Octave 3.8, jcobi/2

Average Error: 23.4 → 11.8

Time: 33.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 2.4342337726402253 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \left(\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.805786311316633 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.4760582663561516 \cdot 10^{+145}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{1}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}} \cdot \frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r3610899 = alpha;
        double r3610900 = beta;
        double r3610901 = r3610899 + r3610900;
        double r3610902 = r3610900 - r3610899;
        double r3610903 = r3610901 * r3610902;
        double r3610904 = 2.0;
        double r3610905 = i;
        double r3610906 = r3610904 * r3610905;
        double r3610907 = r3610901 + r3610906;
        double r3610908 = r3610903 / r3610907;
        double r3610909 = 2.0;
        double r3610910 = r3610907 + r3610909;
        double r3610911 = r3610908 / r3610910;
        double r3610912 = 1.0;
        double r3610913 = r3610911 + r3610912;
        double r3610914 = r3610913 / r3610909;
        return r3610914;
}

↓

double f(double alpha, double beta, double i) {
        double r3610915 = alpha;
        double r3610916 = 2.4342337726402253e+53;
        bool r3610917 = r3610915 <= r3610916;
        double r3610918 = 1.0;
        double r3610919 = beta;
        double r3610920 = r3610919 + r3610915;
        double r3610921 = 1.0;
        double r3610922 = 2.0;
        double r3610923 = 2.0;
        double r3610924 = i;
        double r3610925 = r3610923 * r3610924;
        double r3610926 = r3610920 + r3610925;
        double r3610927 = r3610922 + r3610926;
        double r3610928 = sqrt(r3610927);
        double r3610929 = r3610921 / r3610928;
        double r3610930 = r3610919 - r3610915;
        double r3610931 = r3610930 / r3610926;
        double r3610932 = r3610931 / r3610928;
        double r3610933 = r3610929 * r3610932;
        double r3610934 = r3610920 * r3610933;
        double r3610935 = r3610918 + r3610934;
        double r3610936 = r3610935 * r3610935;
        double r3610937 = r3610935 * r3610936;
        double r3610938 = cbrt(r3610937);
        double r3610939 = r3610938 / r3610922;
        double r3610940 = 5.805786311316633e+78;
        bool r3610941 = r3610915 <= r3610940;
        double r3610942 = 8.0;
        double r3610943 = r3610915 * r3610915;
        double r3610944 = r3610943 * r3610915;
        double r3610945 = r3610942 / r3610944;
        double r3610946 = r3610922 / r3610915;
        double r3610947 = 4.0;
        double r3610948 = r3610947 / r3610943;
        double r3610949 = r3610946 - r3610948;
        double r3610950 = r3610945 + r3610949;
        double r3610951 = r3610950 / r3610922;
        double r3610952 = 3.4760582663561516e+145;
        bool r3610953 = r3610915 <= r3610952;
        double r3610954 = cbrt(r3610927);
        double r3610955 = r3610954 * r3610954;
        double r3610956 = sqrt(r3610955);
        double r3610957 = r3610921 / r3610956;
        double r3610958 = r3610957 * r3610929;
        double r3610959 = sqrt(r3610954);
        double r3610960 = r3610931 / r3610959;
        double r3610961 = r3610958 * r3610960;
        double r3610962 = r3610920 * r3610961;
        double r3610963 = r3610918 + r3610962;
        double r3610964 = r3610963 / r3610922;
        double r3610965 = r3610953 ? r3610964 : r3610951;
        double r3610966 = r3610941 ? r3610951 : r3610965;
        double r3610967 = r3610917 ? r3610939 : r3610966;
        return r3610967;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 2.4342337726402253e+53

   1. Initial program 11.5

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

   3. Applied *-un-lft-identity 11.5

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

   4. Applied *-un-lft-identity 11.5

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

   5. Applied times-frac 1.4

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

   6. Applied times-frac 1.4

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

   7. Simplified 1.4

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

   9. Applied add-sqr-sqrt 1.4

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

   10. Applied *-un-lft-identity 1.4

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

   11. Applied times-frac 1.4

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]
   12. Using strategy rm

   13. Applied add-cbrt-cube 1.4

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right) \cdot \left(\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right)\right) \cdot \left(\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right)}}}{2.0}\]

   if 2.4342337726402253e+53 < alpha < 5.805786311316633e+78 or 3.4760582663561516e+145 < alpha

   1. Initial program 58.5

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

   2. Taylor expanded around inf 41.4

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

   3. Simplified 41.4

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

   if 5.805786311316633e+78 < alpha < 3.4760582663561516e+145

   1. Initial program 42.9

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

   3. Applied *-un-lft-identity 42.9

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

   4. Applied *-un-lft-identity 42.9

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

   5. Applied times-frac 32.2

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

   6. Applied times-frac 32.0

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

   7. Simplified 32.0

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

   9. Applied add-sqr-sqrt 32.0

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

   10. Applied *-un-lft-identity 32.0

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

   11. Applied times-frac 32.0

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 31.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}\right) + 1.0}{2.0}\]

   14. Applied sqrt-prod 31.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}}\right) + 1.0}{2.0}\]

   15. Applied *-un-lft-identity 31.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\color{blue}{1 \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right) + 1.0}{2.0}\]

   16. Applied times-frac 31.8

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \color{blue}{\left(\frac{1}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right)}\right) + 1.0}{2.0}\]

   17. Applied associate-*r* 31.9

       \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\left(\frac{1}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{1}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}\right)} + 1.0}{2.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.4342337726402253 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \left(\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right) \cdot \left(1.0 + \left(\beta + \alpha\right) \cdot \left(\frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}} \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right)\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 5.805786311316633 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.4760582663561516 \cdot 10^{+145}:\\ \;\;\;\;\frac{1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{1}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)} \cdot \sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}} \cdot \frac{1}{\sqrt{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}\right) \cdot \frac{\frac{\beta - \alpha}{\left(\beta + \alpha\right) + 2 \cdot i}}{\sqrt{\sqrt[3]{2.0 + \left(\left(\beta + \alpha\right) + 2 \cdot i\right)}}}\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} + \left(\frac{2.0}{\alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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))