Octave 3.8, jcobi/4

Average Error: 53.7 → 10.1

Time: 28.2s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.919336700596672990219203501876485209361 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\alpha + \left(\beta + i\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - \sqrt{1}} \cdot \frac{\mathsf{fma}\left(i, \alpha + \left(\beta + i\right), \alpha \cdot \beta\right)}{\sqrt{1} + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({e}^{\left(\log \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - \sqrt{1}}\right) \cdot \frac{\mathsf{fma}\left(0.5, i, \frac{1}{i} \cdot 0.125\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r93186 = i;
        double r93187 = alpha;
        double r93188 = beta;
        double r93189 = r93187 + r93188;
        double r93190 = r93189 + r93186;
        double r93191 = r93186 * r93190;
        double r93192 = r93188 * r93187;
        double r93193 = r93192 + r93191;
        double r93194 = r93191 * r93193;
        double r93195 = 2.0;
        double r93196 = r93195 * r93186;
        double r93197 = r93189 + r93196;
        double r93198 = r93197 * r93197;
        double r93199 = r93194 / r93198;
        double r93200 = 1.0;
        double r93201 = r93198 - r93200;
        double r93202 = r93199 / r93201;
        return r93202;
}

↓

double f(double alpha, double beta, double i) {
        double r93203 = i;
        double r93204 = 6.919336700596673e+130;
        bool r93205 = r93203 <= r93204;
        double r93206 = 2.0;
        double r93207 = beta;
        double r93208 = alpha;
        double r93209 = r93207 + r93208;
        double r93210 = fma(r93203, r93206, r93209);
        double r93211 = r93203 / r93210;
        double r93212 = r93207 + r93203;
        double r93213 = r93208 + r93212;
        double r93214 = r93211 * r93213;
        double r93215 = 1.0;
        double r93216 = sqrt(r93215);
        double r93217 = r93210 - r93216;
        double r93218 = r93214 / r93217;
        double r93219 = r93208 * r93207;
        double r93220 = fma(r93203, r93213, r93219);
        double r93221 = r93216 + r93210;
        double r93222 = r93220 / r93221;
        double r93223 = r93218 * r93222;
        double r93224 = r93223 / r93210;
        double r93225 = exp(1.0);
        double r93226 = log(r93211);
        double r93227 = pow(r93225, r93226);
        double r93228 = r93203 + r93209;
        double r93229 = r93228 / r93217;
        double r93230 = r93227 * r93229;
        double r93231 = 0.5;
        double r93232 = r93215 / r93203;
        double r93233 = 0.125;
        double r93234 = r93232 * r93233;
        double r93235 = fma(r93231, r93203, r93234);
        double r93236 = 0.25;
        double r93237 = r93236 * r93216;
        double r93238 = r93235 - r93237;
        double r93239 = r93238 / r93210;
        double r93240 = r93230 * r93239;
        double r93241 = r93205 ? r93224 : r93240;
        return r93241;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 6.919336700596673e+130

   1. Initial program 39.9

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

   2. Simplified 39.9

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

   4. Applied add-sqr-sqrt 39.9

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

   5. Applied difference-of-squares 39.9

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

   6. Applied times-frac 14.6

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

   7. Applied times-frac 10.3

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

   8. Simplified 10.3

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

   9. Simplified 10.3

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

   11. Applied associate-*r/ 10.3

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

   12. Applied associate-*r/ 10.3

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

   13. Simplified 10.3

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

   if 6.919336700596673e+130 < i

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   4. Applied add-sqr-sqrt 64.0

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

   5. Applied difference-of-squares 64.0

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

   6. Applied times-frac 58.3

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

   7. Applied times-frac 58.1

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

   8. Simplified 58.1

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

   9. Simplified 58.1

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

   10. Taylor expanded around inf 9.9

       \[\leadsto \frac{\color{blue}{\left(0.5 \cdot i + 0.125 \cdot \frac{{\left(\sqrt{1}\right)}^{2}}{i}\right) - 0.25 \cdot \sqrt{1}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   11. Simplified 9.9

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]
   12. Using strategy rm

   13. Applied add-exp-log 15.2

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{i}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}\right)\]

   14. Applied add-exp-log 15.1

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \frac{\color{blue}{e^{\log i}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\right)\]

   15. Applied div-exp 15.1

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \color{blue}{e^{\log i - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\right)\]

   16. Simplified 9.9

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot e^{\color{blue}{\log \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}\right)\]
   17. Using strategy rm

   18. Applied pow1 9.9

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot e^{\log \color{blue}{\left({\left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}^{1}\right)}}\right)\]

   19. Applied log-pow 9.9

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot e^{\color{blue}{1 \cdot \log \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}\right)\]

   20. Applied exp-prod 9.9

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\right)}}\right)\]

   21. Simplified 9.9

       \[\leadsto \frac{\mathsf{fma}\left(0.5, i, 0.125 \cdot \frac{1}{i}\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)} \cdot \left(\frac{i + \left(\alpha + \beta\right)}{\mathsf{fma}\left(i, 2, \alpha + \beta\right) - \sqrt{1}} \cdot {\color{blue}{e}}^{\left(\log \left(\frac{i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\right)}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.919336700596672990219203501876485209361 \cdot 10^{130}:\\ \;\;\;\;\frac{\frac{\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)} \cdot \left(\alpha + \left(\beta + i\right)\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - \sqrt{1}} \cdot \frac{\mathsf{fma}\left(i, \alpha + \left(\beta + i\right), \alpha \cdot \beta\right)}{\sqrt{1} + \mathsf{fma}\left(i, 2, \beta + \alpha\right)}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({e}^{\left(\log \left(\frac{i}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\right)\right)} \cdot \frac{i + \left(\beta + \alpha\right)}{\mathsf{fma}\left(i, 2, \beta + \alpha\right) - \sqrt{1}}\right) \cdot \frac{\mathsf{fma}\left(0.5, i, \frac{1}{i} \cdot 0.125\right) - 0.25 \cdot \sqrt{1}}{\mathsf{fma}\left(i, 2, \beta + \alpha\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))