Octave 3.8, jcobi/4

Average Accuracy: 15.9% → 84.5%

Time: 28.3s

Precision: binary64

Cost: 21060

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{1 + t_0} \cdot \frac{i}{\frac{t_0 + -1}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{i}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

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))) (t_1 (/ (+ i (+ beta alpha)) t_0)))
   (if (<= beta 2.7e+150)
     (*
      (/ 1.0 (+ 1.0 t_0))
      (/ i (/ (+ t_0 -1.0) (fma 0.25 (+ beta alpha) (* i 0.25)))))
     (if (<= beta 3.9e+185)
       (/ (/ i beta) (/ beta (+ i alpha)))
       (if (<= beta 9.5e+203)
         (* 0.25 (* (+ 0.5 (* -0.25 (/ (+ beta alpha) i))) t_1))
         (* (* t_1 (/ i 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 t_1 = (i + (beta + alpha)) / t_0;
	double tmp;
	if (beta <= 2.7e+150) {
		tmp = (1.0 / (1.0 + t_0)) * (i / ((t_0 + -1.0) / fma(0.25, (beta + alpha), (i * 0.25))));
	} else if (beta <= 3.9e+185) {
		tmp = (i / beta) / (beta / (i + alpha));
	} else if (beta <= 9.5e+203) {
		tmp = 0.25 * ((0.5 + (-0.25 * ((beta + alpha) / i))) * t_1);
	} else {
		tmp = (t_1 * (i / 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))
	t_1 = Float64(Float64(i + Float64(beta + alpha)) / t_0)
	tmp = 0.0
	if (beta <= 2.7e+150)
		tmp = Float64(Float64(1.0 / Float64(1.0 + t_0)) * Float64(i / Float64(Float64(t_0 + -1.0) / fma(0.25, Float64(beta + alpha), Float64(i * 0.25)))));
	elseif (beta <= 3.9e+185)
		tmp = Float64(Float64(i / beta) / Float64(beta / Float64(i + alpha)));
	elseif (beta <= 9.5e+203)
		tmp = Float64(0.25 * Float64(Float64(0.5 + Float64(-0.25 * Float64(Float64(beta + alpha) / i))) * t_1));
	else
		tmp = Float64(Float64(t_1 * Float64(i / 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]}, Block[{t$95$1 = N[(N[(i + N[(beta + alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[beta, 2.7e+150], N[(N[(1.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(t$95$0 + -1.0), $MachinePrecision] / N[(0.25 * N[(beta + alpha), $MachinePrecision] + N[(i * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 3.9e+185], N[(N[(i / beta), $MachinePrecision] / N[(beta / N[(i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[beta, 9.5e+203], N[(0.25 * N[(N[(0.5 + N[(-0.25 * N[(N[(beta + alpha), $MachinePrecision] / i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(i + alpha), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if beta < 2.70000000000000008e150

   1. Initial program 22.8%

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

   2. Taylor expanded in i around inf 43.0%

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

   3. Simplified 43.0%

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

      [Start] 43.0	\[ \frac{0.25 \cdot {i}^{2} + \left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 43.0	\[ \frac{\color{blue}{\left(0.25 \cdot \left(2 \cdot \beta + 2 \cdot \alpha\right) - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 43.0	\[ \frac{\left(0.25 \cdot \color{blue}{\left(2 \cdot \alpha + 2 \cdot \beta\right)} - 0.25 \cdot \left(\beta + \alpha\right)\right) \cdot i + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [<=] 43.0	\[ \frac{\color{blue}{i \cdot \left(0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right)\right)} + 0.25 \cdot {i}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 43.0	\[ \frac{\color{blue}{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \alpha + 2 \cdot \beta\right) - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [<=] 43.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \color{blue}{\left(2 \cdot \beta + 2 \cdot \alpha\right)} - 0.25 \cdot \left(\beta + \alpha\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out-- [=>] 43.0	\[ \frac{\mathsf{fma}\left(i, \color{blue}{0.25 \cdot \left(\left(2 \cdot \beta + 2 \cdot \alpha\right) - \left(\beta + \alpha\right)\right)}, 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      distribute-lft-out [=>] 43.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(\color{blue}{2 \cdot \left(\beta + \alpha\right)} - \left(\beta + \alpha\right)\right), 0.25 \cdot {i}^{2}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 43.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{{i}^{2} \cdot 0.25}\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 43.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \color{blue}{\left(i \cdot i\right)} \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 43.1%

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

      [Start] 43.0	\[ \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-un-lft-identity [=>] 43.0	\[ \frac{\color{blue}{1 \cdot \mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 43.0	\[ \frac{1 \cdot \mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 42.9	\[ \color{blue}{\frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]
      +-commutative [<=] 42.9	\[ \frac{1}{\left(\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      +-commutative [=>] 42.9	\[ \frac{1}{\color{blue}{\left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 42.9	\[ \frac{1}{\left(\color{blue}{i \cdot 2} + \left(\beta + \alpha\right)\right) + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      fma-def [=>] 42.9	\[ \frac{1}{\color{blue}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} + 1} \cdot \frac{\mathsf{fma}\left(i, 0.25 \cdot \left(2 \cdot \left(\beta + \alpha\right) - \left(\beta + \alpha\right)\right), \left(i \cdot i\right) \cdot 0.25\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   5. Simplified 90.9%

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

      [Start] 43.1	\[ \frac{1}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i \cdot \left(0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      +-commutative [=>] 43.1	\[ \frac{1}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{i \cdot \left(0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      associate-/l* [=>] 90.9	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \color{blue}{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25}}} \]
      +-commutative [=>] 90.9	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{0.25 \cdot \left(\beta + \alpha\right) + i \cdot 0.25}} \]
      fma-def [=>] 90.9	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\color{blue}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}} \]
      *-commutative [=>] 90.9	\[ \frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}{\mathsf{fma}\left(0.25, \beta + \alpha, \color{blue}{0.25 \cdot i}\right)}} \]

   if 2.70000000000000008e150 < beta < 3.8999999999999999e185

   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 6.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \left(\frac{i}{\mathsf{fma}\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right), \alpha + \mathsf{fma}\left(i, 2, \beta\right), -1\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} \]
      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-/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/ [<=] 5.3	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)} \]
      *-commutative [=>] 5.3	\[ \color{blue}{\left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 5.3	\[ \color{blue}{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right) + \beta \cdot \alpha\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 5.3	\[ \color{blue}{\mathsf{fma}\left(i, \left(\alpha + \beta\right) + i, \beta \cdot \alpha\right)} \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 5.3	\[ \mathsf{fma}\left(i, \color{blue}{i + \left(\alpha + \beta\right)}, \beta \cdot \alpha\right) \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      *-commutative [=>] 5.3	\[ \mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \color{blue}{\alpha \cdot \beta}\right) \cdot \frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      times-frac [=>] 6.2	\[ \mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right) \cdot \color{blue}{\left(\frac{i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}\right)} \]

   3. Taylor expanded in beta around inf 9.0%

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

   4. Simplified 10.7%

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

      [Start] 9.0	\[ \frac{i \cdot \left(i + \alpha\right)}{{\beta}^{2}} \]
      associate-/l* [=>] 10.7	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 10.7	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 48.6%

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

      [Start] 10.7	\[ \frac{i}{\frac{\beta \cdot \beta}{i + \alpha}} \]
      clear-num [=>] 10.7	\[ \color{blue}{\frac{1}{\frac{\frac{\beta \cdot \beta}{i + \alpha}}{i}}} \]
      inv-pow [=>] 10.7	\[ \color{blue}{{\left(\frac{\frac{\beta \cdot \beta}{i + \alpha}}{i}\right)}^{-1}} \]
      associate-/l* [=>] 48.6	\[ {\left(\frac{\color{blue}{\frac{\beta}{\frac{i + \alpha}{\beta}}}}{i}\right)}^{-1} \]
      associate-/r/ [=>] 48.6	\[ {\left(\frac{\color{blue}{\frac{\beta}{i + \alpha} \cdot \beta}}{i}\right)}^{-1} \]
      *-commutative [=>] 48.6	\[ {\left(\frac{\color{blue}{\beta \cdot \frac{\beta}{i + \alpha}}}{i}\right)}^{-1} \]

   6. Applied egg-rr 52.3%

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

      [Start] 48.6	\[ {\left(\frac{\beta \cdot \frac{\beta}{i + \alpha}}{i}\right)}^{-1} \]
      unpow-1 [=>] 48.6	\[ \color{blue}{\frac{1}{\frac{\beta \cdot \frac{\beta}{i + \alpha}}{i}}} \]
      clear-num [<=] 48.6	\[ \color{blue}{\frac{i}{\beta \cdot \frac{\beta}{i + \alpha}}} \]
      associate-/r* [=>] 52.3	\[ \color{blue}{\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}} \]

   if 3.8999999999999999e185 < beta < 9.4999999999999995e203

   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 12.0%

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      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 [=>] 12.0	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in i around inf 38.6%

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

   4. Taylor expanded in i around inf 36.0%

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

   if 9.4999999999999995e203 < 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 10.8%

      \[\leadsto \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\mathsf{fma}\left(i, i + \left(\alpha + \beta\right), \alpha \cdot \beta\right)}{\mathsf{fma}\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right), \mathsf{fma}\left(i, 2, \alpha + \beta\right), -1\right)}} \]
      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 [=>] 10.8	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 81.1%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;\frac{1}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1}{\mathsf{fma}\left(0.25, \beta + \alpha, i \cdot 0.25\right)}}\\ \mathbf{elif}\;\beta \leq 3.9 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.3%
Cost	14796

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \beta + \alpha\right)\\ t_1 := \frac{i + \left(\beta + \alpha\right)}{t_0}\\ \mathbf{if}\;\beta \leq 6.1 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625\right) + \frac{\left(\beta \cdot \beta\right) \cdot -0.078125 - \left(-1 + \beta \cdot \beta\right) \cdot 0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.5 \cdot 10^{+203}:\\ \;\;\;\;0.25 \cdot \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t_1 \cdot \frac{i}{t_0}\right) \cdot \frac{i + \alpha}{\beta}\\ \end{array} \]

Alternative 2
Accuracy	84.2%
Cost	8269

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+149}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+185} \lor \neg \left(\beta \leq 1.75 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\\ \end{array} \]

Alternative 3
Accuracy	84.1%
Cost	8269

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.05 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(0.0625, \frac{\beta \cdot \beta}{i \cdot i}, 0.0625\right) + \frac{\left(\beta \cdot \beta\right) \cdot -0.078125 - \left(-1 + \beta \cdot \beta\right) \cdot 0.015625}{i \cdot i}\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+185} \lor \neg \left(\beta \leq 1.12 \cdot 10^{+204}\right):\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \left(\left(0.5 + -0.25 \cdot \frac{\beta + \alpha}{i}\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	83.4%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\beta \leq 9.5 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 2.15 \cdot 10^{+185}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+235}:\\ \;\;\;\;\left(0.0625 + \frac{\beta}{i} \cdot 0.125\right) + \frac{\beta}{i} \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	83.3%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.2 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 4.8 \cdot 10^{+184}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{elif}\;\beta \leq 9.6 \cdot 10^{+235}:\\ \;\;\;\;\left(0.0625 + 0.0625 \cdot \frac{\beta}{i}\right) + \frac{\beta}{i} \cdot -0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	84.2%
Cost	973

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 6.2 \cdot 10^{+184} \lor \neg \left(\beta \leq 9.5 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 7
Accuracy	81.7%
Cost	972

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6 \cdot 10^{+149}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.2 \cdot 10^{+185}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{elif}\;\beta \leq 9 \cdot 10^{+235}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i \cdot \frac{i + \alpha}{\beta}}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	82.4%
Cost	845

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.35 \cdot 10^{+150}:\\ \;\;\;\;0.0625\\ \mathbf{elif}\;\beta \leq 3.1 \cdot 10^{+185} \lor \neg \left(\beta \leq 9.8 \cdot 10^{+203}\right):\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \mathbf{else}:\\ \;\;\;\;0.0625\\ \end{array} \]

Alternative 9
Accuracy	75.6%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2 \cdot 10^{+237}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha}{\beta}\\ \end{array} \]

Alternative 10
Accuracy	74.6%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.7 \cdot 10^{+238}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 11
Accuracy	10.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023137 
(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)))