Octave 3.8, jcobi/4

Average Error: 54.2 → 10.4

Time: 29.7s

Precision: binary64

Cost: 1364

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

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

Math

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

↓

\[\begin{array}{l} t_0 := 0.25 \cdot \left(0.5 \cdot \frac{i}{\beta + i \cdot 2}\right)\\ \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.16 \cdot 10^{+165}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+208}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\alpha + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* 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)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (* 0.25 (* 0.5 (/ i (+ beta (* i 2.0)))))))
   (if (<= beta 3.5e+129)
     t_0
     (if (<= beta 1.2e+155)
       (/ (/ i beta) (/ beta (+ i alpha)))
       (if (<= beta 1.16e+165)
         t_0
         (if (<= beta 9.2e+192)
           (/ (/ i beta) (/ beta i))
           (if (<= beta 6.3e+208)
             (* 0.25 (* 0.5 (/ i (+ alpha (* i 2.0)))))
             (* (/ i beta) (/ (+ i alpha) beta)))))))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = 0.25 * (0.5 * (i / (beta + (i * 2.0))));
	double tmp;
	if (beta <= 3.5e+129) {
		tmp = t_0;
	} else if (beta <= 1.2e+155) {
		tmp = (i / beta) / (beta / (i + alpha));
	} else if (beta <= 1.16e+165) {
		tmp = t_0;
	} else if (beta <= 9.2e+192) {
		tmp = (i / beta) / (beta / i);
	} else if (beta <= 6.3e+208) {
		tmp = 0.25 * (0.5 * (i / (alpha + (i * 2.0))));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(alpha, beta, i)
    real(8), intent (in) :: alpha
    real(8), intent (in) :: beta
    real(8), intent (in) :: i
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.25d0 * (0.5d0 * (i / (beta + (i * 2.0d0))))
    if (beta <= 3.5d+129) then
        tmp = t_0
    else if (beta <= 1.2d+155) then
        tmp = (i / beta) / (beta / (i + alpha))
    else if (beta <= 1.16d+165) then
        tmp = t_0
    else if (beta <= 9.2d+192) then
        tmp = (i / beta) / (beta / i)
    else if (beta <= 6.3d+208) then
        tmp = 0.25d0 * (0.5d0 * (i / (alpha + (i * 2.0d0))))
    else
        tmp = (i / beta) * ((i + alpha) / beta)
    end if
    code = tmp
end function

Java

public static double code(double alpha, double beta, double i) {
	return (((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);
}

↓

public static double code(double alpha, double beta, double i) {
	double t_0 = 0.25 * (0.5 * (i / (beta + (i * 2.0))));
	double tmp;
	if (beta <= 3.5e+129) {
		tmp = t_0;
	} else if (beta <= 1.2e+155) {
		tmp = (i / beta) / (beta / (i + alpha));
	} else if (beta <= 1.16e+165) {
		tmp = t_0;
	} else if (beta <= 9.2e+192) {
		tmp = (i / beta) / (beta / i);
	} else if (beta <= 6.3e+208) {
		tmp = 0.25 * (0.5 * (i / (alpha + (i * 2.0))));
	} else {
		tmp = (i / beta) * ((i + alpha) / beta);
	}
	return tmp;
}

Python

def code(alpha, beta, i):
	return (((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)

↓

def code(alpha, beta, i):
	t_0 = 0.25 * (0.5 * (i / (beta + (i * 2.0))))
	tmp = 0
	if beta <= 3.5e+129:
		tmp = t_0
	elif beta <= 1.2e+155:
		tmp = (i / beta) / (beta / (i + alpha))
	elif beta <= 1.16e+165:
		tmp = t_0
	elif beta <= 9.2e+192:
		tmp = (i / beta) / (beta / i)
	elif beta <= 6.3e+208:
		tmp = 0.25 * (0.5 * (i / (alpha + (i * 2.0))))
	else:
		tmp = (i / beta) * ((i + alpha) / beta)
	return tmp

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(0.25 * Float64(0.5 * Float64(i / Float64(beta + Float64(i * 2.0)))))
	tmp = 0.0
	if (beta <= 3.5e+129)
		tmp = t_0;
	elseif (beta <= 1.2e+155)
		tmp = Float64(Float64(i / beta) / Float64(beta / Float64(i + alpha)));
	elseif (beta <= 1.16e+165)
		tmp = t_0;
	elseif (beta <= 9.2e+192)
		tmp = Float64(Float64(i / beta) / Float64(beta / i));
	elseif (beta <= 6.3e+208)
		tmp = Float64(0.25 * Float64(0.5 * Float64(i / Float64(alpha + Float64(i * 2.0)))));
	else
		tmp = Float64(Float64(i / beta) * Float64(Float64(i + alpha) / beta));
	end
	return tmp
end

MATLAB

function tmp = code(alpha, beta, i)
	tmp = (((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);
end

↓

function tmp_2 = code(alpha, beta, i)
	t_0 = 0.25 * (0.5 * (i / (beta + (i * 2.0))));
	tmp = 0.0;
	if (beta <= 3.5e+129)
		tmp = t_0;
	elseif (beta <= 1.2e+155)
		tmp = (i / beta) / (beta / (i + alpha));
	elseif (beta <= 1.16e+165)
		tmp = t_0;
	elseif (beta <= 9.2e+192)
		tmp = (i / beta) / (beta / i);
	elseif (beta <= 6.3e+208)
		tmp = 0.25 * (0.5 * (i / (alpha + (i * 2.0))));
	else
		tmp = (i / beta) * ((i + alpha) / beta);
	end
	tmp_2 = tmp;
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(0.25 * N[(0.5 * N[(i / N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 3.5e+129], t$95$0, If[LessEqual[beta, 1.2e+155], N[(N[(i / beta), $MachinePrecision] / N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.16e+165], t$95$0, If[LessEqual[beta, 9.2e+192], N[(N[(i / beta), $MachinePrecision] / N[(beta / i), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 6.3e+208], N[(0.25 * N[(0.5 * N[(i / N[(alpha + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if beta < 3.4999999999999998e129 or 1.2000000000000001e155 < beta < 1.1599999999999999e165

   1. Initial program 49.4

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

   2. Simplified 32.7

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

      [Start] 49.4	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 50.7	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 32.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 5.4

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

   4. Taylor expanded in i around inf 5.7

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{0.5}\right) \cdot 0.25 \]

   5. Taylor expanded in alpha around 0 5.7

      \[\leadsto \left(\frac{i}{\color{blue}{\beta + 2 \cdot i}} \cdot 0.5\right) \cdot 0.25 \]

   6. Simplified 5.7

      \[\leadsto \left(\frac{i}{\color{blue}{\beta + i \cdot 2}} \cdot 0.5\right) \cdot 0.25 \]
      Proof

      [Start] 5.7	\[ \left(\frac{i}{\beta + 2 \cdot i} \cdot 0.5\right) \cdot 0.25 \]
      *-commutative [<=] 5.7	\[ \left(\frac{i}{\beta + \color{blue}{i \cdot 2}} \cdot 0.5\right) \cdot 0.25 \]

   if 3.4999999999999998e129 < beta < 1.2000000000000001e155

   1. Initial program 62.8

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

   2. Simplified 63.9

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

      [Start] 62.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 63.9	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 63.9	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 63.9	\[ \frac{i \cdot \left(\color{blue}{\left(i + \left(\alpha + \beta\right)\right)} \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 63.9	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 63.9	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 34.8

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

   4. Simplified 33.4

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

      [Start] 34.8	\[ \frac{\left(i + \alpha\right) \cdot i}{{\beta}^{2}} \]
      *-commutative [<=] 34.8	\[ \frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{{\beta}^{2}} \]
      associate-/l* [=>] 33.4	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 33.4	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 33.3

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

   6. Applied egg-rr 32.8

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

   if 1.1599999999999999e165 < beta < 9.1999999999999997e192

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 64.0	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{i \cdot \left(\color{blue}{\left(i + \left(\alpha + \beta\right)\right)} \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 59.2

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

   4. Simplified 58.2

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

      [Start] 59.2	\[ \frac{\left(i + \alpha\right) \cdot i}{{\beta}^{2}} \]
      *-commutative [<=] 59.2	\[ \frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{{\beta}^{2}} \]
      associate-/l* [=>] 58.2	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 58.2	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 27.0

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

   6. Applied egg-rr 26.3

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

   7. Taylor expanded in i around inf 31.5

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

   if 9.1999999999999997e192 < beta < 6.3000000000000004e208

   1. Initial program 64.0

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

   2. Simplified 56.5

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 56.5	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 39.8

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

   4. Taylor expanded in i around inf 40.4

      \[\leadsto \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \color{blue}{0.5}\right) \cdot 0.25 \]

   5. Taylor expanded in beta around 0 40.5

      \[\leadsto \left(\frac{i}{\color{blue}{\alpha + 2 \cdot i}} \cdot 0.5\right) \cdot 0.25 \]

   6. Simplified 40.5

      \[\leadsto \left(\frac{i}{\color{blue}{\alpha + i \cdot 2}} \cdot 0.5\right) \cdot 0.25 \]
      Proof

      [Start] 40.5	\[ \left(\frac{i}{\alpha + 2 \cdot i} \cdot 0.5\right) \cdot 0.25 \]
      *-commutative [<=] 40.5	\[ \left(\frac{i}{\alpha + \color{blue}{i \cdot 2}} \cdot 0.5\right) \cdot 0.25 \]

   if 6.3000000000000004e208 < beta

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 64.0	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{i \cdot \left(\color{blue}{\left(i + \left(\alpha + \beta\right)\right)} \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 64.0	\[ \frac{i \cdot \left(\left(i + \left(\alpha + \beta\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 43.0

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

   4. Simplified 41.7

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

      [Start] 43.0	\[ \frac{\left(i + \alpha\right) \cdot i}{{\beta}^{2}} \]
      *-commutative [<=] 43.0	\[ \frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{{\beta}^{2}} \]
      associate-/l* [=>] 41.7	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 41.7	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 11.3

      \[\leadsto \color{blue}{\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 3.5 \cdot 10^{+129}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\beta + i \cdot 2}\right)\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.16 \cdot 10^{+165}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\beta + i \cdot 2}\right)\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+208}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\alpha + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.9
Cost	28808

\[\begin{array}{l} t_0 := i + \left(\beta + \alpha\right)\\ t_1 := \beta + i \cdot 2\\ t_2 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_3 := \frac{i}{t_2} \cdot \frac{t_0}{t_2}\\ \mathbf{if}\;\beta \leq 2.65 \cdot 10^{+129}:\\ \;\;\;\;t_3 \cdot 0.25\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;t_3 \cdot \frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{-1 + \left({\left(\beta + \alpha\right)}^{2} + 4 \cdot \left(i \cdot \left(\beta + \left(i + \alpha\right)\right)\right)\right)}\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t_1 + \left(\alpha + -1\right)} \cdot \left(\alpha \cdot \frac{\alpha}{\beta} + \left(i + \alpha\right)\right)}{t_1 + \left(\alpha + 1\right)}\\ \end{array} \]

Alternative 2
Error	8.9
Cost	22152

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := i + \left(\beta + \alpha\right)\\ t_3 := \left(\beta + \alpha\right) + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{t_2}{t_1}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right) \cdot \left(i \cdot \left(t_2 \cdot {t_1}^{-2}\right)\right)}{-1 + t_3 \cdot t_3}\\ \mathbf{elif}\;\beta \leq 1.7 \cdot 10^{+161}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{t_0 + \left(\alpha + -1\right)} \cdot \left(\alpha \cdot \frac{\alpha}{\beta} + \left(i + \alpha\right)\right)}{t_0 + \left(\alpha + 1\right)}\\ \end{array} \]

Alternative 3
Error	8.8
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot 0.25\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+155} \lor \neg \left(\beta \leq 3 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{\frac{i}{t_1 + \left(\alpha + -1\right)} \cdot \left(\alpha \cdot \frac{\alpha}{\beta} + \left(i + \alpha\right)\right)}{t_1 + \left(\alpha + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 4
Error	9.0
Cost	7236

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 3 \cdot 10^{+129}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot 0.5\right)\\ \mathbf{elif}\;\beta \leq 2.8 \cdot 10^{+155} \lor \neg \left(\beta \leq 1.7 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{\frac{i}{t_0 + \left(\alpha + -1\right)} \cdot \left(\alpha \cdot \frac{\alpha}{\beta} + \left(i + \alpha\right)\right)}{t_0 + \left(\alpha + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 5
Error	8.9
Cost	2381

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+129}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{t_0}\right)\\ \mathbf{elif}\;\beta \leq 2.5 \cdot 10^{+155} \lor \neg \left(\beta \leq 1.7 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{\frac{i}{t_0 + \left(\alpha + -1\right)} \cdot \left(\alpha \cdot \frac{\alpha}{\beta} + \left(i + \alpha\right)\right)}{t_0 + \left(\alpha + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 6
Error	10.4
Cost	1364

\[\begin{array}{l} t_0 := 0.25 \cdot \left(0.5 \cdot \frac{i}{\alpha + i \cdot 2}\right)\\ \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 5.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 7.4 \cdot 10^{+163}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+208}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 7
Error	10.3
Cost	1236

\[\begin{array}{l} t_0 := \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \mathbf{if}\;\beta \leq 3.2 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\beta \leq 2.35 \cdot 10^{+165}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.6 \cdot 10^{+208}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	10.3
Cost	1236

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 1.85 \cdot 10^{+164}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+208}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 9
Error	10.3
Cost	1236

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+155}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 3.7 \cdot 10^{+163}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 6.6 \cdot 10^{+208}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 10
Error	11.4
Cost	1110

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+155} \lor \neg \left(\beta \leq 3.2 \cdot 10^{+163}\right) \land \left(\beta \leq 9.2 \cdot 10^{+192} \lor \neg \left(\beta \leq 6.3 \cdot 10^{+208}\right)\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 11
Error	11.3
Cost	1110

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.8 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.9 \cdot 10^{+155} \lor \neg \left(\beta \leq 7 \cdot 10^{+163} \lor \neg \left(\beta \leq 9.2 \cdot 10^{+192}\right) \land \beta \leq 6.3 \cdot 10^{+208}\right):\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 12
Error	11.3
Cost	1109

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.3 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+155}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 1.2 \cdot 10^{+166}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192} \lor \neg \left(\beta \leq 1.6 \cdot 10^{+209}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 13
Error	11.4
Cost	1108

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.6 \cdot 10^{+129}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.55 \cdot 10^{+155}:\\ \;\;\;\;\frac{i}{\beta \cdot \frac{\beta}{i}}\\ \mathbf{elif}\;\beta \leq 4.4 \cdot 10^{+163}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 9.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{i \cdot \frac{i}{\beta}}{\beta}\\ \mathbf{elif}\;\beta \leq 6.3 \cdot 10^{+208}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 14
Error	16.3
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+234}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \frac{\frac{i}{\beta}}{\beta}\\ \end{array} \]

Alternative 15
Error	15.7
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4 \cdot 10^{+223}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 16
Error	18.8
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023012 
(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)))