Octave 3.8, jcobi/2

Average Error: 24.1 → 13.0

Time: 58.2s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 4947264647020423875871637504:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 5.575334987974757590975375438640297190907 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \le 2.371513235940968658731508439006658623794 \cdot 10^{190}:\\ \;\;\;\;\frac{1 + \frac{\frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\sqrt{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}}{\sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}} \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\sqrt{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}}{\sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5440262 = alpha;
        double r5440263 = beta;
        double r5440264 = r5440262 + r5440263;
        double r5440265 = r5440263 - r5440262;
        double r5440266 = r5440264 * r5440265;
        double r5440267 = 2.0;
        double r5440268 = i;
        double r5440269 = r5440267 * r5440268;
        double r5440270 = r5440264 + r5440269;
        double r5440271 = r5440266 / r5440270;
        double r5440272 = r5440270 + r5440267;
        double r5440273 = r5440271 / r5440272;
        double r5440274 = 1.0;
        double r5440275 = r5440273 + r5440274;
        double r5440276 = r5440275 / r5440267;
        return r5440276;
}

↓

double f(double alpha, double beta, double i) {
        double r5440277 = alpha;
        double r5440278 = 4.947264647020424e+27;
        bool r5440279 = r5440277 <= r5440278;
        double r5440280 = beta;
        double r5440281 = r5440280 + r5440277;
        double r5440282 = r5440280 - r5440277;
        double r5440283 = i;
        double r5440284 = 2.0;
        double r5440285 = r5440283 * r5440284;
        double r5440286 = r5440285 + r5440281;
        double r5440287 = r5440282 / r5440286;
        double r5440288 = r5440284 + r5440286;
        double r5440289 = r5440287 / r5440288;
        double r5440290 = r5440281 * r5440289;
        double r5440291 = 1.0;
        double r5440292 = r5440290 + r5440291;
        double r5440293 = r5440292 * r5440292;
        double r5440294 = r5440292 * r5440293;
        double r5440295 = cbrt(r5440294);
        double r5440296 = r5440295 / r5440284;
        double r5440297 = 5.575334987974758e+121;
        bool r5440298 = r5440277 <= r5440297;
        double r5440299 = r5440284 / r5440277;
        double r5440300 = 8.0;
        double r5440301 = r5440277 * r5440277;
        double r5440302 = r5440301 * r5440277;
        double r5440303 = r5440300 / r5440302;
        double r5440304 = 4.0;
        double r5440305 = r5440304 / r5440301;
        double r5440306 = r5440303 - r5440305;
        double r5440307 = r5440299 + r5440306;
        double r5440308 = r5440307 / r5440284;
        double r5440309 = 2.3715132359409687e+190;
        bool r5440310 = r5440277 <= r5440309;
        double r5440311 = cbrt(r5440287);
        double r5440312 = sqrt(r5440288);
        double r5440313 = sqrt(r5440312);
        double r5440314 = r5440311 / r5440313;
        double r5440315 = cbrt(r5440312);
        double r5440316 = r5440314 / r5440315;
        double r5440317 = r5440311 * r5440311;
        double r5440318 = r5440317 / r5440313;
        double r5440319 = r5440315 * r5440315;
        double r5440320 = r5440318 / r5440319;
        double r5440321 = r5440281 * r5440320;
        double r5440322 = r5440316 * r5440321;
        double r5440323 = r5440291 + r5440322;
        double r5440324 = r5440323 / r5440284;
        double r5440325 = r5440310 ? r5440324 : r5440308;
        double r5440326 = r5440298 ? r5440308 : r5440325;
        double r5440327 = r5440279 ? r5440296 : r5440326;
        return r5440327;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 4.947264647020424e+27

   1. Initial program 11.2

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

   3. Applied *-un-lft-identity 11.2

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

   4. Applied *-un-lft-identity 11.2

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

   5. Applied times-frac 0.7

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

   6. Applied times-frac 0.7

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

   7. Simplified 0.7

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

   9. Applied add-cbrt-cube 0.7

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

   if 4.947264647020424e+27 < alpha < 5.575334987974758e+121 or 2.3715132359409687e+190 < alpha

   1. Initial program 52.1

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

   2. Taylor expanded around inf 40.7

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

   3. Simplified 40.7

      \[\leadsto \frac{\color{blue}{\left(\frac{8}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4}{\alpha \cdot \alpha}\right) + \frac{2}{\alpha}}}{2}\]

   if 5.575334987974758e+121 < alpha < 2.3715132359409687e+190

   1. Initial program 56.7

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

   3. Applied *-un-lft-identity 56.7

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

   4. Applied *-un-lft-identity 56.7

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

   5. Applied times-frac 40.0

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

   6. Applied times-frac 39.9

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

   7. Simplified 39.9

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

   9. Applied add-sqr-sqrt 40.0

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

   10. Applied associate-/r* 39.9

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

   12. Applied add-cube-cbrt 40.0

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

   13. Applied add-sqr-sqrt 40.0

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

   14. Applied sqrt-prod 40.1

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

   15. Applied add-cube-cbrt 40.1

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

   16. Applied times-frac 40.0

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

   17. Applied times-frac 40.1

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

   18. Applied associate-*r* 40.0

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

4. Final simplification 13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4947264647020423875871637504:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)} + 1\right)\right)}}{2}\\ \mathbf{elif}\;\alpha \le 5.575334987974757590975375438640297190907 \cdot 10^{121}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \mathbf{elif}\;\alpha \le 2.371513235940968658731508439006658623794 \cdot 10^{190}:\\ \;\;\;\;\frac{1 + \frac{\frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\sqrt{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}}{\sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}} \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\frac{\beta - \alpha}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\sqrt{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}}{\sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}} \cdot \sqrt[3]{\sqrt{2 + \left(i \cdot 2 + \left(\beta + \alpha\right)\right)}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\alpha} + \left(\frac{8}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :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))