Octave 3.8, jcobi/2

Average Error: 23.3 → 1.3

Time: 26.5s

Precision: binary64

Cost: 28868

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

Math

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

↓

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)} + 1}\right)}{2}\\ \end{array} \]

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 (+ (+ alpha beta) (* 2.0 i))))
   (if (<= (/ (/ (* (+ alpha beta) (- beta alpha)) t_0) (+ 2.0 t_0)) -0.999998)
     (/
      (-
       (/ (+ 2.0 (* 2.0 i)) alpha)
       (+ (* -2.0 (/ i alpha)) (* -2.0 (/ beta alpha))))
      2.0)
     (/
      (log
       (exp
        (+
         (/
          (+ alpha beta)
          (*
           (/ (+ alpha (fma 2.0 i beta)) (- 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 = (alpha + beta) + (2.0 * i);
	double tmp;
	if (((((alpha + beta) * (beta - alpha)) / t_0) / (2.0 + t_0)) <= -0.999998) {
		tmp = (((2.0 + (2.0 * i)) / alpha) - ((-2.0 * (i / alpha)) + (-2.0 * (beta / alpha)))) / 2.0;
	} else {
		tmp = log(exp((((alpha + beta) / (((alpha + fma(2.0, i, beta)) / (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(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)) <= -0.999998)
		tmp = Float64(Float64(Float64(Float64(2.0 + Float64(2.0 * i)) / alpha) - Float64(Float64(-2.0 * Float64(i / alpha)) + Float64(-2.0 * Float64(beta / alpha)))) / 2.0);
	else
		tmp = Float64(log(exp(Float64(Float64(Float64(alpha + beta) / Float64(Float64(Float64(alpha + fma(2.0, i, beta)) / 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[(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], -0.999998], N[(N[(N[(N[(2.0 + N[(2.0 * i), $MachinePrecision]), $MachinePrecision] / alpha), $MachinePrecision] - N[(N[(-2.0 * N[(i / alpha), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(beta / alpha), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[Log[N[Exp[N[(N[(N[(alpha + beta), $MachinePrecision] / N[(N[(N[(alpha + N[(2.0 * i + beta), $MachinePrecision]), $MachinePrecision] / N[(beta - alpha), $MachinePrecision]), $MachinePrecision] * N[(alpha + N[(beta + N[(2.0 * i + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $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)) < -0.999998000000000054

   1. Initial program 62.1

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

   2. Simplified 54.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\alpha + \beta, \frac{\frac{\beta - \alpha}{\alpha + \mathsf{fma}\left(2, i, \beta\right)}}{\alpha + \mathsf{fma}\left(2, i, \beta + 2\right)}, 1\right)}{2}} \]
      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): 1 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): 64 points increase in error, 17 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): 13 points increase in error, 11 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): 11 points increase in error, 13 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): 13 points increase in error, 11 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): 86 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): 2 points increase in error, 0 points decrease in error

   3. Taylor expanded in alpha around -inf 5.7

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

   4. Taylor expanded in beta around inf 5.6

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

   if -0.999998000000000054 < (/.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 12.2

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

   2. Simplified 0.1

      \[\leadsto \color{blue}{\frac{\frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + \left(2 \cdot i + 2\right)} + 1}{2}} \]
      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): 83 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): 1 points increase in error, 83 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): 1 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): 0 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): 82 points increase in error, 0 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): 2 points increase in error, 0 points decrease in error

   3. Applied egg-rr 0.1

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{\frac{\alpha + \beta}{\frac{\alpha + \mathsf{fma}\left(2, i, \beta\right)}{\beta - \alpha} \cdot \left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right)} + 1}\right)}{2}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	16068

\[\begin{array}{l} t_0 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_0}}{2 + t_0} \leq -0.999998:\\ \;\;\;\;\frac{\frac{2 + 2 \cdot i}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\alpha + \beta}{\left(\alpha + \left(\beta + \mathsf{fma}\left(2, i, 2\right)\right)\right) \cdot \frac{\mathsf{fma}\left(2, i, \alpha + \beta\right)}{\beta - \alpha}}}{2}\\ \end{array} \]

Alternative 2
Error	1.3
Cost	3524

\[\begin{array}{l} t_0 := 2 + 2 \cdot i\\ t_1 := \left(\alpha + \beta\right) + 2 \cdot i\\ \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{t_1}}{2 + t_1} \leq -0.999998:\\ \;\;\;\;\frac{\frac{t_0}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\alpha + \left(\beta + 2 \cdot i\right)}{\beta - \alpha}}}{\left(\alpha + \beta\right) + t_0}}{2}\\ \end{array} \]

Alternative 3
Error	1.9
Cost	3268

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

Alternative 4
Error	8.1
Cost	1612

\[\begin{array}{l} t_0 := \frac{\left(2 \cdot \frac{1}{\alpha} - \frac{i}{\alpha} \cdot -4\right) - -2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 1.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	8.1
Cost	1612

\[\begin{array}{l} t_0 := \frac{\left(2 \cdot \frac{1}{\alpha} - \frac{i}{\alpha} \cdot -4\right) - -2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 2.25 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{\beta + 2}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	8.1
Cost	1612

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ t_1 := \frac{\left(2 \cdot \frac{1}{\alpha} - \frac{i}{\alpha} \cdot -4\right) - -2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{if}\;\alpha \leq 2 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 3.3 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	7.7
Cost	1608

\[\begin{array}{l} t_0 := 2 + 2 \cdot i\\ \mathbf{if}\;\alpha \leq 3.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{\beta + t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{t_0}{\alpha} - \left(-2 \cdot \frac{i}{\alpha} + -2 \cdot \frac{\beta}{\alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 8
Error	7.7
Cost	1604

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 3.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} - \frac{i}{\alpha} \cdot -4\right) - -2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 9
Error	8.1
Cost	1484

\[\begin{array}{l} t_0 := \beta + \left(\alpha + 2\right)\\ \mathbf{if}\;\alpha \leq 3.3 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\frac{\alpha + \beta}{\frac{\beta + 2 \cdot i}{\beta}}}{t_0}}{2}\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} - \frac{i}{\alpha} \cdot -4\right) - -2 \cdot \frac{\beta}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;\frac{1 + \frac{\beta}{t_0}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\beta - \beta\right) + \left(i \cdot 4 + \left(2 + \beta \cdot 2\right)\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 10
Error	14.0
Cost	1100

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;\alpha \leq 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 11
Error	14.0
Cost	1100

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{if}\;\alpha \leq 2.45 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 2.35 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 12
Error	10.7
Cost	1100

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.7 \cdot 10^{+85}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(2 + 2 \cdot i\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 3.1 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+239}:\\ \;\;\;\;\frac{1 + \frac{\beta}{\beta + \left(\alpha + 2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + 2 \cdot \frac{\beta}{\alpha}}{2}\\ \end{array} \]

Alternative 13
Error	16.2
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 2.35 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 8.5 \cdot 10^{+192}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 6.9 \cdot 10^{+239}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 14
Error	13.1
Cost	972

\[\begin{array}{l} t_0 := \frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ t_1 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 1.32 \cdot 10^{+234}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	14.2
Cost	972

\[\begin{array}{l} t_0 := \frac{1 + \frac{\beta}{\beta + 2}}{2}\\ \mathbf{if}\;\alpha \leq 2.8 \cdot 10^{+85}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\alpha \leq 3.5 \cdot 10^{+194}:\\ \;\;\;\;\frac{\frac{2 + i \cdot 4}{\alpha}}{2}\\ \mathbf{elif}\;\alpha \leq 4.8 \cdot 10^{+239}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 + \left(\beta + \beta\right)}{\alpha}}{2}\\ \end{array} \]

Alternative 16
Error	18.3
Cost	196

\[\begin{array}{l} \mathbf{if}\;\beta \leq 1.15 \cdot 10^{+143}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 17
Error	24.3
Cost	64

\[0.5 \]

Reproduce

herbie shell --seed 2022321 
(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))