Octave 3.8, jcobi/4

Average Error: 53.7 → 15.3

Time: 11.4s

Precision: binary64

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(i + \alpha\right)\\ t_1 := \alpha + \mathsf{fma}\left(i, 2, \beta\right)\\ t_2 := \frac{\frac{t_0 \cdot 0.25}{t_1}}{\frac{t_1}{i}}\\ t_3 := \frac{i}{t_1} \cdot \left(i + \left(\beta + \alpha\right)\right)\\ t_4 := t_3 \cdot \left(\frac{1}{t_1} \cdot \frac{i + \alpha}{\beta}\right)\\ \mathbf{if}\;\beta \leq 7.234486167206169 \cdot 10^{+72}:\\ \;\;\;\;\left(0.0625 + 0.015625 \cdot \frac{1}{i \cdot i}\right) - \mathsf{fma}\left(0.03125, \frac{\beta \cdot \beta}{i \cdot i}, 0.03125 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 2.2489104551141814 \cdot 10^{+89}:\\ \;\;\;\;t_3 \cdot {\left(\sqrt{\frac{\frac{\mathsf{fma}\left(i, t_0, \beta \cdot \alpha\right)}{t_1}}{{t_1}^{2} + -1}}\right)}^{2}\\ \mathbf{elif}\;\beta \leq 4.1268322475984935 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 1.6033367610360856 \cdot 10^{+168}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\beta \leq 3.435789572452624 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\beta \leq 8.377992451191864 \cdot 10^{+229}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\beta \leq 4.960988974597858 \cdot 10^{+261}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (+ beta (+ i alpha)))
        (t_1 (+ alpha (fma i 2.0 beta)))
        (t_2 (/ (/ (* t_0 0.25) t_1) (/ t_1 i)))
        (t_3 (* (/ i t_1) (+ i (+ beta alpha))))
        (t_4 (* t_3 (* (/ 1.0 t_1) (/ (+ i alpha) beta)))))
   (if (<= beta 7.234486167206169e+72)
     (-
      (+ 0.0625 (* 0.015625 (/ 1.0 (* i i))))
      (fma
       0.03125
       (/ (* beta beta) (* i i))
       (* 0.03125 (/ (* alpha alpha) (* i i)))))
     (if (<= beta 2.2489104551141814e+89)
       (*
        t_3
        (pow
         (sqrt (/ (/ (fma i t_0 (* beta alpha)) t_1) (+ (pow t_1 2.0) -1.0)))
         2.0))
       (if (<= beta 4.1268322475984935e+116)
         t_2
         (if (<= beta 1.6033367610360856e+168)
           t_4
           (if (<= beta 3.435789572452624e+190)
             t_2
             (if (<= beta 8.377992451191864e+229)
               t_4
               (if (<= beta 4.960988974597858e+261) t_2 t_4)))))))))

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 = beta + (i + alpha);
	double t_1 = alpha + fma(i, 2.0, beta);
	double t_2 = ((t_0 * 0.25) / t_1) / (t_1 / i);
	double t_3 = (i / t_1) * (i + (beta + alpha));
	double t_4 = t_3 * ((1.0 / t_1) * ((i + alpha) / beta));
	double tmp;
	if (beta <= 7.234486167206169e+72) {
		tmp = (0.0625 + (0.015625 * (1.0 / (i * i)))) - fma(0.03125, ((beta * beta) / (i * i)), (0.03125 * ((alpha * alpha) / (i * i))));
	} else if (beta <= 2.2489104551141814e+89) {
		tmp = t_3 * pow(sqrt(((fma(i, t_0, (beta * alpha)) / t_1) / (pow(t_1, 2.0) + -1.0))), 2.0);
	} else if (beta <= 4.1268322475984935e+116) {
		tmp = t_2;
	} else if (beta <= 1.6033367610360856e+168) {
		tmp = t_4;
	} else if (beta <= 3.435789572452624e+190) {
		tmp = t_2;
	} else if (beta <= 8.377992451191864e+229) {
		tmp = t_4;
	} else if (beta <= 4.960988974597858e+261) {
		tmp = t_2;
	} else {
		tmp = t_4;
	}
	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(beta + Float64(i + alpha))
	t_1 = Float64(alpha + fma(i, 2.0, beta))
	t_2 = Float64(Float64(Float64(t_0 * 0.25) / t_1) / Float64(t_1 / i))
	t_3 = Float64(Float64(i / t_1) * Float64(i + Float64(beta + alpha)))
	t_4 = Float64(t_3 * Float64(Float64(1.0 / t_1) * Float64(Float64(i + alpha) / beta)))
	tmp = 0.0
	if (beta <= 7.234486167206169e+72)
		tmp = Float64(Float64(0.0625 + Float64(0.015625 * Float64(1.0 / Float64(i * i)))) - fma(0.03125, Float64(Float64(beta * beta) / Float64(i * i)), Float64(0.03125 * Float64(Float64(alpha * alpha) / Float64(i * i)))));
	elseif (beta <= 2.2489104551141814e+89)
		tmp = Float64(t_3 * (sqrt(Float64(Float64(fma(i, t_0, Float64(beta * alpha)) / t_1) / Float64((t_1 ^ 2.0) + -1.0))) ^ 2.0));
	elseif (beta <= 4.1268322475984935e+116)
		tmp = t_2;
	elseif (beta <= 1.6033367610360856e+168)
		tmp = t_4;
	elseif (beta <= 3.435789572452624e+190)
		tmp = t_2;
	elseif (beta <= 8.377992451191864e+229)
		tmp = t_4;
	elseif (beta <= 4.960988974597858e+261)
		tmp = t_2;
	else
		tmp = t_4;
	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[(beta + N[(i + alpha), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$0 * 0.25), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(t$95$1 / i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(i / t$95$1), $MachinePrecision] * N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.234486167206169e+72], N[(N[(0.0625 + N[(0.015625 * N[(1.0 / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.03125 * N[(N[(beta * beta), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision] + N[(0.03125 * N[(N[(alpha * alpha), $MachinePrecision] / N[(i * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 2.2489104551141814e+89], N[(t$95$3 * N[Power[N[Sqrt[N[(N[(N[(i * t$95$0 + N[(beta * alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(N[Power[t$95$1, 2.0], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 4.1268322475984935e+116], t$95$2, If[LessEqual[beta, 1.6033367610360856e+168], t$95$4, If[LessEqual[beta, 3.435789572452624e+190], t$95$2, If[LessEqual[beta, 8.377992451191864e+229], t$95$4, If[LessEqual[beta, 4.960988974597858e+261], t$95$2, t$95$4]]]]]]]]]]]]

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 4 regimes

2. if beta < 7.2344861672061687e72

   1. Initial program 47.7

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

   2. Simplified 42.4

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

   3. Applied egg-rr 32.5

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

   4. Taylor expanded in i around inf 1.6

      \[\leadsto \color{blue}{\left(0.0625 + 0.015625 \cdot \frac{1}{{i}^{2}}\right) - \left(0.03125 \cdot \frac{{\beta}^{2}}{{i}^{2}} + 0.03125 \cdot \frac{{\alpha}^{2}}{{i}^{2}}\right)} \]

   5. Simplified 1.6

      \[\leadsto \color{blue}{\left(0.0625 + 0.015625 \cdot \frac{1}{i \cdot i}\right) - \mathsf{fma}\left(0.03125, \frac{\beta \cdot \beta}{i \cdot i}, 0.03125 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right)} \]

   if 7.2344861672061687e72 < beta < 2.2489104551141814e89

   1. Initial program 48.7

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

   2. Simplified 40.8

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

   3. Applied egg-rr 29.0

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

   if 2.2489104551141814e89 < beta < 4.1268322475984935e116 or 1.603336761036086e168 < beta < 3.43578957245262408e190 or 8.37799245119186437e229 < beta < 4.96098897459785793e261

   1. Initial program 61.2

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

   2. Simplified 55.2

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

   3. Applied egg-rr 49.3

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

   4. Taylor expanded in i around inf 35.1

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

   5. Applied egg-rr 35.0

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

   if 4.1268322475984935e116 < beta < 1.603336761036086e168 or 3.43578957245262408e190 < beta < 8.37799245119186437e229 or 4.96098897459785793e261 < beta

   1. Initial program 63.2

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

   2. Simplified 59.1

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

   3. Applied egg-rr 51.4

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

   4. Taylor expanded in beta around inf 32.5

      \[\leadsto \left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \color{blue}{\frac{i + \alpha}{\beta}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.234486167206169 \cdot 10^{+72}:\\ \;\;\;\;\left(0.0625 + 0.015625 \cdot \frac{1}{i \cdot i}\right) - \mathsf{fma}\left(0.03125, \frac{\beta \cdot \beta}{i \cdot i}, 0.03125 \cdot \frac{\alpha \cdot \alpha}{i \cdot i}\right)\\ \mathbf{elif}\;\beta \leq 2.2489104551141814 \cdot 10^{+89}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot {\left(\sqrt{\frac{\frac{\mathsf{fma}\left(i, \beta + \left(i + \alpha\right), \beta \cdot \alpha\right)}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}^{2} + -1}}\right)}^{2}\\ \mathbf{elif}\;\beta \leq 4.1268322475984935 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(i + \alpha\right)\right) \cdot 0.25}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i}}\\ \mathbf{elif}\;\beta \leq 1.6033367610360856 \cdot 10^{+168}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \alpha}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 3.435789572452624 \cdot 10^{+190}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(i + \alpha\right)\right) \cdot 0.25}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i}}\\ \mathbf{elif}\;\beta \leq 8.377992451191864 \cdot 10^{+229}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \alpha}{\beta}\right)\\ \mathbf{elif}\;\beta \leq 4.960988974597858 \cdot 10^{+261}:\\ \;\;\;\;\frac{\frac{\left(\beta + \left(i + \alpha\right)\right) \cdot 0.25}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \left(i + \left(\beta + \alpha\right)\right)\right) \cdot \left(\frac{1}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \alpha}{\beta}\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022159 
(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)))