Octave 3.8, jcobi/2

Average Error: 23.9 → 7.7

Time: 29.3s

Precision: 64

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

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999963:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right)}\right) \cdot \left(\left(\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \left(\beta + \alpha\right) + 1.0\right) \cdot \left(\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \left(\beta + \alpha\right) + 1.0\right)\right)}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r5131202 = alpha;
        double r5131203 = beta;
        double r5131204 = r5131202 + r5131203;
        double r5131205 = r5131203 - r5131202;
        double r5131206 = r5131204 * r5131205;
        double r5131207 = 2.0;
        double r5131208 = i;
        double r5131209 = r5131207 * r5131208;
        double r5131210 = r5131204 + r5131209;
        double r5131211 = r5131206 / r5131210;
        double r5131212 = 2.0;
        double r5131213 = r5131210 + r5131212;
        double r5131214 = r5131211 / r5131213;
        double r5131215 = 1.0;
        double r5131216 = r5131214 + r5131215;
        double r5131217 = r5131216 / r5131212;
        return r5131217;
}

↓

double f(double alpha, double beta, double i) {
        double r5131218 = beta;
        double r5131219 = alpha;
        double r5131220 = r5131218 + r5131219;
        double r5131221 = r5131218 - r5131219;
        double r5131222 = r5131220 * r5131221;
        double r5131223 = 2.0;
        double r5131224 = i;
        double r5131225 = r5131223 * r5131224;
        double r5131226 = r5131225 + r5131220;
        double r5131227 = r5131222 / r5131226;
        double r5131228 = 2.0;
        double r5131229 = r5131228 + r5131226;
        double r5131230 = r5131227 / r5131229;
        double r5131231 = -0.9999999999999963;
        bool r5131232 = r5131230 <= r5131231;
        double r5131233 = 8.0;
        double r5131234 = r5131219 * r5131219;
        double r5131235 = r5131219 * r5131234;
        double r5131236 = r5131233 / r5131235;
        double r5131237 = r5131228 / r5131219;
        double r5131238 = 4.0;
        double r5131239 = r5131238 / r5131234;
        double r5131240 = r5131237 - r5131239;
        double r5131241 = r5131236 + r5131240;
        double r5131242 = r5131241 / r5131228;
        double r5131243 = 1.0;
        double r5131244 = r5131228 + r5131220;
        double r5131245 = r5131225 + r5131244;
        double r5131246 = r5131221 / r5131245;
        double r5131247 = r5131246 / r5131226;
        double r5131248 = r5131220 * r5131247;
        double r5131249 = r5131248 * r5131248;
        double r5131250 = r5131249 * r5131248;
        double r5131251 = cbrt(r5131250);
        double r5131252 = r5131243 + r5131251;
        double r5131253 = r5131221 / r5131226;
        double r5131254 = r5131253 / r5131229;
        double r5131255 = r5131254 * r5131220;
        double r5131256 = r5131255 + r5131243;
        double r5131257 = r5131256 * r5131256;
        double r5131258 = r5131252 * r5131257;
        double r5131259 = cbrt(r5131258);
        double r5131260 = r5131259 / r5131228;
        double r5131261 = r5131232 ? r5131242 : r5131260;
        return r5131261;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) < -0.9999999999999963

   1. Initial program 62.7

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Taylor expanded around inf 32.7

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   3. Simplified 32.7

      \[\leadsto \frac{\color{blue}{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]

   if -0.9999999999999963 < (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0))

   1. Initial program 12.5

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   2. Using strategy rm

   3. Applied *-un-lft-identity 12.5

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 12.5

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 0.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 0.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 0.3

      \[\leadsto \frac{\color{blue}{\left(\alpha + \beta\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
   8. Using strategy rm

   9. Applied add-cbrt-cube 0.3

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]
   10. Using strategy rm

   11. Applied add-cbrt-cube 11.3

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}}} + 1.0\right)}}{2.0}\]

   12. Applied add-cbrt-cube 15.9

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0\right)}}{2.0}\]

   13. Applied add-cbrt-cube 22.2

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0\right)}}{2.0}\]

   14. Applied cbrt-undiv 22.2

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}}{\sqrt[3]{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0\right)}}{2.0}\]

   15. Applied cbrt-undiv 22.2

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}}} + 1.0\right)}}{2.0}\]

   16. Applied add-cbrt-cube 22.3

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)}} \cdot \sqrt[3]{\frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0\right)}}{2.0}\]

   17. Applied cbrt-unprod 22.2

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right)\right) \cdot \left(\alpha + \beta\right)\right) \cdot \frac{\frac{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)\right) \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}}} + 1.0\right)}}{2.0}\]

   18. Simplified 0.3

       \[\leadsto \frac{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\sqrt[3]{\color{blue}{\left(\left(\frac{\frac{\beta - \alpha}{\left(2.0 + \left(\beta + \alpha\right)\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(2.0 + \left(\beta + \alpha\right)\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right)\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(2.0 + \left(\beta + \alpha\right)\right) + 2 \cdot i}}{\left(\beta + \alpha\right) + 2 \cdot i} \cdot \left(\beta + \alpha\right)\right)}} + 1.0\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \le -0.9999999999999963:\\ \;\;\;\;\frac{\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \frac{\frac{\beta - \alpha}{2 \cdot i + \left(2.0 + \left(\beta + \alpha\right)\right)}}{2 \cdot i + \left(\beta + \alpha\right)}\right)}\right) \cdot \left(\left(\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \left(\beta + \alpha\right) + 1.0\right) \cdot \left(\frac{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)} \cdot \left(\beta + \alpha\right) + 1.0\right)\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))