Octave 3.8, jcobi/4

Average Accuracy: 15.9% → 97.9%

Time: 32.0s

Precision: binary64

Cost: 21444

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

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

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\beta + i \cdot 2}{i + \beta}\\ t_1 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ t_2 := t_1 + -1\\ t_3 := 1 + t_1\\ \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{i}{t_3}}{t_0} \cdot \frac{\frac{i}{t_0}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_1}{i + \left(\alpha + \beta\right)}}}{t_3} \cdot \frac{\alpha + i}{t_2}\\ \end{array} \]

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 (/ (+ beta (* i 2.0)) (+ i beta)))
        (t_1 (fma i 2.0 (+ alpha beta)))
        (t_2 (+ t_1 -1.0))
        (t_3 (+ 1.0 t_1)))
   (if (<= alpha 3.1e+103)
     (* (/ (/ i t_3) t_0) (/ (/ i t_0) t_2))
     (* (/ (/ i (/ t_1 (+ i (+ alpha beta)))) t_3) (/ (+ alpha i) t_2)))))

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 = (beta + (i * 2.0)) / (i + beta);
	double t_1 = fma(i, 2.0, (alpha + beta));
	double t_2 = t_1 + -1.0;
	double t_3 = 1.0 + t_1;
	double tmp;
	if (alpha <= 3.1e+103) {
		tmp = ((i / t_3) / t_0) * ((i / t_0) / t_2);
	} else {
		tmp = ((i / (t_1 / (i + (alpha + beta)))) / t_3) * ((alpha + i) / t_2);
	}
	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(beta + Float64(i * 2.0)) / Float64(i + beta))
	t_1 = fma(i, 2.0, Float64(alpha + beta))
	t_2 = Float64(t_1 + -1.0)
	t_3 = Float64(1.0 + t_1)
	tmp = 0.0
	if (alpha <= 3.1e+103)
		tmp = Float64(Float64(Float64(i / t_3) / t_0) * Float64(Float64(i / t_0) / t_2));
	else
		tmp = Float64(Float64(Float64(i / Float64(t_1 / Float64(i + Float64(alpha + beta)))) / t_3) * Float64(Float64(alpha + i) / t_2));
	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[(beta + N[(i * 2.0), $MachinePrecision]), $MachinePrecision] / N[(i + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(i * 2.0 + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 + t$95$1), $MachinePrecision]}, If[LessEqual[alpha, 3.1e+103], N[(N[(N[(i / t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(i / t$95$0), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(i / N[(t$95$1 / N[(i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] * N[(N[(alpha + i), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if alpha < 3.1000000000000002e103

   1. Initial program 16.6%

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

   2. Applied egg-rr 43.7%

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

      [Start] 16.6	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 40.3	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 40.3	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 43.7	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in alpha around 0 43.3%

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

   4. Simplified 99.0%

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

      [Start] 43.3	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i \cdot \left(\beta + i\right)}{\beta + 2 \cdot i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-/l* [=>] 99.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\color{blue}{\frac{i}{\frac{\beta + 2 \cdot i}{\beta + i}}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      *-commutative [=>] 99.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i}{\frac{\beta + \color{blue}{i \cdot 2}}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   5. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{i}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}\right)} - 1\right)} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      Proof

      [Start] 99.0	\[ \frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      expm1-log1p-u [=>] 99.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}\right)\right)} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      expm1-udef [=>] 81.4	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}\right)} - 1\right)} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-/l/ [=>] 81.4	\[ \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}\right)} - 1\right) \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   6. Simplified 99.0%

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

      [Start] 81.4	\[ \left(e^{\mathsf{log1p}\left(\frac{i}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}\right)} - 1\right) \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      expm1-def [=>] 98.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{i}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}\right)\right)} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      expm1-log1p [=>] 98.9	\[ \color{blue}{\frac{i}{\left(\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1\right) \cdot \frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      associate-/r* [=>] 99.0	\[ \color{blue}{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + 1}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{i}{\color{blue}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\frac{\mathsf{fma}\left(i, 2, \color{blue}{\beta + \alpha}\right)}{i + \left(\alpha + \beta\right)}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]
      +-commutative [=>] 99.0	\[ \frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\frac{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}{i + \color{blue}{\left(\beta + \alpha\right)}}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{\beta + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \]

   7. Taylor expanded in alpha around 0 99.0%

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

   if 3.1000000000000002e103 < alpha

   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. Applied egg-rr 7.9%

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

      [Start] 0.0	\[ \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      times-frac [=>] 5.5	\[ \frac{\color{blue}{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1} \]
      difference-of-sqr-1 [=>] 5.5	\[ \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right)}} \]
      times-frac [=>] 7.9	\[ \color{blue}{\frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 1} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1}} \]

   3. Taylor expanded in beta around inf 75.0%

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

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \leq 3.1 \cdot 10^{+103}:\\ \;\;\;\;\frac{\frac{i}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\frac{\beta + i \cdot 2}{i + \beta}} \cdot \frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{i + \left(\alpha + \beta\right)}}}{1 + \mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \frac{\alpha + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.8%
Cost	15428

\[\begin{array}{l} t_0 := \frac{\beta + i \cdot 2}{i + \beta}\\ t_1 := \frac{\beta}{\sqrt{\alpha + i}}\\ t_2 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i}{1 + t_2}}{t_0} \cdot \frac{\frac{i}{t_0}}{t_2 + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_1} \cdot \frac{i}{t_1}\\ \end{array} \]

Alternative 2
Accuracy	97.8%
Cost	15300

\[\begin{array}{l} t_0 := \frac{\beta}{\sqrt{\alpha + i}}\\ \mathbf{if}\;\alpha \leq 8.5 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{i}{\frac{\beta + i \cdot 2}{i + \beta}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \cdot \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{i + \beta}{\beta + \left(1 + i \cdot 2\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_0} \cdot \frac{i}{t_0}\\ \end{array} \]

Alternative 3
Accuracy	84.6%
Cost	14924

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ t_1 := i + \left(\alpha + \beta\right)\\ t_2 := \mathsf{fma}\left(i, 2, \alpha + \beta\right)\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{i}{t_0 + -1} \cdot \left(0.25 \cdot t_1\right)}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \mathbf{elif}\;\beta \leq 6 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \beta}}}{t_2 + -1} \cdot \frac{i \cdot \left(i + \beta\right)}{t_0 \cdot \left(\beta + \left(1 + i \cdot 2\right)\right)}\\ \mathbf{elif}\;\beta \leq 3.8 \cdot 10^{+152}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_2}{t_1}}}{1 + t_2} \cdot \frac{\alpha + i}{\beta}\\ \end{array} \]

Alternative 4
Accuracy	84.5%
Cost	9160

\[\begin{array}{l} t_0 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{i}{t_0 + -1} \cdot \left(0.25 \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \mathbf{elif}\;\beta \leq 4.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{\frac{i}{\frac{t_0}{i + \beta}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) + -1} \cdot \frac{i \cdot \left(i + \beta\right)}{t_0 \cdot \left(\beta + \left(1 + i \cdot 2\right)\right)}\\ \mathbf{elif}\;\beta \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 5
Accuracy	84.5%
Cost	9096

\[\begin{array}{l} t_0 := i + \left(\alpha + \beta\right)\\ t_1 := i \cdot 2 + \left(\alpha + \beta\right)\\ t_2 := \beta + i \cdot 2\\ \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{i}{t_2 + -1} \cdot \left(0.25 \cdot t_0\right)}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \mathbf{elif}\;\beta \leq 5.8 \cdot 10^{+144}:\\ \;\;\;\;\frac{\left(i \cdot t_0\right) \cdot \frac{i \cdot \left(i + \beta\right)}{{t_2}^{2}}}{t_1 \cdot t_1 + -1}\\ \mathbf{elif}\;\beta \leq 8.8 \cdot 10^{+155}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 6
Accuracy	84.1%
Cost	8132

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;\frac{\frac{i}{\left(\beta + i \cdot 2\right) + -1} \cdot \left(0.25 \cdot \left(i + \left(\alpha + \beta\right)\right)\right)}{1 + \left(\beta + \mathsf{fma}\left(i, 2, \alpha\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 7
Accuracy	83.6%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{\alpha + i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 8
Accuracy	81.5%
Cost	580

\[\begin{array}{l} \mathbf{if}\;\beta \leq 4.2 \cdot 10^{+107}:\\ \;\;\;\;0.0625\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\beta} \cdot \frac{i}{\beta}\\ \end{array} \]

Alternative 9
Accuracy	70.7%
Cost	64

\[0.0625 \]

Reproduce

herbie shell --seed 2023151 
(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)))