Octave 3.8, jcobi/4

Average Accuracy: 16.3% → 84.7%

Time: 30.5s

Precision: binary64

Cost: 14532

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+197}:\\ \;\;\;\;{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(0.25 + \beta \cdot \frac{0.25}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\beta + \alpha\right)}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

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 (fma i 2.0 (+ beta alpha))))
   (if (<= beta 7.2e+197)
     (* (pow (/ i (fma i 2.0 beta)) 2.0) (+ 0.25 (* beta (/ 0.25 i))))
     (* (* (/ i t_0) (/ (+ i (+ beta alpha)) t_0)) (/ (+ 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 = fma(i, 2.0, (beta + alpha));
	double tmp;
	if (beta <= 7.2e+197) {
		tmp = pow((i / fma(i, 2.0, beta)), 2.0) * (0.25 + (beta * (0.25 / i)));
	} else {
		tmp = ((i / t_0) * ((i + (beta + alpha)) / t_0)) * ((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 = fma(i, 2.0, Float64(beta + alpha))
	tmp = 0.0
	if (beta <= 7.2e+197)
		tmp = Float64((Float64(i / fma(i, 2.0, beta)) ^ 2.0) * Float64(0.25 + Float64(beta * Float64(0.25 / i))));
	else
		tmp = Float64(Float64(Float64(i / t_0) * Float64(Float64(i + Float64(beta + alpha)) / t_0)) * Float64(Float64(i + alpha) / beta));
	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[(i * 2.0 + N[(beta + alpha), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[beta, 7.2e+197], N[(N[Power[N[(i / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(0.25 + N[(beta * N[(0.25 / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / t$95$0), $MachinePrecision] * N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if beta < 7.19999999999999965e197

   1. Initial program 20.8%

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

   2. Simplified 45.4%

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

      [Start] 20.8	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/r* [<=] 18.9	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 45.4	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in alpha around 0 18.9%

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

   4. Simplified 44.1%

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

      [Start] 18.9	\[ \frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} \cdot \left({\left(\beta + 2 \cdot i\right)}^{2} - 1\right)} \]
      times-frac [=>] 44.1	\[ \color{blue}{\frac{{i}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1}} \]
      unpow2 [=>] 44.1	\[ \frac{\color{blue}{i \cdot i}}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} - 1} \]
      sub-neg [=>] 44.1	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{2} + \left(-1\right)}} \]
      metadata-eval [=>] 44.1	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \frac{{\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2} + \color{blue}{-1}} \]

   5. Taylor expanded in i around -inf 36.6%

      \[\leadsto \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \color{blue}{\left(0.25 + -1 \cdot \frac{-0.5 \cdot \beta - -0.25 \cdot \beta}{i}\right)} \]

   6. Simplified 36.6%

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

      [Start] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + -1 \cdot \frac{-0.5 \cdot \beta - -0.25 \cdot \beta}{i}\right) \]
      distribute-rgt-out-- [=>] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + -1 \cdot \frac{\color{blue}{\beta \cdot \left(-0.5 - -0.25\right)}}{i}\right) \]
      metadata-eval [=>] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + -1 \cdot \frac{\beta \cdot \color{blue}{-0.25}}{i}\right) \]
      *-commutative [<=] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + -1 \cdot \frac{\color{blue}{-0.25 \cdot \beta}}{i}\right) \]
      associate-*r/ [<=] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + -1 \cdot \color{blue}{\left(-0.25 \cdot \frac{\beta}{i}\right)}\right) \]
      associate-*r* [=>] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + \color{blue}{\left(-1 \cdot -0.25\right) \cdot \frac{\beta}{i}}\right) \]
      metadata-eval [=>] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + \color{blue}{0.25} \cdot \frac{\beta}{i}\right) \]

   7. Applied egg-rr 85.8%

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

      [Start] 36.6	\[ \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 + 0.25 \cdot \frac{\beta}{i}\right) \]
      distribute-lft-in [=>] 36.6	\[ \color{blue}{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right)} \]
      add-sqr-sqrt [=>] 36.6	\[ \color{blue}{\left(\sqrt{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}}} \cdot \sqrt{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      pow2 [=>] 36.6	\[ \color{blue}{{\left(\sqrt{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)}^{2}} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      sqrt-div [=>] 36.6	\[ {\color{blue}{\left(\frac{\sqrt{i \cdot i}}{\sqrt{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)}}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      sqrt-prod [=>] 36.2	\[ {\left(\frac{\color{blue}{\sqrt{i} \cdot \sqrt{i}}}{\sqrt{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      add-sqr-sqrt [<=] 36.6	\[ {\left(\frac{\color{blue}{i}}{\sqrt{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      sqrt-pow1 [=>] 37.2	\[ {\left(\frac{i}{\color{blue}{{\left(\beta + 2 \cdot i\right)}^{\left(\frac{2}{2}\right)}}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      metadata-eval [=>] 37.2	\[ {\left(\frac{i}{{\left(\beta + 2 \cdot i\right)}^{\color{blue}{1}}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      pow1 [<=] 37.2	\[ {\left(\frac{i}{\color{blue}{\beta + 2 \cdot i}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      +-commutative [=>] 37.2	\[ {\left(\frac{i}{\color{blue}{2 \cdot i + \beta}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      *-commutative [=>] 37.2	\[ {\left(\frac{i}{\color{blue}{i \cdot 2} + \beta}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      fma-def [=>] 37.2	\[ {\left(\frac{i}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}\right)}^{2} \cdot 0.25 + \frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]
      add-sqr-sqrt [=>] 37.2	\[ {\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot 0.25 + \color{blue}{\left(\sqrt{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}}} \cdot \sqrt{\frac{i \cdot i}{{\left(\beta + 2 \cdot i\right)}^{2}}}\right)} \cdot \left(0.25 \cdot \frac{\beta}{i}\right) \]

   8. Simplified 85.8%

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

      [Start] 85.8	\[ {\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot 0.25 + {\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(\frac{0.25}{i} \cdot \beta\right) \]
      distribute-lft-out [=>] 85.8	\[ \color{blue}{{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(0.25 + \frac{0.25}{i} \cdot \beta\right)} \]
      *-commutative [=>] 85.8	\[ {\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(0.25 + \color{blue}{\beta \cdot \frac{0.25}{i}}\right) \]

   if 7.19999999999999965e197 < 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 11.0%

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

      [Start] 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-/r* [<=] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 11.0	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 80.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+197}:\\ \;\;\;\;{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(0.25 + \beta \cdot \frac{0.25}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.6%
Cost	13828

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.2 \cdot 10^{+197}:\\ \;\;\;\;{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)}\right)}^{2} \cdot \left(0.25 + \beta \cdot \frac{0.25}{i}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 2
Accuracy	84.5%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.6 \cdot 10^{+198}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta \cdot 2 + 2 \cdot \alpha}{i}\right) + \frac{\beta + \alpha}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 3
Accuracy	85.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 7.6 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i + \alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	85.1%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 10^{+159}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \end{array} \]

Alternative 5
Accuracy	75.3%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.12 \cdot 10^{+204}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	83.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.25 \cdot 10^{+159}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	83.1%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.4 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 8
Accuracy	74.5%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.5 \cdot 10^{+232}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 9
Accuracy	9.7%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023135 
(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)))