Octave 3.8, jcobi/4

Average Error: 53.6 → 1.4

Time: 27.0s

Precision: binary64

Cost: 27844

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

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+171}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right) + 1}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\beta}{\alpha + i} \cdot \frac{\beta}{i}\right)}^{-1}\\ \end{array} \]

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
 (if (<= alpha 1.4e+171)
   (*
    (* (/ i (fma i 2.0 beta)) (/ (+ i beta) (+ (fma i 2.0 beta) 1.0)))
    (/
     (/ i (/ (fma i 2.0 beta) (+ i beta)))
     (+ alpha (+ (fma i 2.0 beta) -1.0))))
   (pow (* (/ beta (+ alpha i)) (/ beta i)) -1.0)))

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 tmp;
	if (alpha <= 1.4e+171) {
		tmp = ((i / fma(i, 2.0, beta)) * ((i + beta) / (fma(i, 2.0, beta) + 1.0))) * ((i / (fma(i, 2.0, beta) / (i + beta))) / (alpha + (fma(i, 2.0, beta) + -1.0)));
	} else {
		tmp = pow(((beta / (alpha + i)) * (beta / i)), -1.0);
	}
	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)
	tmp = 0.0
	if (alpha <= 1.4e+171)
		tmp = Float64(Float64(Float64(i / fma(i, 2.0, beta)) * Float64(Float64(i + beta) / Float64(fma(i, 2.0, beta) + 1.0))) * Float64(Float64(i / Float64(fma(i, 2.0, beta) / Float64(i + beta))) / Float64(alpha + Float64(fma(i, 2.0, beta) + -1.0))));
	else
		tmp = Float64(Float64(beta / Float64(alpha + i)) * Float64(beta / i)) ^ -1.0;
	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_] := If[LessEqual[alpha, 1.4e+171], N[(N[(N[(i / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / N[(N[(i * 2.0 + beta), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(i / N[(N[(i * 2.0 + beta), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision] * N[(beta / i), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if alpha < 1.40000000000000002e171

   1. Initial program 53.4

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

   2. Taylor expanded in alpha around 0 53.5

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

   3. Simplified 41.4

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

      [Start] 53.5	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 41.4	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 41.4	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 41.4	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 1.1

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

   5. Taylor expanded in alpha around 0 39.2

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

   6. Simplified 1.1

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

      [Start] 39.2	\[ \frac{i \cdot \left(\beta + i\right)}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      +-commutative [<=] 39.2	\[ \frac{i \cdot \color{blue}{\left(i + \beta\right)}}{\left(\beta + 2 \cdot i\right) \cdot \left(\beta + \left(1 + 2 \cdot i\right)\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      times-frac [=>] 1.1	\[ \color{blue}{\left(\frac{i}{\beta + 2 \cdot i} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right)} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      +-commutative [=>] 1.1	\[ \left(\frac{i}{\color{blue}{2 \cdot i + \beta}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      *-commutative [=>] 1.1	\[ \left(\frac{i}{\color{blue}{i \cdot 2} + \beta} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      fma-udef [<=] 1.1	\[ \left(\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{i + \beta}{\beta + \left(1 + 2 \cdot i\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      +-commutative [=>] 1.1	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{\left(1 + 2 \cdot i\right) + \beta}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      associate-+l+ [=>] 1.1	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\color{blue}{1 + \left(2 \cdot i + \beta\right)}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      *-commutative [=>] 1.1	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \left(\color{blue}{i \cdot 2} + \beta\right)}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      fma-udef [<=] 1.1	\[ \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{1 + \color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]

   if 1.40000000000000002e171 < alpha

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

   3. Taylor expanded in beta around inf 59.2

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

   4. Simplified 57.4

      \[\leadsto \color{blue}{\frac{i}{\frac{\beta \cdot \beta}{i + \alpha}}} \]
      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* [=>] 57.4	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 57.4	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 31.8

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

   6. Applied egg-rr 13.7

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+171}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right) + 1}\right) \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\beta}{\alpha + i} \cdot \frac{\beta}{i}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.5
Cost	27524

\[\begin{array}{l} \mathbf{if}\;i \leq 1.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{{\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}\right)}^{2}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 2
Error	9.3
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_1 := \frac{i}{t_0} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\\ \mathbf{if}\;\beta \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;t_1 \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 3
Error	9.4
Cost	14276

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 3.9 \cdot 10^{+181}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot 0.25\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Error	9.5
Cost	1348

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 9.8 \cdot 10^{+182}:\\ \;\;\;\;0.25 \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Error	9.4
Cost	836

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.2 \cdot 10^{+163}:\\ \;\;\;\;0.25 \cdot \left(0.5 \cdot \frac{i}{\alpha + i \cdot 2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	9.4
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+163}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	10.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+163}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	18.4
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2022354 
(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)))