Octave 3.8, jcobi/2

Average Error: 24.2 → 1.6

Time: 30.3s

Precision: binary64

Cost: 25156

\[\left(\alpha > -1 \land \beta > -1\right) \land i > 0\]

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ t_2 := 2 \cdot i + \left(\beta + 2\right)\\ t_3 := \beta + t_2\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{i}{\frac{\alpha \cdot \alpha}{\beta + 2 \cdot i}}, \mathsf{fma}\left(-2, \frac{t_3}{\frac{\alpha}{\frac{i}{\alpha}}}, \frac{t_2 - \mathsf{fma}\left(-1, \beta, i \cdot -2\right)}{\alpha} - \frac{t_3}{\frac{\alpha \cdot \alpha}{t_2}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1} + 1}{2}\\ \end{array} \]

FPCore

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

↓

(FPCore (alpha beta i)
 :precision binary64
 (let* ((t_0 (+ (+ alpha beta) (* 2.0 i)))
        (t_1 (+ 2.0 t_0))
        (t_2 (+ (* 2.0 i) (+ beta 2.0)))
        (t_3 (+ beta t_2)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) t_1) -0.5)
     (/
      (fma
       -2.0
       (/ i (/ (* alpha alpha) (+ beta (* 2.0 i))))
       (fma
        -2.0
        (/ t_3 (/ alpha (/ i alpha)))
        (-
         (/ (- t_2 (fma -1.0 beta (* i -2.0))) alpha)
         (/ t_3 (/ (* alpha alpha) t_2)))))
      2.0)
     (/
      (+
       (/ (* (- beta alpha) (/ (+ alpha beta) (fma 2.0 i (+ alpha beta)))) t_1)
       1.0)
      2.0))))

C

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

↓

double code(double alpha, double beta, double i) {
	double t_0 = (alpha + beta) + (2.0 * i);
	double t_1 = 2.0 + t_0;
	double t_2 = (2.0 * i) + (beta + 2.0);
	double t_3 = beta + t_2;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / t_1) <= -0.5) {
		tmp = fma(-2.0, (i / ((alpha * alpha) / (beta + (2.0 * i)))), fma(-2.0, (t_3 / (alpha / (i / alpha))), (((t_2 - fma(-1.0, beta, (i * -2.0))) / alpha) - (t_3 / ((alpha * alpha) / t_2))))) / 2.0;
	} else {
		tmp = ((((beta - alpha) * ((alpha + beta) / fma(2.0, i, (alpha + beta)))) / t_1) + 1.0) / 2.0;
	}
	return tmp;
}

Julia

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

↓

function code(alpha, beta, i)
	t_0 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_1 = Float64(2.0 + t_0)
	t_2 = Float64(Float64(2.0 * i) + Float64(beta + 2.0))
	t_3 = Float64(beta + t_2)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / t_1) <= -0.5)
		tmp = Float64(fma(-2.0, Float64(i / Float64(Float64(alpha * alpha) / Float64(beta + Float64(2.0 * i)))), fma(-2.0, Float64(t_3 / Float64(alpha / Float64(i / alpha))), Float64(Float64(Float64(t_2 - fma(-1.0, beta, Float64(i * -2.0))) / alpha) - Float64(t_3 / Float64(Float64(alpha * alpha) / t_2))))) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) / fma(2.0, i, Float64(alpha + beta)))) / t_1) + 1.0) / 2.0);
	end
	return tmp
end

Wolfram

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

↓

code[alpha_, beta_, i_] := Block[{t$95$0 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 * i), $MachinePrecision] + N[(beta + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(beta + t$95$2), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$1), $MachinePrecision], -0.5], N[(N[(-2.0 * N[(i / N[(N[(alpha * alpha), $MachinePrecision] / N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(t$95$3 / N[(alpha / N[(i / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$2 - N[(-1.0 * beta + N[(i * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(t$95$3 / N[(N[(alpha * alpha), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] / N[(2.0 * i + N[(alpha + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] / 2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) < -0.5

   1. Initial program 61.3

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

   2. Simplified 52.9

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

      [Start] 61.3	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 61.3	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 52.9	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 52.9	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 52.9	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Taylor expanded in alpha around inf 13.3

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

   4. Taylor expanded in alpha around -inf 13.3

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

   5. Simplified 6.6

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

      [Start] 13.3	\[ \frac{-2 \cdot \frac{i \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}} + \left(-2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(-1 \cdot \frac{\left(-1 \cdot \beta + -2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      fma-def [=>] 13.3	\[ \frac{\color{blue}{\mathsf{fma}\left(-2, \frac{i \cdot \left(\beta + 2 \cdot i\right)}{{\alpha}^{2}}, -2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(-1 \cdot \frac{\left(-1 \cdot \beta + -2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)}}{2} \]
      associate-/l* [=>] 13.3	\[ \frac{\mathsf{fma}\left(-2, \color{blue}{\frac{i}{\frac{{\alpha}^{2}}{\beta + 2 \cdot i}}}, -2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(-1 \cdot \frac{\left(-1 \cdot \beta + -2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      unpow2 [=>] 13.3	\[ \frac{\mathsf{fma}\left(-2, \frac{i}{\frac{\color{blue}{\alpha \cdot \alpha}}{\beta + 2 \cdot i}}, -2 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}} + \left(-1 \cdot \frac{\left(-1 \cdot \beta + -2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)\right)}{2} \]
      fma-def [=>] 13.3	\[ \frac{\mathsf{fma}\left(-2, \frac{i}{\frac{\alpha \cdot \alpha}{\beta + 2 \cdot i}}, \color{blue}{\mathsf{fma}\left(-2, \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot i}{{\alpha}^{2}}, -1 \cdot \frac{\left(-1 \cdot \beta + -2 \cdot i\right) - \left(\beta + \left(2 + 2 \cdot i\right)\right)}{\alpha} + -1 \cdot \frac{\left(\beta - -1 \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)\right) \cdot \left(\beta + \left(2 + 2 \cdot i\right)\right)}{{\alpha}^{2}}\right)}\right)}{2} \]

   if -0.5 < (/.f64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))

   1. Initial program 12.9

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

   2. Applied egg-rr 0.0

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

4. Final simplification 1.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, \frac{i}{\frac{\alpha \cdot \alpha}{\beta + 2 \cdot i}}, \mathsf{fma}\left(-2, \frac{\beta + \left(2 \cdot i + \left(\beta + 2\right)\right)}{\frac{\alpha}{\frac{i}{\alpha}}}, \frac{\left(2 \cdot i + \left(\beta + 2\right)\right) - \mathsf{fma}\left(-1, \beta, i \cdot -2\right)}{\alpha} - \frac{\beta + \left(2 \cdot i + \left(\beta + 2\right)\right)}{\frac{\alpha \cdot \alpha}{2 \cdot i + \left(\beta + 2\right)}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	9796

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.99999996:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \alpha\right) \cdot \frac{\alpha + \beta}{\mathsf{fma}\left(2, i, \alpha + \beta\right)}}{t_1} + 1}{2}\\ \end{array} \]

Alternative 2
Error	2.4
Cost	2756

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.5:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	7.9
Cost	1484

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{if}\;\alpha \leq 1.4 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 9 \cdot 10^{+97}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta + 2 \cdot i\right) + \left(2 \cdot i + \left(\beta + 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	10.9
Cost	1357

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.45 \cdot 10^{+75} \lor \neg \left(\alpha \leq 3.15 \cdot 10^{+97}\right) \land \alpha \leq 1.95 \cdot 10^{+169}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	14.1
Cost	1101

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+76} \lor \neg \left(\alpha \leq 2.3 \cdot 10^{+97}\right) \land \alpha \leq 1.6 \cdot 10^{+170}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	14.8
Cost	973

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.55 \cdot 10^{+76}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 4.4 \cdot 10^{+98} \lor \neg \left(\alpha \leq 5.5 \cdot 10^{+170}\right):\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 7
Error	15.7
Cost	708

\[\begin{array}{l} \mathbf{if}\;i \leq 1.6 \cdot 10^{+168}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	17.5
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.38 \cdot 10^{+45}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	25.1
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023067 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (and (> alpha -1.0) (> beta -1.0)) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))