Octave 3.8, jcobi/2

Average Error: 24.2 → 2.7

Time: 23.3s

Precision: binary64

Cost: 44996

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

Math

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

↓

\[\begin{array}{l} t_0 := \beta + \left(i \cdot -2 - \beta\right)\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ t_2 := 2 + t_1\\ t_3 := \beta + \mathsf{fma}\left(2, i, 2\right)\\ t_4 := \beta + t_3\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{t_2} \leq -0.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \beta, t_0\right) - t_3}{\alpha}, \frac{\left(\beta + 2 \cdot i\right) \cdot t_0}{\alpha \cdot \alpha} - \left(\frac{t_4}{\frac{\alpha}{\frac{t_3}{\alpha}}} + \frac{t_4}{\frac{\frac{\alpha \cdot \alpha}{-1}}{t_0}}\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\beta - \alpha\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{1}{t_1}\right)}{t_2}}{2}\\ \end{array} \]

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 (+ beta (- (* i -2.0) beta)))
        (t_1 (+ (+ alpha beta) (* 2.0 i)))
        (t_2 (+ 2.0 t_1))
        (t_3 (+ beta (fma 2.0 i 2.0)))
        (t_4 (+ beta t_3)))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_1) t_2) -0.5)
     (/
      (fma
       -1.0
       (/ (- (fma -1.0 beta t_0) t_3) alpha)
       (-
        (/ (* (+ beta (* 2.0 i)) t_0) (* alpha alpha))
        (+
         (/ t_4 (/ alpha (/ t_3 alpha)))
         (/ t_4 (/ (/ (* alpha alpha) -1.0) t_0)))))
      2.0)
     (/
      (+ 1.0 (/ (* (- beta alpha) (* (+ alpha beta) (/ 1.0 t_1))) t_2))
      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 = beta + ((i * -2.0) - beta);
	double t_1 = (alpha + beta) + (2.0 * i);
	double t_2 = 2.0 + t_1;
	double t_3 = beta + fma(2.0, i, 2.0);
	double t_4 = beta + t_3;
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_1) / t_2) <= -0.5) {
		tmp = fma(-1.0, ((fma(-1.0, beta, t_0) - t_3) / alpha), ((((beta + (2.0 * i)) * t_0) / (alpha * alpha)) - ((t_4 / (alpha / (t_3 / alpha))) + (t_4 / (((alpha * alpha) / -1.0) / t_0))))) / 2.0;
	} else {
		tmp = (1.0 + (((beta - alpha) * ((alpha + beta) * (1.0 / t_1))) / t_2)) / 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(beta + Float64(Float64(i * -2.0) - beta))
	t_1 = Float64(Float64(alpha + beta) + Float64(2.0 * i))
	t_2 = Float64(2.0 + t_1)
	t_3 = Float64(beta + fma(2.0, i, 2.0))
	t_4 = Float64(beta + t_3)
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_1) / t_2) <= -0.5)
		tmp = Float64(fma(-1.0, Float64(Float64(fma(-1.0, beta, t_0) - t_3) / alpha), Float64(Float64(Float64(Float64(beta + Float64(2.0 * i)) * t_0) / Float64(alpha * alpha)) - Float64(Float64(t_4 / Float64(alpha / Float64(t_3 / alpha))) + Float64(t_4 / Float64(Float64(Float64(alpha * alpha) / -1.0) / t_0))))) / 2.0);
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(beta - alpha) * Float64(Float64(alpha + beta) * Float64(1.0 / t_1))) / t_2)) / 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[(beta + N[(N[(i * -2.0), $MachinePrecision] - beta), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(alpha + beta), $MachinePrecision] + N[(2.0 * i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(beta + t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[(alpha + beta), $MachinePrecision] * N[(beta - alpha), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$2), $MachinePrecision], -0.5], N[(N[(-1.0 * N[(N[(N[(-1.0 * beta + t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision] / alpha), $MachinePrecision] + N[(N[(N[(N[(beta + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] / N[(alpha * alpha), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$4 / N[(alpha / N[(t$95$3 / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / N[(N[(N[(alpha * alpha), $MachinePrecision] / -1.0), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(1.0 + N[(N[(N[(beta - alpha), $MachinePrecision] * N[(N[(alpha + beta), $MachinePrecision] * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $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 61.7

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

   2. Simplified 53.1

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

      [Start] 61.7	\[ \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2} \]
      associate-/r* [<=] 61.7	\[ \frac{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2\right)}} + 1}{2} \]
      times-frac [=>] 53.1	\[ \frac{\color{blue}{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2} \]
      fma-def [=>] 53.1	\[ \frac{\color{blue}{\mathsf{fma}\left(\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}}{2} \]
      +-commutative [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\left(\beta + \alpha\right)} + 2 \cdot i}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\color{blue}{\beta + \left(\alpha + 2 \cdot i\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      +-commutative [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\left(2 \cdot i + \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      fma-def [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \color{blue}{\mathsf{fma}\left(2, i, \alpha\right)}}, \frac{\beta - \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}, 1\right)}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)}}, 1\right)}{2} \]
      associate-+l+ [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\color{blue}{\alpha + \left(\beta + \left(2 \cdot i + 2\right)\right)}}, 1\right)}{2} \]
      fma-def [=>] 53.1	\[ \frac{\mathsf{fma}\left(\frac{\alpha + \beta}{\beta + \mathsf{fma}\left(2, i, \alpha\right)}, \frac{\beta - \alpha}{\alpha + \left(\beta + \color{blue}{\mathsf{fma}\left(2, i, 2\right)}\right)}, 1\right)}{2} \]

   3. Taylor expanded in alpha around -inf 14.7

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

   4. Simplified 11.7

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

      [Start] 14.7

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

      fma-def [=>] 14.7

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

   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

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

4. Final simplification 2.7

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

Alternatives

Alternative 1
Error	1.5
Cost	22340

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

Alternative 2
Error	1.5
Cost	3652

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

Alternative 3
Error	1.5
Cost	3524

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

Alternative 4
Error	6.7
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.2 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 \cdot i + \left(\beta + 2\right)} \cdot \frac{\beta}{\beta + 2 \cdot i}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 5
Error	7.0
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 6
Error	10.2
Cost	964

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 4.7 \cdot 10^{+67}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 7
Error	15.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;2 \cdot i \leq 2 \cdot 10^{+146}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \]

Alternative 8
Error	15.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2.15 \cdot 10^{+68}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot i + \left(\beta + 2\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	17.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 6.4 \cdot 10^{+96}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10
Error	24.4
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2023011 
(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))