Octave 3.8, jcobi/4

Percentage Accurate: 16.8% → 97.7%

Time: 29.3s

Precision: binary64

Cost: 28292

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

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+130}:\\ \;\;\;\;\left({\left(\frac{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 1}{i}\right)}^{-1} \cdot \frac{i + \beta}{t_0}\right) \cdot \left(\frac{i + \beta}{-1 + \left(\beta + i \cdot 2\right)} \cdot \frac{i}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

FPCore

(FPCore (alpha beta i)
 :precision binary64
 (/
  (/
   (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i))))
   (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))))
  (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (fma 2.0 i (+ alpha beta))))
   (if (<= alpha 9.5e+130)
     (*
      (*
       (pow (/ (+ (+ alpha (fma i 2.0 beta)) 1.0) i) -1.0)
       (/ (+ i beta) t_0))
      (* (/ (+ i beta) (+ -1.0 (+ beta (* i 2.0)))) (/ i t_0)))
     (/ (/ i beta) (/ beta (+ alpha i))))))

C

double code(double alpha, double beta, double i) {
	return (((i * ((alpha + beta) + i)) * ((beta * alpha) + (i * ((alpha + beta) + i)))) / (((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i)))) / ((((alpha + beta) + (2.0 * i)) * ((alpha + beta) + (2.0 * i))) - 1.0);
}

↓

double code(double alpha, double beta, double i) {
	double t_0 = fma(2.0, i, (alpha + beta));
	double tmp;
	if (alpha <= 9.5e+130) {
		tmp = (pow((((alpha + fma(i, 2.0, beta)) + 1.0) / i), -1.0) * ((i + beta) / t_0)) * (((i + beta) / (-1.0 + (beta + (i * 2.0)))) * (i / t_0));
	} else {
		tmp = (i / beta) / (beta / (alpha + i));
	}
	return tmp;
}

Julia

function code(alpha, beta, i)
	return Float64(Float64(Float64(Float64(i * Float64(Float64(alpha + beta) + i)) * Float64(Float64(beta * alpha) + Float64(i * Float64(Float64(alpha + beta) + i)))) / Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i)))) / Float64(Float64(Float64(Float64(alpha + beta) + Float64(2.0 * i)) * Float64(Float64(alpha + beta) + Float64(2.0 * i))) - 1.0))
end

↓

function code(alpha, beta, i)
	t_0 = fma(2.0, i, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 9.5e+130)
		tmp = Float64(Float64((Float64(Float64(Float64(alpha + fma(i, 2.0, beta)) + 1.0) / i) ^ -1.0) * Float64(Float64(i + beta) / t_0)) * Float64(Float64(Float64(i + beta) / Float64(-1.0 + Float64(beta + Float64(i * 2.0)))) * Float64(i / t_0)));
	else
		tmp = Float64(Float64(i / beta) / Float64(beta / Float64(alpha + i)));
	end
	return tmp
end

Wolfram

code[alpha_, beta_, i_] := N[(N[(N[(N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] * N[(N[(beta * alpha), $MachinePrecision] + N[(i * N[(N[(alpha + beta), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 9.5e+130], N[(N[(N[Power[N[(N[(N[(alpha + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / i), $MachinePrecision], -1.0], $MachinePrecision] * N[(N[(i + beta), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(i + beta), $MachinePrecision] / N[(-1.0 + N[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(i / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i / beta), $MachinePrecision] / N[(beta / N[(alpha + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 9.5000000000000009e130

   1. Initial program 18.6%

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

   2. Taylor expanded in alpha around 0 17.4%

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

   3. Simplified 17.4%

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

      [Start] 17.4	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 17.4	\[ \frac{\frac{\color{blue}{\left(i \cdot i\right)} \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 45.4%

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

      [Start] 17.4	\[ \frac{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      add-sqr-sqrt [=>] 17.4	\[ \frac{\color{blue}{\sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 17.4	\[ \frac{\sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 17.4	\[ \color{blue}{\frac{\sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\sqrt{\frac{\left(i \cdot i\right) \cdot {\left(\beta + i\right)}^{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   5. Simplified 97.0%

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

      [Start] 45.4	\[ \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      associate-/l/ [=>] 41.9	\[ \color{blue}{\frac{i \cdot \left(i + \beta\right)}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      times-frac [=>] 45.4	\[ \color{blue}{\left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)} \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      +-commutative [=>] 45.4	\[ \left(\frac{i}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      fma-udef [=>] 45.4	\[ \left(\frac{i}{1 + \color{blue}{\left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      *-commutative [=>] 45.4	\[ \left(\frac{i}{1 + \left(\color{blue}{2 \cdot i} + \left(\beta + \alpha\right)\right)} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      fma-def [=>] 45.4	\[ \left(\frac{i}{1 + \color{blue}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}} \cdot \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      +-commutative [<=] 45.4	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\color{blue}{\beta + i}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      fma-udef [=>] 45.4	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\color{blue}{i \cdot 2 + \left(\beta + \alpha\right)}}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      *-commutative [=>] 45.4	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\color{blue}{2 \cdot i} + \left(\beta + \alpha\right)}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      fma-def [=>] 45.4	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\color{blue}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}}\right) \cdot \frac{\frac{i \cdot \left(i + \beta\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \]
      associate-/l/ [=>] 41.9	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \color{blue}{\frac{i \cdot \left(i + \beta\right)}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)}} \]
      *-commutative [=>] 41.9	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \frac{\color{blue}{\left(i + \beta\right) \cdot i}}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1\right) \cdot \mathsf{fma}\left(i, 2, \beta + \alpha\right)} \]
      times-frac [=>] 97.0	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \color{blue}{\left(\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + -1} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)} \]

   6. Applied egg-rr 97.1%

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

      [Start] 97.0	\[ \left(\frac{i}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      clear-num [=>] 97.1	\[ \left(\color{blue}{\frac{1}{\frac{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}{i}}} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      inv-pow [=>] 97.1	\[ \left(\color{blue}{{\left(\frac{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}{i}\right)}^{-1}} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      *-un-lft-identity [=>] 97.1	\[ \left({\left(\frac{\color{blue}{1 \cdot \left(1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      *-un-lft-identity [<=] 97.1	\[ \left({\left(\frac{\color{blue}{1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)}}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      +-commutative [=>] 97.1	\[ \left({\left(\frac{\color{blue}{\mathsf{fma}\left(2, i, \beta + \alpha\right) + 1}}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      fma-udef [=>] 97.1	\[ \left({\left(\frac{\color{blue}{\left(2 \cdot i + \left(\beta + \alpha\right)\right)} + 1}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      associate-+r+ [=>] 97.1	\[ \left({\left(\frac{\color{blue}{\left(\left(2 \cdot i + \beta\right) + \alpha\right)} + 1}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      *-commutative [=>] 97.1	\[ \left({\left(\frac{\left(\left(\color{blue}{i \cdot 2} + \beta\right) + \alpha\right) + 1}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]
      fma-def [=>] 97.1	\[ \left({\left(\frac{\left(\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \alpha\right) + 1}{i}\right)}^{-1} \cdot \frac{\beta + i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \cdot \left(\frac{\beta + i}{-1 + \mathsf{fma}\left(2, i, \beta + \alpha\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \beta + \alpha\right)}\right) \]

   7. Taylor expanded in alpha around 0 97.1%

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

   if 9.5000000000000009e130 < alpha

   1. Initial program 0.2%

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

   2. Simplified 0.0%

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

      [Start] 0.2	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l/ [=>] 0.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      fma-def [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]
      +-commutative [=>] 0.0	\[ \frac{\left(i \cdot \left(i + \left(\alpha + \beta\right)\right)\right) \cdot \mathsf{fma}\left(\beta, \alpha, i \cdot \color{blue}{\left(i + \left(\alpha + \beta\right)\right)}\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)} \]

   3. Taylor expanded in beta around inf 2.6%

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

   4. Simplified 3.1%

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

      [Start] 2.6	\[ \frac{\left(i + \alpha\right) \cdot i}{{\beta}^{2}} \]
      *-commutative [<=] 2.6	\[ \frac{\color{blue}{i \cdot \left(i + \alpha\right)}}{{\beta}^{2}} \]
      associate-/l* [=>] 3.1	\[ \color{blue}{\frac{i}{\frac{{\beta}^{2}}{i + \alpha}}} \]
      unpow2 [=>] 3.1	\[ \frac{i}{\frac{\color{blue}{\beta \cdot \beta}}{i + \alpha}} \]

   5. Applied egg-rr 11.6%

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

      [Start] 3.1	\[ \frac{i}{\frac{\beta \cdot \beta}{i + \alpha}} \]
      associate-/l* [=>] 7.4	\[ \frac{i}{\color{blue}{\frac{\beta}{\frac{i + \alpha}{\beta}}}} \]
      associate-/r/ [=>] 11.6	\[ \color{blue}{\frac{i}{\beta} \cdot \frac{i + \alpha}{\beta}} \]

   6. Applied egg-rr 11.7%

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

      [Start] 11.6	\[ \frac{i}{\beta} \cdot \frac{i + \alpha}{\beta} \]
      clear-num [=>] 11.6	\[ \frac{i}{\beta} \cdot \color{blue}{\frac{1}{\frac{\beta}{i + \alpha}}} \]
      un-div-inv [=>] 11.7	\[ \color{blue}{\frac{\frac{i}{\beta}}{\frac{\beta}{i + \alpha}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+130}:\\ \;\;\;\;\left({\left(\frac{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 1}{i}\right)}^{-1} \cdot \frac{i + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right) \cdot \left(\frac{i + \beta}{-1 + \left(\beta + i \cdot 2\right)} \cdot \frac{i}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.7%
Cost	21828

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, i, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 9.5 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{i + \beta}{t_0} \cdot \frac{i}{\beta + \left(1 + i \cdot 2\right)}\right) \cdot \left(\frac{i}{t_0} \cdot \frac{i + \beta}{-1 + t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 2
Accuracy	83.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 3
Accuracy	83.8%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{\alpha + i}}\\ \end{array} \]

Alternative 4
Accuracy	82.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	82.2%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 8.2 \cdot 10^{+200}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 6
Accuracy	70.4%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023161 
(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)))