Octave 3.8, jcobi/2

Average Error: 24.0 → 1.3

Time: 8.6min

Precision: binary64

Cost: 6145

\[\alpha > -1 \land \beta > -1 \land i > 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999120837:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(\frac{2}{\alpha} + 1.375 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right) - \left(\frac{4}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + 6 \cdot \frac{\beta}{\alpha \cdot \alpha}\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

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
 (if (<=
      (/
       (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i)))
       (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))
      -0.9999999999120837)
   (/
    (-
     (+
      (* 2.0 (/ beta alpha))
      (+
       (* 4.0 (/ i alpha))
       (+ (/ 2.0 alpha) (* 1.375 (* (/ beta alpha) (/ beta alpha))))))
     (+
      (/ 4.0 (* alpha alpha))
      (+
       (* 12.0 (/ i (* alpha alpha)))
       (+
        (* 12.0 (+ (* (/ beta alpha) (/ i alpha)) (* (/ i alpha) (/ i alpha))))
        (* 6.0 (/ beta (* alpha alpha)))))))
    2.0)
   (/
    (+
     1.0
     (*
      (* (+ alpha beta) (/ (- beta alpha) (+ (+ alpha beta) (* 2.0 i))))
      (/ 1.0 (+ 2.0 (+ (+ alpha beta) (* 2.0 i))))))
    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 tmp;
	if (((((alpha + beta) * (beta - alpha)) / ((alpha + beta) + (2.0 * i))) / (2.0 + ((alpha + beta) + (2.0 * i)))) <= -0.9999999999120837) {
		tmp = (((2.0 * (beta / alpha)) + ((4.0 * (i / alpha)) + ((2.0 / alpha) + (1.375 * ((beta / alpha) * (beta / alpha)))))) - ((4.0 / (alpha * alpha)) + ((12.0 * (i / (alpha * alpha))) + ((12.0 * (((beta / alpha) * (i / alpha)) + ((i / alpha) * (i / alpha)))) + (6.0 * (beta / (alpha * alpha))))))) / 2.0;
	} else {
		tmp = (1.0 + (((alpha + beta) * ((beta - alpha) / ((alpha + beta) + (2.0 * i)))) * (1.0 / (2.0 + ((alpha + beta) + (2.0 * i)))))) / 2.0;
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Alternatives

Alternative 1
Error	1.3
Cost	3841

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999120837:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Alternative 2
Error	1.3
Cost	3713

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999120837:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \frac{\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Alternative 3
Error	5.7
Cost	2177

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 8.169303527528793 \cdot 10^{+115}:\\ \;\;\;\;\frac{1 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 4
Error	6.3
Cost	1537

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 9.647760065235352 \cdot 10^{+115}:\\ \;\;\;\;\frac{1 + \left(\beta - \alpha\right) \cdot \frac{1}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 5
Error	6.3
Cost	1409

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 1.7832028453706053 \cdot 10^{+117}:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 6
Error	7.0
Cost	1409

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.409244970131082 \cdot 10^{+147}:\\ \;\;\;\;\frac{1 + \beta \cdot \frac{1}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 7
Error	7.0
Cost	1281

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 7.409244970131082 \cdot 10^{+147}:\\ \;\;\;\;\frac{1 + \frac{\beta}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 8
Error	10.8
Cost	1474

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 2577959.6744566467:\\ \;\;\;\;\frac{1 + \frac{\beta - \alpha}{\alpha + \left(\beta + 2\right)}}{2}\\ \mathbf{elif}\;\alpha \leq 5.60925540053646 \cdot 10^{+97}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 9
Error	16.7
Cost	1153

\[\begin{array}{l} \mathbf{if}\;\alpha \leq 5.60925540053646 \cdot 10^{+97}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i \cdot 4 + \left(2 + \beta \cdot 2\right)}{\alpha}}{2}\\ \end{array}\]

Alternative 10
Error	17.3
Cost	385

\[\begin{array}{l} \mathbf{if}\;\beta \leq 2.0010033081354466 \cdot 10^{+65}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array}\]

Alternative 11
Error	42.9
Cost	64

\[1\]

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)) < -0.99999999991208366

   1. Initial program 62.8

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

   3. Applied add-sqr-sqrt_binary64_1805 62.8

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

   4. Applied *-un-lft-identity_binary64_1783 62.8

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

   5. Applied times-frac_binary64_1789 55.1

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

   6. Applied times-frac_binary64_1789 54.9

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

   7. Simplified 54.9

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

   8. Simplified 54.9

      \[\leadsto \frac{\frac{\alpha + \beta}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot \color{blue}{\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}} + 1}{2}\]
   9. Using strategy rm

   10. Applied *-un-lft-identity_binary64_1783 54.9

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

   11. Applied sqrt-prod_binary64_1799 54.9

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

   12. Applied add-sqr-sqrt_binary64_1805 54.7

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

   13. Applied times-frac_binary64_1789 54.7

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

   14. Applied associate-*l*_binary64_1724 54.7

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

   15. Simplified 54.7

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

   16. Taylor expanded around 0 56.1

       \[\leadsto \frac{\frac{\sqrt{\alpha + \beta}}{\sqrt{1}} \cdot \left(\color{blue}{\left(\left(0.5 \cdot \left(\frac{\alpha \cdot \beta}{\alpha + 2 \cdot i} \cdot \sqrt{\frac{1}{{\left(\alpha + \left(2 \cdot i + 2\right)\right)}^{3}}}\right) + \left(\sqrt{\frac{1}{\alpha + \left(2 \cdot i + 2\right)}} \cdot \frac{\beta}{\alpha + 2 \cdot i} + \frac{\alpha \cdot \beta}{{\left(\alpha + 2 \cdot i\right)}^{2}} \cdot \sqrt{\frac{1}{\alpha + \left(2 \cdot i + 2\right)}}\right)\right) - \frac{\alpha}{\alpha + 2 \cdot i} \cdot \sqrt{\frac{1}{\alpha + \left(2 \cdot i + 2\right)}}\right)} \cdot \frac{\sqrt{\alpha + \beta}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) + 1}{2}\]

   17. Simplified 56.1

       \[\leadsto \frac{\frac{\sqrt{\alpha + \beta}}{\sqrt{1}} \cdot \left(\color{blue}{\left(\left(0.5 \cdot \left(\frac{\alpha \cdot \beta}{\alpha + 2 \cdot i} \cdot \sqrt{\frac{1}{{\left(\alpha + \left(2 + 2 \cdot i\right)\right)}^{3}}}\right) + \sqrt{\frac{1}{\alpha + \left(2 + 2 \cdot i\right)}} \cdot \left(\frac{\beta}{\alpha + 2 \cdot i} + \frac{\alpha \cdot \beta}{{\left(\alpha + 2 \cdot i\right)}^{2}}\right)\right) - \frac{\alpha}{\alpha + 2 \cdot i} \cdot \sqrt{\frac{1}{\alpha + \left(2 + 2 \cdot i\right)}}\right)} \cdot \frac{\sqrt{\alpha + \beta}}{\sqrt{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}\right) + 1}{2}\]

   18. Taylor expanded around inf 14.3

       \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(2 \cdot \frac{1}{\alpha} + 1.375 \cdot \frac{{\beta}^{2}}{{\alpha}^{2}}\right)\right)\right) - \left(4 \cdot \frac{1}{{\alpha}^{2}} + \left(12 \cdot \frac{i}{{\alpha}^{2}} + \left(12 \cdot \frac{i \cdot \beta}{{\alpha}^{2}} + \left(12 \cdot \frac{{i}^{2}}{{\alpha}^{2}} + 6 \cdot \frac{\beta}{{\alpha}^{2}}\right)\right)\right)\right)}}{2}\]

   19. Simplified 5.5

       \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(\frac{2}{\alpha} + 1.375 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right) - \left(\frac{4}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + 6 \cdot \frac{\beta}{\alpha \cdot \alpha}\right)\right)\right)}}{2}\]

   20. Simplified 5.5

       \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(\frac{2}{\alpha} + 1.375 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right) - \left(\frac{4}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + 6 \cdot \frac{\beta}{\alpha \cdot \alpha}\right)\right)\right)}{2}}\]

   if -0.99999999991208366 < (/.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 13.1

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

   3. Applied *-un-lft-identity_binary64_1783 13.1

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

   4. Applied *-un-lft-identity_binary64_1783 13.1

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

   5. Applied times-frac_binary64_1789 0.2

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

   6. Applied times-frac_binary64_1789 0.2

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

   7. Simplified 0.2

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

   8. Simplified 0.2

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

   10. Applied div-inv_binary64_1780 0.2

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

   11. Applied associate-*r*_binary64_1723 0.2

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

   12. Simplified 0.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \leq -0.9999999999120837:\\ \;\;\;\;\frac{\left(2 \cdot \frac{\beta}{\alpha} + \left(4 \cdot \frac{i}{\alpha} + \left(\frac{2}{\alpha} + 1.375 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{\beta}{\alpha}\right)\right)\right)\right) - \left(\frac{4}{\alpha \cdot \alpha} + \left(12 \cdot \frac{i}{\alpha \cdot \alpha} + \left(12 \cdot \left(\frac{\beta}{\alpha} \cdot \frac{i}{\alpha} + \frac{i}{\alpha} \cdot \frac{i}{\alpha}\right) + 6 \cdot \frac{\beta}{\alpha \cdot \alpha}\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + \left(\left(\alpha + \beta\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{2 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2021014 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (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))