Octave 3.8, jcobi/2

Average Error: 24.2 → 1.8

Time: 16.2s

Precision: binary64

Cost: 22340

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

Math

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

↓

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

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 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -1.0)
     (/ (/ (+ (- beta beta) (+ (* i 4.0) (- 2.0 (* beta -2.0)))) alpha) 2.0)
     (/
      (fma
       (+ alpha beta)
       (/
        (/ (- beta alpha) (+ alpha (fma 2.0 i beta)))
        (+ alpha (fma 2.0 i (+ beta 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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -1.0) {
		tmp = (((beta - beta) + ((i * 4.0) + (2.0 - (beta * -2.0)))) / alpha) / 2.0;
	} else {
		tmp = fma((alpha + beta), (((beta - alpha) / (alpha + fma(2.0, i, beta))) / (alpha + fma(2.0, i, (beta + 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(alpha + beta) + Float64(2.0 * i))
	tmp = 0.0
	if (Float64(Float64(Float64(Float64(alpha + beta) * Float64(beta - alpha)) / t_0) / Float64(2.0 + t_0)) <= -1.0)
		tmp = Float64(Float64(Float64(Float64(beta - beta) + Float64(Float64(i * 4.0) + Float64(2.0 - Float64(beta * -2.0)))) / alpha) / 2.0);
	else
		tmp = Float64(fma(Float64(alpha + beta), Float64(Float64(Float64(beta - alpha) / Float64(alpha + fma(2.0, i, beta))) / Float64(alpha + fma(2.0, i, Float64(beta + 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[(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$0), $MachinePrecision] / N[(2.0 + t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[(N[(beta - beta), $MachinePrecision] + N[(N[(i * 4.0), $MachinePrecision] + N[(2.0 - N[(beta * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(alpha + beta), $MachinePrecision] * N[(N[(N[(beta - alpha), $MachinePrecision] / N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(alpha + N[(2.0 * i + N[(beta + 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)) < -1

   1. Initial program 63.3

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

   2. Simplified 54.4

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2}} \]
      Proof
      (/.f64 (+.f64 (/.f64 (/.f64 (+.f64 alpha beta) (/.f64 (+.f64 alpha (+.f64 beta (*.f64 2 i))) (-.f64 beta alpha))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (/.f64 (+.f64 alpha beta) (/.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (-.f64 beta alpha))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 77 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (/.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) (+.f64 (+.f64 alpha beta) (+.f64 (*.f64 2 i) 2))) 1) 2): 0 points increase in error, 77 points decrease in error
      (/.f64 (+.f64 (/.f64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 (+.f64 alpha beta) (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2)) (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 1 points increase in error, 1 points decrease in error
      (/.f64 (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 76 points increase in error, 2 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.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) 2): 3 points increase in error, 1 points decrease in error

   3. Taylor expanded in alpha around inf 6.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-1 \cdot \beta + \beta\right) - -1 \cdot \left(4 \cdot i + \left(2 + 2 \cdot \beta\right)\right)}{\alpha}}}{2} \]

   if -1 < (/.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 13.0

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}} \]
      Proof
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (fma.f64 2 i beta))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) beta)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 alpha (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha (fma.f64 2 i (+.f64 beta 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 i) (+.f64 beta 2))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 (*.f64 2 i) beta) 2)))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 alpha (+.f64 (Rewrite<= +-commutative_binary64 (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha (+.f64 beta (*.f64 2 i))) 2))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (/.f64 (-.f64 beta alpha) (+.f64 (+.f64 alpha beta) (*.f64 2 i))) (+.f64 (Rewrite<= associate-+l+_binary64 (+.f64 (+.f64 alpha beta) (*.f64 2 i))) 2)) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (Rewrite=> associate-/l/_binary64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 58 points increase in error, 16 points decrease in error
      (/.f64 (fma.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (Rewrite<= metadata-eval (neg.f64 -1))) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) -1)) 2): 15 points increase in error, 9 points decrease in error
      (/.f64 (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta))) -1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> fma-neg_binary64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (neg.f64 -1))) 2): 9 points increase in error, 15 points decrease in error
      (/.f64 (fma.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta) (Rewrite=> metadata-eval 1)) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))) (+.f64 alpha beta)) 1)) 2): 15 points increase in error, 9 points decrease in error
      (/.f64 (+.f64 (Rewrite=> *-commutative_binary64 (*.f64 (+.f64 alpha beta) (/.f64 (-.f64 beta alpha) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i)))))) 1) 2): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 alpha beta) (-.f64 beta alpha)) (*.f64 (+.f64 (+.f64 (+.f64 alpha beta) (*.f64 2 i)) 2) (+.f64 (+.f64 alpha beta) (*.f64 2 i))))) 1) 2): 81 points increase in error, 3 points decrease in error
      (/.f64 (+.f64 (Rewrite<= associate-/l/_binary64 (/.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) 2): 3 points increase in error, 1 points decrease in error
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

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

Alternatives

Alternative 1
Error	1.8
Cost	3524

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

Alternative 2
Error	2.4
Cost	2756

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

Alternative 3
Error	6.5
Cost	1348

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.9 \cdot 10^{+57}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + 2 \cdot \frac{1}{\alpha}\right)}{2}\\ \end{array} \]

Alternative 4
Error	12.3
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 2.8 \cdot 10^{+197}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.26 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	9.3
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 1.65 \cdot 10^{+74}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.02 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.82 \cdot 10^{+241}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.9
Cost	1228

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 3.9 \cdot 10^{+73}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 6 \cdot 10^{+196}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 8.8 \cdot 10^{+240}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \left(\beta + 2 \cdot i\right)\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.4
Cost	972

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{2 + 2 \cdot i}{\alpha}\\ \mathbf{if}\;\alpha \leq 2.32 \cdot 10^{+167}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2.2 \cdot 10^{+244}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 5.6 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	14.5
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 3.65 \cdot 10^{+177}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 2 \cdot 10^{+245}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.95 \cdot 10^{+264}:\\ \;\;\;\;\frac{\frac{i \cdot 4}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	12.5
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 1.2 \cdot 10^{+73}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.05 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 10^{+241}:\\ \;\;\;\;\frac{\frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	12.4
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+72}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.75 \cdot 10^{+202}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.3 \cdot 10^{+242}:\\ \;\;\;\;\frac{\frac{2 - \beta \cdot -2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	17.7
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.25 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 12
Error	24.7
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022331 
(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))