Octave 3.8, jcobi/4

Percentage Accurate: 15.3% → 84.5%

Time: 29.7s

Precision: binary64

Cost: 35144

• Unsound rule application detected in e-graph. Results may not be sound.

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

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

Math

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

↓

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 7 \cdot 10^{+132}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(t_3 + \beta \cdot \alpha\right)}{i \cdot i} - \frac{\mathsf{fma}\left(4, t_3 + -1, t_3 \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{i}{\left(t_1 + 1\right) \cdot \frac{t_1}{t_2}} \cdot \frac{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right)}{t_1}}{-1 + t_1}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

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 (* 0.125 (/ beta i)))
        (t_1 (fma i 2.0 (+ beta alpha)))
        (t_2 (+ (+ beta alpha) i))
        (t_3 (pow (+ beta alpha) 2.0)))
   (if (<= beta 7e+132)
     (+
      0.0625
      (-
       (/ (* 0.0625 (+ t_3 (* beta alpha))) (* i i))
       (/ (* (fma 4.0 (+ t_3 -1.0) (* t_3 20.0)) 0.00390625) (* i i))))
     (if (<= beta 1.8e+156)
       (/
        (*
         (/ i (* (+ t_1 1.0) (/ t_1 t_2)))
         (/ (fma i t_2 (* beta alpha)) t_1))
        (+ -1.0 t_1))
       (if (<= beta 2.9e+202)
         (- (+ 0.0625 t_0) t_0)
         (* (/ (+ alpha i) beta) (/ (/ 1.0 beta) (/ 1.0 i))))))))

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 = 0.125 * (beta / i);
	double t_1 = fma(i, 2.0, (beta + alpha));
	double t_2 = (beta + alpha) + i;
	double t_3 = pow((beta + alpha), 2.0);
	double tmp;
	if (beta <= 7e+132) {
		tmp = 0.0625 + (((0.0625 * (t_3 + (beta * alpha))) / (i * i)) - ((fma(4.0, (t_3 + -1.0), (t_3 * 20.0)) * 0.00390625) / (i * i)));
	} else if (beta <= 1.8e+156) {
		tmp = ((i / ((t_1 + 1.0) * (t_1 / t_2))) * (fma(i, t_2, (beta * alpha)) / t_1)) / (-1.0 + t_1);
	} else if (beta <= 2.9e+202) {
		tmp = (0.0625 + t_0) - t_0;
	} else {
		tmp = ((alpha + i) / beta) * ((1.0 / beta) / (1.0 / i));
	}
	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(0.125 * Float64(beta / i))
	t_1 = fma(i, 2.0, Float64(beta + alpha))
	t_2 = Float64(Float64(beta + alpha) + i)
	t_3 = Float64(beta + alpha) ^ 2.0
	tmp = 0.0
	if (beta <= 7e+132)
		tmp = Float64(0.0625 + Float64(Float64(Float64(0.0625 * Float64(t_3 + Float64(beta * alpha))) / Float64(i * i)) - Float64(Float64(fma(4.0, Float64(t_3 + -1.0), Float64(t_3 * 20.0)) * 0.00390625) / Float64(i * i))));
	elseif (beta <= 1.8e+156)
		tmp = Float64(Float64(Float64(i / Float64(Float64(t_1 + 1.0) * Float64(t_1 / t_2))) * Float64(fma(i, t_2, Float64(beta * alpha)) / t_1)) / Float64(-1.0 + t_1));
	elseif (beta <= 2.9e+202)
		tmp = Float64(Float64(0.0625 + t_0) - t_0);
	else
		tmp = Float64(Float64(Float64(alpha + i) / beta) * Float64(Float64(1.0 / beta) / Float64(1.0 / i)));
	end
	return 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[(0.125 * N[(beta / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(beta + alpha), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(beta + alpha), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[beta, 7e+132], N[(0.0625 + N[(N[(N[(0.0625 * N[(t$95$3 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(4.0 * N[(t$95$3 + -1.0), $MachinePrecision] + N[(t$95$3 * 20.0), $MachinePrecision]), $MachinePrecision] * 0.00390625), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 1.8e+156], N[(N[(N[(i / N[(N[(t$95$1 + 1.0), $MachinePrecision] * N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i * t$95$2 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.9e+202], N[(N[(0.0625 + t$95$0), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision] * N[(N[(1.0 / beta), $MachinePrecision] / N[(1.0 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 7.00000000000000041e132

   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 i around -inf 18.5%

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

   3. Simplified 18.5%

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

      [Start] 18.5	\[ \frac{\frac{-1 \cdot \left(\left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}\right) + \left(\left(2 \cdot \beta + 2 \cdot \alpha\right) \cdot {i}^{3} + {i}^{4}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 18.5	\[ \frac{\frac{\color{blue}{\left(\left(2 \cdot \beta + 2 \cdot \alpha\right) \cdot {i}^{3} + {i}^{4}\right) + -1 \cdot \left(\left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      mul-1-neg [=>] 18.5	\[ \frac{\frac{\left(\left(2 \cdot \beta + 2 \cdot \alpha\right) \cdot {i}^{3} + {i}^{4}\right) + \color{blue}{\left(-\left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unsub-neg [=>] 18.5	\[ \frac{\frac{\color{blue}{\left(\left(2 \cdot \beta + 2 \cdot \alpha\right) \cdot {i}^{3} + {i}^{4}\right) - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 18.5	\[ \frac{\frac{\left(\color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)} \cdot {i}^{3} + {i}^{4}\right) - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [<=] 18.5	\[ \frac{\frac{\left(\color{blue}{{i}^{3} \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right)} + {i}^{4}\right) - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 18.5	\[ \frac{\frac{\color{blue}{\mathsf{fma}\left({i}^{3}, 2 \cdot \alpha + 2 \cdot \beta, {i}^{4}\right)} - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, \color{blue}{2 \cdot \beta + 2 \cdot \alpha}, {i}^{4}\right) - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out [=>] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, \color{blue}{2 \cdot \left(\beta + \alpha\right)}, {i}^{4}\right) - \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right) \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, 2 \cdot \left(\beta + \alpha\right), {i}^{4}\right) - \color{blue}{{i}^{2} \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, 2 \cdot \left(\beta + \alpha\right), {i}^{4}\right) - \color{blue}{\left(i \cdot i\right)} \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + -1 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      mul-1-neg [=>] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, 2 \cdot \left(\beta + \alpha\right), {i}^{4}\right) - \left(i \cdot i\right) \cdot \left(-1 \cdot \left(\beta \cdot \alpha\right) + \color{blue}{\left(-{\left(\beta + \alpha\right)}^{2}\right)}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unsub-neg [=>] 18.5	\[ \frac{\frac{\mathsf{fma}\left({i}^{3}, 2 \cdot \left(\beta + \alpha\right), {i}^{4}\right) - \left(i \cdot i\right) \cdot \color{blue}{\left(-1 \cdot \left(\beta \cdot \alpha\right) - {\left(\beta + \alpha\right)}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Taylor expanded in i around inf 77.2%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{{\left(\beta + \alpha\right)}^{2} - -1 \cdot \left(\beta \cdot \alpha\right)}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}} \]

   5. Simplified 77.2%

      \[\leadsto \color{blue}{0.0625 + \left(\frac{\left({\left(\beta + \alpha\right)}^{2} - \left(-\beta\right) \cdot \alpha\right) \cdot 0.0625}{i \cdot i} - \frac{\mathsf{fma}\left(4, {\left(\beta + \alpha\right)}^{2} + -1, {\left(\beta + \alpha\right)}^{2} \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)} \]
      Step-by-step derivation

      [Start] 77.2	\[ \left(0.0625 + 0.0625 \cdot \frac{{\left(\beta + \alpha\right)}^{2} - -1 \cdot \left(\beta \cdot \alpha\right)}{{i}^{2}}\right) - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}} \]
      associate--l+ [=>] 77.2	\[ \color{blue}{0.0625 + \left(0.0625 \cdot \frac{{\left(\beta + \alpha\right)}^{2} - -1 \cdot \left(\beta \cdot \alpha\right)}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right)} \]
      associate-*r/ [=>] 77.2	\[ 0.0625 + \left(\color{blue}{\frac{0.0625 \cdot \left({\left(\beta + \alpha\right)}^{2} - -1 \cdot \left(\beta \cdot \alpha\right)\right)}{{i}^{2}}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right) \]
      *-commutative [=>] 77.2	\[ 0.0625 + \left(\frac{\color{blue}{\left({\left(\beta + \alpha\right)}^{2} - -1 \cdot \left(\beta \cdot \alpha\right)\right) \cdot 0.0625}}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right) \]
      associate-*r* [=>] 77.2	\[ 0.0625 + \left(\frac{\left({\left(\beta + \alpha\right)}^{2} - \color{blue}{\left(-1 \cdot \beta\right) \cdot \alpha}\right) \cdot 0.0625}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right) \]
      mul-1-neg [=>] 77.2	\[ 0.0625 + \left(\frac{\left({\left(\beta + \alpha\right)}^{2} - \color{blue}{\left(-\beta\right)} \cdot \alpha\right) \cdot 0.0625}{{i}^{2}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right) \]
      unpow2 [=>] 77.2	\[ 0.0625 + \left(\frac{\left({\left(\beta + \alpha\right)}^{2} - \left(-\beta\right) \cdot \alpha\right) \cdot 0.0625}{\color{blue}{i \cdot i}} - 0.00390625 \cdot \frac{4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)}{{i}^{2}}\right) \]
      associate-*r/ [=>] 77.2	\[ 0.0625 + \left(\frac{\left({\left(\beta + \alpha\right)}^{2} - \left(-\beta\right) \cdot \alpha\right) \cdot 0.0625}{i \cdot i} - \color{blue}{\frac{0.00390625 \cdot \left(4 \cdot {\left(\beta + \alpha\right)}^{2} + \left(4 \cdot \left({\left(\beta + \alpha\right)}^{2} - 1\right) + 16 \cdot {\left(\beta + \alpha\right)}^{2}\right)\right)}{{i}^{2}}}\right) \]

   if 7.00000000000000041e132 < beta < 1.79999999999999989e156

   1. Initial program 2.2%

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

   2. Applied egg-rr 99.2%

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

      [Start] 2.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 84.1	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 84.1	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 99.2	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Simplified 99.7%

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

      [Start] 99.2	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-*r/ [=>] 99.7	\[ \color{blue}{\frac{\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}} \]

   if 1.79999999999999989e156 < beta < 2.8999999999999999e202

   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. Simplified 0.7%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      associate-*l* [=>] 0.0	\[ \frac{\color{blue}{i \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \]

   3. Taylor expanded in i around inf 63.7%

      \[\leadsto \color{blue}{\left(0.0625 + 0.0625 \cdot \frac{2 \cdot \beta + 2 \cdot \alpha}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   4. Taylor expanded in beta around inf 63.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\left(2 \cdot \frac{\beta}{i}\right)}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   5. Simplified 63.4%

      \[\leadsto \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{\beta}{\frac{i}{2}}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      Step-by-step derivation

      [Start] 63.4	\[ \left(0.0625 + 0.0625 \cdot \left(2 \cdot \frac{\beta}{i}\right)\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      associate-*r/ [=>] 63.4	\[ \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{2 \cdot \beta}{i}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      *-commutative [=>] 63.4	\[ \left(0.0625 + 0.0625 \cdot \frac{\color{blue}{\beta \cdot 2}}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]
      associate-/l* [=>] 63.4	\[ \left(0.0625 + 0.0625 \cdot \color{blue}{\frac{\beta}{\frac{i}{2}}}\right) - 0.125 \cdot \frac{\beta + \alpha}{i} \]

   6. Taylor expanded in i around 0 63.4%

      \[\leadsto \color{blue}{\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta + \alpha}{i}} \]

   7. Taylor expanded in beta around inf 63.8%

      \[\leadsto \left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \color{blue}{\frac{\beta}{i}} \]

   if 2.8999999999999999e202 < 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. Simplified 0.0%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 34.8%

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

   4. Simplified 36.9%

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

      [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* [=>] 36.9	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 36.9	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 79.6%

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

      [Start] 36.9	\[ \frac{i}{\frac{\beta \cdot \beta}{i + \alpha}} \]
      clear-num [=>] 36.9	\[ \color{blue}{\frac{1}{\frac{\frac{\beta \cdot \beta}{i + \alpha}}{i}}} \]
      inv-pow [=>] 36.9	\[ \color{blue}{{\left(\frac{\frac{\beta \cdot \beta}{i + \alpha}}{i}\right)}^{-1}} \]
      associate-/l* [=>] 50.3	\[ {\left(\frac{\color{blue}{\frac{\beta}{\frac{i + \alpha}{\beta}}}}{i}\right)}^{-1} \]
      associate-/l/ [=>] 79.6	\[ {\color{blue}{\left(\frac{\beta}{i \cdot \frac{i + \alpha}{\beta}}\right)}}^{-1} \]

   6. Applied egg-rr 83.4%

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

      [Start] 79.6	\[ {\left(\frac{\beta}{i \cdot \frac{i + \alpha}{\beta}}\right)}^{-1} \]
      unpow-1 [=>] 79.6	\[ \color{blue}{\frac{1}{\frac{\beta}{i \cdot \frac{i + \alpha}{\beta}}}} \]
      div-inv [=>] 79.5	\[ \frac{1}{\color{blue}{\beta \cdot \frac{1}{i \cdot \frac{i + \alpha}{\beta}}}} \]
      associate-/r* [=>] 83.4	\[ \color{blue}{\frac{\frac{1}{\beta}}{\frac{1}{i \cdot \frac{i + \alpha}{\beta}}}} \]
      *-commutative [=>] 83.4	\[ \frac{\frac{1}{\beta}}{\frac{1}{\color{blue}{\frac{i + \alpha}{\beta} \cdot i}}} \]
      associate-/r* [=>] 83.4	\[ \frac{\frac{1}{\beta}}{\color{blue}{\frac{\frac{1}{\frac{i + \alpha}{\beta}}}{i}}} \]
      clear-num [<=] 83.4	\[ \frac{\frac{1}{\beta}}{\frac{\color{blue}{\frac{\beta}{i + \alpha}}}{i}} \]

   7. Applied egg-rr 83.8%

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

      [Start] 83.4	\[ \frac{\frac{1}{\beta}}{\frac{\frac{\beta}{i + \alpha}}{i}} \]
      *-un-lft-identity [=>] 83.4	\[ \frac{\color{blue}{1 \cdot \frac{1}{\beta}}}{\frac{\frac{\beta}{i + \alpha}}{i}} \]
      div-inv [=>] 83.3	\[ \frac{1 \cdot \frac{1}{\beta}}{\color{blue}{\frac{\beta}{i + \alpha} \cdot \frac{1}{i}}} \]
      times-frac [=>] 83.7	\[ \color{blue}{\frac{1}{\frac{\beta}{i + \alpha}} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}} \]
      clear-num [<=] 83.8	\[ \color{blue}{\frac{i + \alpha}{\beta}} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7 \cdot 10^{+132}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left({\left(\beta + \alpha\right)}^{2} + \beta \cdot \alpha\right)}{i \cdot i} - \frac{\mathsf{fma}\left(4, {\left(\beta + \alpha\right)}^{2} + -1, {\left(\beta + \alpha\right)}^{2} \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\left(\beta + \alpha\right) + i}} \cdot \frac{\mathsf{fma}\left(i, \left(\beta + \alpha\right) + i, \beta \cdot \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{elif}\;\beta \leq 2.9 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + 0.125 \cdot \frac{\beta}{i}\right) - 0.125 \cdot \frac{\beta}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.5%
Cost	35144

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_2 := \left(\beta + \alpha\right) + i\\ t_3 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 4.7 \cdot 10^{+132}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(t_3 + \beta \cdot \alpha\right)}{i \cdot i} - \frac{\mathsf{fma}\left(4, t_3 + -1, t_3 \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 2.6 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_1}{t_2}}}{t_1 + 1} \cdot \frac{\frac{\mathsf{fma}\left(i, t_2, \beta \cdot \alpha\right)}{t_1}}{-1 + t_1}\\ \mathbf{elif}\;\beta \leq 2.2 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 2
Accuracy	84.4%
Cost	34824

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ t_1 := \left(\beta + \alpha\right) + i\\ t_2 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_3 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 1.4 \cdot 10^{+134}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(t_3 + \beta \cdot \alpha\right)}{i \cdot i} - \frac{\mathsf{fma}\left(4, t_3 + -1, t_3 \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 1.8 \cdot 10^{+156}:\\ \;\;\;\;\frac{\frac{\left(i \cdot t_1\right) \cdot \left(\mathsf{fma}\left(i, t_1, \beta \cdot \alpha\right) \cdot {t_2}^{-2}\right)}{-1 + t_2}}{t_2 + 1}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 3
Accuracy	84.3%
Cost	28036

\[\begin{array}{l} t_0 := \frac{\alpha + i}{\beta}\\ t_1 := 0.125 \cdot \frac{\beta}{i}\\ t_2 := {\left(\beta + \alpha\right)}^{2}\\ \mathbf{if}\;\beta \leq 6.6 \cdot 10^{+132}:\\ \;\;\;\;0.0625 + \left(\frac{0.0625 \cdot \left(t_2 + \beta \cdot \alpha\right)}{i \cdot i} - \frac{\mathsf{fma}\left(4, t_2 + -1, t_2 \cdot 20\right) \cdot 0.00390625}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 2.85 \cdot 10^{+156}:\\ \;\;\;\;{\left(\frac{\beta}{i \cdot t_0}\right)}^{-1}\\ \mathbf{elif}\;\beta \leq 7.3 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + t_1\right) - t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 4
Accuracy	84.7%
Cost	1732

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + 0.015625 \cdot \frac{\left(\beta + \alpha\right) \cdot -4 + \left(\beta + \alpha\right) \cdot 8}{i}\right) - 0.0625 \cdot \frac{\beta + \alpha}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 5
Accuracy	84.2%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.6 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 6
Accuracy	84.7%
Cost	964

\[\begin{array}{l} t_0 := 0.125 \cdot \frac{\beta}{i}\\ \mathbf{if}\;\beta \leq 2.45 \cdot 10^{+202}:\\ \;\;\;\;\left(0.0625 + t_0\right) - t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{\frac{1}{\beta}}{\frac{1}{i}}\\ \end{array} \]

Alternative 7
Accuracy	84.2%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.6 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	75.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+242}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	82.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.5 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	82.4%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 3.7 \cdot 10^{+202}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 11
Accuracy	74.2%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+242}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Accuracy	10.4%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023159 
(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)))