Octave 3.8, jcobi/2

Average Error: 36.97% → 2.15%

Time: 21.6s

Precision: binary64

Cost: 25284

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

Math

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

↓

\[\begin{array}{l} t_0 := 2 \cdot i - \left(-2 - \beta\right)\\ t_1 := \frac{\beta + t_0}{\alpha}\\ t_2 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_2}}{2 + t_2} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{t_0}{\alpha} + \mathsf{fma}\left(-2, i \cdot \frac{\beta - i \cdot -2}{\alpha \cdot \alpha}, \mathsf{fma}\left(2, \frac{i}{\alpha}, \mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot t_1, t_1 \cdot \frac{i \cdot -2 + \left(-2 - \beta\right)}{\alpha}\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

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 (- (* 2.0 i) (- -2.0 beta)))
        (t_1 (/ (+ beta t_0) alpha))
        (t_2 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_2) (+ 2.0 t_2)) -0.999995)
     (/
      (+
       (/ beta alpha)
       (+
        (/ t_0 alpha)
        (fma
         -2.0
         (* i (/ (- beta (* i -2.0)) (* alpha alpha)))
         (fma
          2.0
          (/ i alpha)
          (fma
           -2.0
           (* (/ i alpha) t_1)
           (* t_1 (/ (+ (* i -2.0) (- -2.0 beta)) alpha)))))))
      2.0)
     (/
      (fma
       (/ (+ alpha beta) (+ beta (fma 2.0 i alpha)))
       (/ (- beta alpha) (+ alpha (+ beta (fma 2.0 i 2.0))))
       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 = (2.0 * i) - (-2.0 - beta);
	double t_1 = (beta + t_0) / alpha;
	double t_2 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_2) / (2.0 + t_2)) <= -0.999995) {
		tmp = ((beta / alpha) + ((t_0 / alpha) + fma(-2.0, (i * ((beta - (i * -2.0)) / (alpha * alpha))), fma(2.0, (i / alpha), fma(-2.0, ((i / alpha) * t_1), (t_1 * (((i * -2.0) + (-2.0 - beta)) / alpha))))))) / 2.0;
	} else {
		tmp = fma(((alpha + beta) / (beta + fma(2.0, i, alpha))), ((beta - alpha) / (alpha + (beta + fma(2.0, i, 2.0)))), 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(2.0 * i) - Float64(-2.0 - beta))
	t_1 = Float64(Float64(beta + t_0) / alpha)
	t_2 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_2) / Float64(2.0 + t_2)) <= -0.999995)
		tmp = Float64(Float64(Float64(beta / alpha) + Float64(Float64(t_0 / alpha) + fma(-2.0, Float64(i * Float64(Float64(beta - Float64(i * -2.0)) / Float64(alpha * alpha))), fma(2.0, Float64(i / alpha), fma(-2.0, Float64(Float64(i / alpha) * t_1), Float64(t_1 * Float64(Float64(Float64(i * -2.0) + Float64(-2.0 - beta)) / alpha))))))) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(alpha + beta) / Float64(beta + fma(2.0, i, alpha))), Float64(Float64(beta - alpha) / Float64(alpha + Float64(beta + fma(2.0, i, 2.0)))), 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[(2.0 * i), $MachinePrecision] - N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(beta + t$95$0), $MachinePrecision] / alpha), $MachinePrecision]}, Block[{t$95$2 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision], -0.999995], N[(N[(N[(beta / alpha), $MachinePrecision] + N[(N[(t$95$0 / alpha), $MachinePrecision] + N[(-2.0 * N[(i * N[(N[(beta - N[(i * -2.0), $MachinePrecision]), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(i / alpha), $MachinePrecision] + N[(-2.0 * N[(N[(i / alpha), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$1 * N[(N[(N[(i * -2.0), $MachinePrecision] + N[(-2.0 - beta), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(alpha + beta), $MachinePrecision] / N[(beta + N[(2.0 * i + alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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)) < -0.99999499999999997

   1. Initial program 96.99

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

   2. Simplified 84.19

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

      [Start] 96.99	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 97.05	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 84.19	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 84.19	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 84.19	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Taylor expanded in alpha around inf 20.97

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

   4. Applied egg-rr 20.97

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

   5. Simplified 9.24

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

      [Start] 20.97

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

   if -0.99999499999999997 < (/.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 19.38

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

   2. Simplified 0.07

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

      [Start] 19.38	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 19.39	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 0.08	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 0.07	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 0.07	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 2.15

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\beta}{\alpha} + \left(\frac{2 \cdot i - \left(-2 - \beta\right)}{\alpha} + \mathsf{fma}\left(-2, i \cdot \frac{\beta - i \cdot -2}{\alpha \cdot \alpha}, \mathsf{fma}\left(2, \frac{i}{\alpha}, \mathsf{fma}\left(-2, \frac{i}{\alpha} \cdot \frac{\beta + \left(2 \cdot i - \left(-2 - \beta\right)\right)}{\alpha}, \frac{\beta + \left(2 \cdot i - \left(-2 - \beta\right)\right)}{\alpha} \cdot \frac{i \cdot -2 + \left(-2 - \beta\right)}{\alpha}\right)\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	2.26%
Cost	22340

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)}, 1\right)}{2}\\ \end{array} \]

Alternative 2
Error	3.16%
Cost	2756

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_1 := 2 + t_0\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{t_1} \leq -0.999995:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(\left(2 + \beta \cdot 2\right) + i \cdot 4\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{t_1}}{2}\\ \end{array} \]

Alternative 3
Error	16.46%
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+138} \lor \neg \left(\alpha \leq 2.6 \cdot 10^{+166}\right) \land \alpha \leq 4.5 \cdot 10^{+201}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - -2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 4
Error	10.75%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.1 \cdot 10^{+134}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(i \cdot 4 - \left(-2 - \beta\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	11.57%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2 \cdot 10^{+138}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(i \cdot 4 - \left(-2 - \beta\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	11.57%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.85 \cdot 10^{+138}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta + \left(i \cdot 4 - \left(-2 - \beta\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	24.31%
Cost	836

\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 5 \cdot 10^{+114}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	21.67%
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 - -2 \cdot \left(\beta + i\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	22.38%
Cost	708

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \beta \cdot 2}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	27.46%
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.65 \cdot 10^{+59}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 11
Error	67.75%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023089 
(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))