 76.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.6 \cdot 10^{+153}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+190}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5: 83.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+189}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.8e+102)
   0.0625
   (if (<= beta 9.5e+152)
     (* (+ i alpha) (/ (/ i beta) beta))
     (if (<= beta 3.8e+189) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 9.5e+152) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 3.8e+189) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 8.8d+102) then
        tmp = 0.0625d0
    else if (beta <= 9.5d+152) then
        tmp = (i + alpha) * ((i / beta) / beta)
    else if (beta <= 3.8d+189) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 9.5e+152) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 3.8e+189) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.8e+102:
		tmp = 0.0625
	elif beta <= 9.5e+152:
		tmp = (i + alpha) * ((i / beta) / beta)
	elif beta <= 3.8e+189:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 9.5e+152)
		tmp = Float64(Float64(i + alpha) * Float64(Float64(i / beta) / beta));
	elseif (beta <= 3.8e+189)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 9.5e+152)
		tmp = (i + alpha) * ((i / beta) / beta);
	elseif (beta <= 3.8e+189)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.8e+102], 0.0625, If[LessEqual[beta, 9.5e+152], N[(N[(i + alpha), $MachinePrecision] * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.8e+189], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.8000000000000003e102 or 9.49999999999999916e152 < beta < 3.7999999999999998e189

   1. Initial program 22.9%

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

      1. associate-/l/ 21.0%

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

      2. associate-*l* 20.9%

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

      3. times-frac 26.5%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in i around inf 77.8%

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

   if 8.8000000000000003e102 < beta < 9.49999999999999916e152

   1. Initial program 20.7%

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

      1. times-frac 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. fma-def 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. *-commutative 59.4%

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

      9. fma-udef 59.4%

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

      10. +-commutative 59.4%

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

      11. *-commutative 59.4%

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

      12. fma-def 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. *-commutative 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. +-commutative 59.4%

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

      6. associate-/l* 59.4%

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

      7. +-commutative 59.4%

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

      8. +-commutative 59.4%

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

      9. +-commutative 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in beta around inf 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*r/ 58.6%

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

      3. unpow2 58.6%

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

      4. associate-/r* 58.8%

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

   8. Simplified 58.8%

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

   if 3.7999999999999998e189 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in beta around inf 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/l* 34.9%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 34.9%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 34.9%

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

   6. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 34.9%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 49.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 49.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 49.1%

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

      2. associate-/r/ 49.2%

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

   10. Simplified 49.2%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 49.1%

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

       2. add-cube-cbrt 49.1%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 48.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 48.3%

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

       5. associate-/r/ 48.3%

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

       6. +-commutative 48.3%

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

       7. *-commutative 48.3%

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

   12. Applied egg-rr 48.3%

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

       1. add-exp-log 49.2%

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

       2. associate-/r* 76.3%

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

   14. Applied egg-rr 76.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+152}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+189}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 6: 83.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+189}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.5e+103)
   0.0625
   (if (<= beta 2.6e+153)
     (/ 1.0 (/ (* beta (/ beta i)) (+ i alpha)))
     (if (<= beta 8e+189) 0.0625 (/ (/ (+ i alpha) beta) (/ beta i))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.6e+153) {
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	} else if (beta <= 8e+189) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 2.5d+103) then
        tmp = 0.0625d0
    else if (beta <= 2.6d+153) then
        tmp = 1.0d0 / ((beta * (beta / i)) / (i + alpha))
    else if (beta <= 8d+189) then
        tmp = 0.0625d0
    else
        tmp = ((i + alpha) / beta) / (beta / i)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.6e+153) {
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	} else if (beta <= 8e+189) {
		tmp = 0.0625;
	} else {
		tmp = ((i + alpha) / beta) / (beta / i);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.5e+103:
		tmp = 0.0625
	elif beta <= 2.6e+153:
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha))
	elif beta <= 8e+189:
		tmp = 0.0625
	else:
		tmp = ((i + alpha) / beta) / (beta / i)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.5e+103)
		tmp = 0.0625;
	elseif (beta <= 2.6e+153)
		tmp = Float64(1.0 / Float64(Float64(beta * Float64(beta / i)) / Float64(i + alpha)));
	elseif (beta <= 8e+189)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(Float64(i + alpha) / beta) / Float64(beta / i));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.5e+103)
		tmp = 0.0625;
	elseif (beta <= 2.6e+153)
		tmp = 1.0 / ((beta * (beta / i)) / (i + alpha));
	elseif (beta <= 8e+189)
		tmp = 0.0625;
	else
		tmp = ((i + alpha) / beta) / (beta / i);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.5e+103], 0.0625, If[LessEqual[beta, 2.6e+153], N[(1.0 / N[(N[(beta * N[(beta / i), $MachinePrecision]), $MachinePrecision] / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 8e+189], 0.0625, N[(N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5e103 or 2.5999999999999999e153 < beta < 8.0000000000000002e189

   1. Initial program 22.9%

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

      1. associate-/l/ 21.0%

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

      2. associate-*l* 20.9%

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

      3. times-frac 26.5%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in i around inf 77.8%

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

   if 2.5e103 < beta < 2.5999999999999999e153

   1. Initial program 20.7%

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

      1. associate-/l/ 1.1%

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

      2. associate-*l* 1.1%

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

      3. times-frac 20.8%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in beta around inf 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-/l* 58.6%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 58.6%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 58.6%

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

   6. Simplified 58.6%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 58.6%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 58.6%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 58.6%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 58.6%

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

      2. associate-/r/ 58.8%

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

   10. Simplified 58.8%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 58.6%

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

       2. add-cube-cbrt 57.9%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 54.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 54.3%

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

       5. associate-/r/ 54.3%

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

       6. +-commutative 54.3%

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

       7. *-commutative 54.3%

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

   12. Applied egg-rr 54.3%

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

       1. add-exp-log 58.8%

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

       2. clear-num 59.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}} \]

   14. Applied egg-rr 59.0%

       \[\leadsto \color{blue}{\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}} \]

   if 8.0000000000000002e189 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 19.9%

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

   4. Taylor expanded in beta around inf 33.3%

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

      1. *-commutative 33.3%

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

      2. associate-/l* 34.9%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 34.9%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 34.9%

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

   6. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{\alpha + i}{\frac{\beta \cdot \beta}{i}}} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 34.9%

         \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta \cdot \beta}{i}}} \]

      2. associate-/l* 49.1%

         \[\leadsto \frac{\alpha + i}{1 \cdot \color{blue}{\frac{\beta}{\frac{i}{\beta}}}} \]

   8. Applied egg-rr 49.1%

      \[\leadsto \frac{\alpha + i}{\color{blue}{1 \cdot \frac{\beta}{\frac{i}{\beta}}}} \]
   9. Step-by-step derivation

      1. *-lft-identity 49.1%

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

      2. associate-/r/ 49.2%

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

   10. Simplified 49.2%

       \[\leadsto \frac{\alpha + i}{\color{blue}{\frac{\beta}{i} \cdot \beta}} \]
   11. Step-by-step derivation

       1. associate-/r/ 49.1%

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

       2. add-cube-cbrt 49.1%

          \[\leadsto \frac{\alpha + i}{\color{blue}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}} \]

       3. add-exp-log 48.3%

          \[\leadsto \color{blue}{e^{\log \left(\frac{\alpha + i}{\left(\sqrt[3]{\frac{\beta}{\frac{i}{\beta}}} \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}\right) \cdot \sqrt[3]{\frac{\beta}{\frac{i}{\beta}}}}\right)}} \]

       4. add-cube-cbrt 48.3%

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

       5. associate-/r/ 48.3%

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

       6. +-commutative 48.3%

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

       7. *-commutative 48.3%

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

   12. Applied egg-rr 48.3%

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

       1. add-exp-log 49.2%

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

       2. associate-/r* 76.3%

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

   14. Applied egg-rr 76.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{\frac{\beta \cdot \frac{\beta}{i}}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 8 \cdot 10^{+189}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 7: 81.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+154} \lor \neg \left(\beta \leq 6.5 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.4e+103)
   0.0625
   (if (or (<= beta 3e+154) (not (<= beta 6.5e+190)))
     (* (/ i beta) (/ i beta))
     0.0625)))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+103) {
		tmp = 0.0625;
	} else if ((beta <= 3e+154) || !(beta <= 6.5e+190)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 2.4d+103) then
        tmp = 0.0625d0
    else if ((beta <= 3d+154) .or. (.not. (beta <= 6.5d+190))) then
        tmp = (i / beta) * (i / beta)
    else
        tmp = 0.0625d0
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.4e+103) {
		tmp = 0.0625;
	} else if ((beta <= 3e+154) || !(beta <= 6.5e+190)) {
		tmp = (i / beta) * (i / beta);
	} else {
		tmp = 0.0625;
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.4e+103:
		tmp = 0.0625
	elif (beta <= 3e+154) or not (beta <= 6.5e+190):
		tmp = (i / beta) * (i / beta)
	else:
		tmp = 0.0625
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.4e+103)
		tmp = 0.0625;
	elseif ((beta <= 3e+154) || !(beta <= 6.5e+190))
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	else
		tmp = 0.0625;
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.4e+103)
		tmp = 0.0625;
	elseif ((beta <= 3e+154) || ~((beta <= 6.5e+190)))
		tmp = (i / beta) * (i / beta);
	else
		tmp = 0.0625;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.4e+103], 0.0625, If[Or[LessEqual[beta, 3e+154], N[Not[LessEqual[beta, 6.5e+190]], $MachinePrecision]], N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision], 0.0625]]

Derivation

1. Split input into 2 regimes

2. if beta < 2.3999999999999998e103 or 3.00000000000000026e154 < beta < 6.5000000000000001e190

   1. Initial program 23.0%

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

      1. associate-/l/ 21.1%

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

      2. associate-*l* 21.0%

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

      3. times-frac 26.7%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in i around inf 78.2%

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

   if 2.3999999999999998e103 < beta < 3.00000000000000026e154 or 6.5000000000000001e190 < beta

   1. Initial program 4.0%

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

      1. times-frac 27.1%

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

      2. +-commutative 27.1%

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

      3. +-commutative 27.1%

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

      4. *-commutative 27.1%

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

      5. fma-def 27.1%

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

      6. +-commutative 27.1%

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

      7. +-commutative 27.1%

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

      8. *-commutative 27.1%

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

      9. fma-udef 27.1%

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

      10. +-commutative 27.1%

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

      11. *-commutative 27.1%

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

      12. fma-def 27.1%

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

   3. Applied egg-rr 27.1%

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

      1. *-commutative 27.1%

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

      2. +-commutative 27.1%

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

      3. +-commutative 27.1%

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

      4. *-commutative 27.1%

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

      5. +-commutative 27.1%

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

      6. associate-/l* 27.1%

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

      7. +-commutative 27.1%

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

      8. +-commutative 27.1%

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

      9. +-commutative 27.1%

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

   5. Simplified 27.1%

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

   6. Taylor expanded in alpha around 0 29.2%

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

   7. Taylor expanded in beta around inf 37.6%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 37.6%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 37.6%

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

      3. times-frac 69.5%

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

   9. Simplified 69.5%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3 \cdot 10^{+154} \lor \neg \left(\beta \leq 6.5 \cdot 10^{+190}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 8: 81.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 8.8e+102)
   0.0625
   (if (<= beta 6.4e+153)
     (/ (* i i) (* beta beta))
     (if (<= beta 3.1e+191) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 6.4e+153) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 3.1e+191) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 8.8d+102) then
        tmp = 0.0625d0
    else if (beta <= 6.4d+153) then
        tmp = (i * i) / (beta * beta)
    else if (beta <= 3.1d+191) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 8.8e+102) {
		tmp = 0.0625;
	} else if (beta <= 6.4e+153) {
		tmp = (i * i) / (beta * beta);
	} else if (beta <= 3.1e+191) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 8.8e+102:
		tmp = 0.0625
	elif beta <= 6.4e+153:
		tmp = (i * i) / (beta * beta)
	elif beta <= 3.1e+191:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 6.4e+153)
		tmp = Float64(Float64(i * i) / Float64(beta * beta));
	elseif (beta <= 3.1e+191)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 8.8e+102)
		tmp = 0.0625;
	elseif (beta <= 6.4e+153)
		tmp = (i * i) / (beta * beta);
	elseif (beta <= 3.1e+191)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 8.8e+102], 0.0625, If[LessEqual[beta, 6.4e+153], N[(N[(i * i), $MachinePrecision] / N[(beta * beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.1e+191], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 8.8000000000000003e102 or 6.4000000000000003e153 < beta < 3.09999999999999999e191

   1. Initial program 23.0%

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

      1. associate-/l/ 21.1%

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

      2. associate-*l* 21.0%

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

      3. times-frac 26.7%

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

   3. Simplified 43.5%

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

   4. Taylor expanded in i around inf 78.2%

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

   if 8.8000000000000003e102 < beta < 6.4000000000000003e153

   1. Initial program 18.8%

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

      1. associate-/l/ 1.0%

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

      2. associate-*l* 1.0%

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

      3. times-frac 18.9%

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

   3. Simplified 54.3%

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

   4. Taylor expanded in beta around inf 53.3%

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

      1. *-commutative 53.3%

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

      2. associate-/l* 53.6%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 53.6%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 53.6%

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

   6. Simplified 53.6%

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

   7. Taylor expanded in alpha around 0 53.5%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 53.5%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 53.5%

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

   9. Simplified 53.5%

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

   if 3.09999999999999999e191 < 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. times-frac 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. fma-def 19.9%

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

      6. +-commutative 19.9%

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

      7. +-commutative 19.9%

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

      8. *-commutative 19.9%

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

      9. fma-udef 19.9%

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

      10. +-commutative 19.9%

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

      11. *-commutative 19.9%

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

      12. fma-def 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. *-commutative 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. +-commutative 19.9%

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

      6. associate-/l* 19.9%

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

      7. +-commutative 19.9%

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

      8. +-commutative 19.9%

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

      9. +-commutative 19.9%

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

   5. Simplified 19.9%

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

   6. Taylor expanded in alpha around 0 22.4%

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

   7. Taylor expanded in beta around inf 33.3%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 33.3%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 33.3%

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

      3. times-frac 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 8.8 \cdot 10^{+102}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{i \cdot i}{\beta \cdot \beta}\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+191}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9: 81.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+193}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 2.5e+103)
   0.0625
   (if (<= beta 2.8e+153)
     (* (+ i alpha) (/ (/ i beta) beta))
     (if (<= beta 1.6e+193) 0.0625 (* (/ i beta) (/ i beta))))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.8e+153) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 1.6e+193) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 2.5d+103) then
        tmp = 0.0625d0
    else if (beta <= 2.8d+153) then
        tmp = (i + alpha) * ((i / beta) / beta)
    else if (beta <= 1.6d+193) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (i / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 2.5e+103) {
		tmp = 0.0625;
	} else if (beta <= 2.8e+153) {
		tmp = (i + alpha) * ((i / beta) / beta);
	} else if (beta <= 1.6e+193) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (i / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 2.5e+103:
		tmp = 0.0625
	elif beta <= 2.8e+153:
		tmp = (i + alpha) * ((i / beta) / beta)
	elif beta <= 1.6e+193:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (i / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 2.5e+103)
		tmp = 0.0625;
	elseif (beta <= 2.8e+153)
		tmp = Float64(Float64(i + alpha) * Float64(Float64(i / beta) / beta));
	elseif (beta <= 1.6e+193)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(i / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 2.5e+103)
		tmp = 0.0625;
	elseif (beta <= 2.8e+153)
		tmp = (i + alpha) * ((i / beta) / beta);
	elseif (beta <= 1.6e+193)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (i / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 2.5e+103], 0.0625, If[LessEqual[beta, 2.8e+153], N[(N[(i + alpha), $MachinePrecision] * N[(N[(i / beta), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.6e+193], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(i / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if beta < 2.5e103 or 2.79999999999999985e153 < beta < 1.60000000000000007e193

   1. Initial program 22.9%

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

      1. associate-/l/ 21.0%

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

      2. associate-*l* 20.9%

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

      3. times-frac 26.5%

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

   3. Simplified 43.3%

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

   4. Taylor expanded in i around inf 77.8%

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

   if 2.5e103 < beta < 2.79999999999999985e153

   1. Initial program 20.7%

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

      1. times-frac 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. fma-def 59.4%

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

      6. +-commutative 59.4%

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

      7. +-commutative 59.4%

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

      8. *-commutative 59.4%

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

      9. fma-udef 59.4%

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

      10. +-commutative 59.4%

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

      11. *-commutative 59.4%

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

      12. fma-def 59.4%

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

   3. Applied egg-rr 59.4%

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

      1. *-commutative 59.4%

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

      2. +-commutative 59.4%

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

      3. +-commutative 59.4%

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

      4. *-commutative 59.4%

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

      5. +-commutative 59.4%

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

      6. associate-/l* 59.4%

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

      7. +-commutative 59.4%

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

      8. +-commutative 59.4%

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

      9. +-commutative 59.4%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in beta around inf 58.4%

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

      1. *-commutative 58.4%

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

      2. associate-*r/ 58.6%

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

      3. unpow2 58.6%

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

      4. associate-/r* 58.8%

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

   8. Simplified 58.8%

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

   if 1.60000000000000007e193 < 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. times-frac 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. fma-def 19.9%

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

      6. +-commutative 19.9%

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

      7. +-commutative 19.9%

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

      8. *-commutative 19.9%

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

      9. fma-udef 19.9%

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

      10. +-commutative 19.9%

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

      11. *-commutative 19.9%

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

      12. fma-def 19.9%

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

   3. Applied egg-rr 19.9%

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

      1. *-commutative 19.9%

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

      2. +-commutative 19.9%

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

      3. +-commutative 19.9%

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

      4. *-commutative 19.9%

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

      5. +-commutative 19.9%

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

      6. associate-/l* 19.9%

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

      7. +-commutative 19.9%

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

      8. +-commutative 19.9%

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

      9. +-commutative 19.9%

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

   5. Simplified 19.9%

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

   6. Taylor expanded in alpha around 0 22.4%

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

   7. Taylor expanded in beta around inf 33.3%

      \[\leadsto \color{blue}{\frac{{i}^{2}}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. unpow2 33.3%

         \[\leadsto \frac{\color{blue}{i \cdot i}}{{\beta}^{2}} \]

      2. unpow2 33.3%

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

      3. times-frac 73.8%

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

   9. Simplified 73.8%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+103}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+153}:\\ \;\;\;\;\left(i + \alpha\right) \cdot \frac{\frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 1.6 \cdot 10^{+193}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10: 75.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} [alpha, beta] = \mathsf{sort}([alpha, beta])\\ \\ \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \end{array} \]

FPCore

NOTE: alpha and beta should be sorted in increasing order before calling this function.
(FPCore (alpha beta i)
 :precision binary64
 (if (<= beta 9.4e+204) 0.0625 (* (/ i beta) (/ alpha beta))))

C

assert(alpha < beta);
double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Fortran

NOTE: alpha and beta 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 <= 9.4d+204) then
        tmp = 0.0625d0
    else
        tmp = (i / beta) * (alpha / beta)
    end if
    code = tmp
end function

Java

assert alpha < beta;
public static double code(double alpha, double beta, double i) {
	double tmp;
	if (beta <= 9.4e+204) {
		tmp = 0.0625;
	} else {
		tmp = (i / beta) * (alpha / beta);
	}
	return tmp;
}

Python

[alpha, beta] = sort([alpha, beta])
def code(alpha, beta, i):
	tmp = 0
	if beta <= 9.4e+204:
		tmp = 0.0625
	else:
		tmp = (i / beta) * (alpha / beta)
	return tmp

Julia

alpha, beta = sort([alpha, beta])
function code(alpha, beta, i)
	tmp = 0.0
	if (beta <= 9.4e+204)
		tmp = 0.0625;
	else
		tmp = Float64(Float64(i / beta) * Float64(alpha / beta));
	end
	return tmp
end

MATLAB

alpha, beta = num2cell(sort([alpha, beta])){:}
function tmp_2 = code(alpha, beta, i)
	tmp = 0.0;
	if (beta <= 9.4e+204)
		tmp = 0.0625;
	else
		tmp = (i / beta) * (alpha / beta);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: alpha and beta should be sorted in increasing order before calling this function.
code[alpha_, beta_, i_] := If[LessEqual[beta, 9.4e+204], 0.0625, N[(N[(i / beta), $MachinePrecision] * N[(alpha / beta), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if beta < 9.4000000000000003e204

   1. Initial program 21.9%

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

      1. associate-/l/ 19.3%

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

      2. associate-*l* 19.2%

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

      3. times-frac 25.2%

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

   3. Simplified 43.2%

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

   4. Taylor expanded in i around inf 74.7%

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

   if 9.4000000000000003e204 < 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. associate-/l/ 0.0%

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

      2. associate-*l* 0.0%

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

      3. times-frac 0.0%

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

   3. Simplified 19.1%

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

   4. Taylor expanded in beta around inf 36.1%

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

      1. *-commutative 36.1%

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

      2. associate-/l* 37.7%

         \[\leadsto \color{blue}{\frac{i + \alpha}{\frac{{\beta}^{2}}{i}}} \]

      3. +-commutative 37.7%

         \[\leadsto \frac{\color{blue}{\alpha + i}}{\frac{{\beta}^{2}}{i}} \]

      4. unpow2 37.7%

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

   6. Simplified 37.7%

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

   7. Taylor expanded in alpha around inf 37.6%

      \[\leadsto \color{blue}{\frac{i \cdot \alpha}{{\beta}^{2}}} \]
   8. Step-by-step derivation

      1. associate-/l* 37.7%

         \[\leadsto \color{blue}{\frac{i}{\frac{{\beta}^{2}}{\alpha}}} \]

      2. unpow2 37.7%

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

   9. Simplified 37.7%

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

   10. Taylor expanded in i around 0 37.6%

       \[\leadsto \color{blue}{\frac{i \cdot \alpha}{{\beta}^{2}}} \]
   11. Step-by-step derivation

       1. *-commutative 37.6%

          \[\leadsto \frac{\color{blue}{\alpha \cdot i}}{{\beta}^{2}} \]

       2. unpow2 37.6%

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

       3. times-frac 40.9%

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

   12. Simplified 40.9%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 9.4 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 11: 70.6% accurate, 53.0× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: alpha and beta 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;
public static double code(double alpha, double beta, double i) {
	return 0.0625;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.1%

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

   1. associate-/l/ 16.9%

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

   2. associate-*l* 16.8%

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

   3. times-frac 22.1%

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

3. Simplified 40.2%

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

4. Taylor expanded in i around inf 67.2%

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

5. Final simplification 67.2%

   \[\leadsto 0.0625 \]

Reproduce

herbie shell --seed 2023199 
(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)))