Octave 3.8, jcobi/4

Average Error: 54.2 → 1.4

Time: 28.4s

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

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

↓

\[\begin{array}{l} t_0 := \frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{i}{\frac{\left(\alpha + 1\right) + \mathsf{fma}\left(i, 2, \beta\right)}{t_0}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) + -1}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \left(\alpha + \beta\right)}{t_1}\right) \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

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 (/ (+ i beta) (fma i 2.0 beta))) (t_1 (fma i 2.0 (+ alpha beta))))
   (if (<= alpha 2.4e+105)
     (*
      (/ i (/ (+ (+ alpha 1.0) (fma i 2.0 beta)) t_0))
      (/ i (/ (+ (fma i 2.0 beta) -1.0) t_0)))
     (* (* (/ i t_1) (/ (+ i (+ alpha beta)) t_1)) (/ (+ alpha i) 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 = (i + beta) / fma(i, 2.0, beta);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double tmp;
	if (alpha <= 2.4e+105) {
		tmp = (i / (((alpha + 1.0) + fma(i, 2.0, beta)) / t_0)) * (i / ((fma(i, 2.0, beta) + -1.0) / t_0));
	} else {
		tmp = ((i / t_1) * ((i + (alpha + beta)) / t_1)) * ((alpha + i) / 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 = Float64(Float64(i + beta) / fma(i, 2.0, beta))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	tmp = 0.0
	if (alpha <= 2.4e+105)
		tmp = Float64(Float64(i / Float64(Float64(Float64(alpha + 1.0) + fma(i, 2.0, beta)) / t_0)) * Float64(i / Float64(Float64(fma(i, 2.0, beta) + -1.0) / t_0)));
	else
		tmp = Float64(Float64(Float64(i / t_1) * Float64(Float64(i + Float64(alpha + beta)) / t_1)) * Float64(Float64(alpha + i) / 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[(N[(i + beta), $MachinePrecision] / N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[alpha, 2.4e+105], N[(N[(i / N[(N[(N[(alpha + 1.0), $MachinePrecision] + N[(i * 2.0 + beta), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(i / N[(N[(N[(i * 2.0 + beta), $MachinePrecision] + -1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / t$95$1), $MachinePrecision] * N[(N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / beta), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 2.39999999999999975e105

   1. Initial program 53.7

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

   2. Taylor expanded in alpha around 0 53.8

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

   3. Simplified 41.3

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

      [Start] 53.8	\[ \frac{\frac{{i}^{2} \cdot {\left(\beta + i\right)}^{2}}{{\left(\beta + 2 \cdot i\right)}^{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      associate-/l* [=>] 41.3	\[ \frac{\color{blue}{\frac{{i}^{2}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      unpow2 [=>] 41.3	\[ \frac{\frac{\color{blue}{i \cdot i}}{\frac{{\left(\beta + 2 \cdot i\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      *-commutative [=>] 41.3	\[ \frac{\frac{i \cdot i}{\frac{{\left(\beta + \color{blue}{i \cdot 2}\right)}^{2}}{{\left(\beta + i\right)}^{2}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]

   4. Applied egg-rr 0.6

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

   5. Simplified 0.6

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

      [Start] 0.6	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      associate-/l/ [=>] 0.6	\[ \color{blue}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \beta + \alpha\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      +-commutative [=>] 0.6	\[ \frac{i}{\color{blue}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right)} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)} \]
      associate-/l/ [=>] 0.6	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \color{blue}{\frac{i}{\left(\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) - 1\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}}} \]
      sub-neg [=>] 0.6	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\left(\alpha + \color{blue}{\left(\mathsf{fma}\left(i, 2, \beta\right) + \left(-1\right)\right)}\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]
      metadata-eval [=>] 0.6	\[ \frac{i}{\left(1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \cdot \frac{i}{\left(\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{-1}\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{\beta + i}} \]

   6. Applied egg-rr 36.7

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

   7. Simplified 0.6

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

      [Start] 36.7	\[ \frac{\frac{i \cdot i}{\left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-/r* [=>] 36.7	\[ \frac{\color{blue}{\frac{\frac{i \cdot i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-*r/ [<=] 35.6	\[ \frac{\frac{\color{blue}{i \cdot \frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-/r* [<=] 35.6	\[ \color{blue}{\frac{i \cdot \frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}}} \]
      *-commutative [=>] 35.6	\[ \frac{\color{blue}{\frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)} \cdot i}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta} \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      times-frac [=>] 35.7	\[ \color{blue}{\frac{\frac{i}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}}} \]
      associate-/r* [<=] 35.7	\[ \color{blue}{\frac{i}{\left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right) \cdot \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-*r/ [=>] 35.7	\[ \frac{i}{\color{blue}{\frac{\left(1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)\right) \cdot \mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-/l* [=>] 35.7	\[ \frac{i}{\color{blue}{\frac{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]
      associate-+r+ [=>] 35.7	\[ \frac{i}{\frac{\color{blue}{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) \cdot \left(\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)\right)}{i + \beta}} \]

   8. Taylor expanded in alpha around 0 35.7

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

   9. Simplified 0.6

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

      [Start] 35.7	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)}{\beta + i}} \]
      +-commutative [=>] 35.7	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\left(\left(\beta + 2 \cdot i\right) - 1\right) \cdot \left(\beta + 2 \cdot i\right)}{\color{blue}{i + \beta}}} \]
      associate-/l* [=>] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\color{blue}{\frac{\left(\beta + 2 \cdot i\right) - 1}{\frac{i + \beta}{\beta + 2 \cdot i}}}} \]
      sub-neg [=>] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\color{blue}{\left(\beta + 2 \cdot i\right) + \left(-1\right)}}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      +-commutative [=>] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\color{blue}{\left(2 \cdot i + \beta\right)} + \left(-1\right)}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      *-commutative [<=] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\left(\color{blue}{i \cdot 2} + \beta\right) + \left(-1\right)}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      fma-udef [<=] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)} + \left(-1\right)}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      metadata-eval [=>] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) + \color{blue}{-1}}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      +-commutative [<=] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\color{blue}{-1 + \mathsf{fma}\left(i, 2, \beta\right)}}{\frac{i + \beta}{\beta + 2 \cdot i}}} \]
      +-commutative [=>] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\color{blue}{2 \cdot i + \beta}}}} \]
      *-commutative [<=] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\color{blue}{i \cdot 2} + \beta}}} \]
      fma-udef [<=] 0.6	\[ \frac{i}{\frac{\left(1 + \alpha\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{-1 + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\color{blue}{\mathsf{fma}\left(i, 2, \beta\right)}}}} \]

   if 2.39999999999999975e105 < 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 62.6

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

      [Start] 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-/r* [<=] 64.0	\[ \color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 62.6	\[ \color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 16.8

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{i}{\frac{\left(\alpha + 1\right) + \mathsf{fma}\left(i, 2, \beta\right)}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right) + -1}{\frac{i + \beta}{\mathsf{fma}\left(i, 2, \beta\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right) \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	27844

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 2.4 \cdot 10^{+105}:\\ \;\;\;\;\frac{i}{\left(1 + t_1\right) \cdot t_0} \cdot \frac{i}{\left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right) \cdot t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_1} \cdot \frac{i + \left(\alpha + \beta\right)}{t_1}\right) \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 2
Error	3.4
Cost	27524

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 2.1 \cdot 10^{+225}:\\ \;\;\;\;\frac{i}{\left(\left(\alpha + 1\right) + \mathsf{fma}\left(i, 2, \beta\right)\right) \cdot \frac{{\left(\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}\right)}^{2}}{\frac{i}{\mathsf{fma}\left(i, 2, \beta\right) + \left(\alpha + -1\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot \left(\frac{i}{\frac{\beta \cdot \beta}{\alpha + i}} + \left(\left(\frac{\alpha}{\beta} + \frac{i}{\beta}\right) - \frac{\alpha + i}{\frac{\beta \cdot \beta}{\mathsf{fma}\left(4, i, \alpha \cdot 2\right)}}\right)\right)\\ \end{array} \]

Alternative 3
Error	6.5
Cost	27396

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;i \leq 4 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{{\left(\frac{i}{\frac{\mathsf{fma}\left(i, 2, \beta\right)}{i + \beta}}\right)}^{2}}{\alpha + \left(\mathsf{fma}\left(i, 2, \beta\right) + -1\right)}}{1 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot 0.25\\ \end{array} \]

Alternative 4
Error	8.8
Cost	14532

\[\begin{array}{l} t_0 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{i}{i \cdot 16 - \frac{4}{i}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{i}{t_0} \cdot \frac{i + \left(\alpha + \beta\right)}{t_0}\right) \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 5
Error	8.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.3 \cdot 10^{+151}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Error	8.8
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.1 \cdot 10^{+151}:\\ \;\;\;\;\frac{i}{i \cdot 16 - \frac{4}{i}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Error	15.8
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.9 \cdot 10^{+236}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Error	10.2
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.5 \cdot 10^{+158}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Error	10.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.85 \cdot 10^{+158}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 10
Error	10.0
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 5.8 \cdot 10^{+157}:\\ \;\;\;\;0.0625 + \frac{\frac{0.015625}{i}}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\beta}}{\frac{\beta}{i}}\\ \end{array} \]

Alternative 11
Error	16.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.05 \cdot 10^{+236}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 12
Error	57.4
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023237 
(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)))