Octave 3.8, jcobi/2

Average Error: 23.7 → 13.1

Time: 8.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 19393672272838702021867389529627543207940:\\ \;\;\;\;\frac{\frac{\log \left(e^{{\left(\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}}\right)}^{3} + {1}^{3}}\right)}{\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} \cdot \frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} + \left(1 \cdot 1 - \frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} \cdot 1\right)}}{2}\\ \mathbf{elif}\;\alpha \le 1.262557062960945982227949818049794246774 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \mathbf{elif}\;\alpha \le 1.468201707068765789213963828469773650324 \cdot 10^{209}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{1}}{\alpha + \beta}}}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{1}{\sqrt{1}}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r794286 = alpha;
        double r794287 = beta;
        double r794288 = r794286 + r794287;
        double r794289 = r794287 - r794286;
        double r794290 = r794288 * r794289;
        double r794291 = 2.0;
        double r794292 = i;
        double r794293 = r794291 * r794292;
        double r794294 = r794288 + r794293;
        double r794295 = r794290 / r794294;
        double r794296 = r794294 + r794291;
        double r794297 = r794295 / r794296;
        double r794298 = 1.0;
        double r794299 = r794297 + r794298;
        double r794300 = r794299 / r794291;
        return r794300;
}

↓

double f(double alpha, double beta, double i) {
        double r794301 = alpha;
        double r794302 = 1.9393672272838702e+40;
        bool r794303 = r794301 <= r794302;
        double r794304 = 1.0;
        double r794305 = sqrt(r794304);
        double r794306 = r794304 / r794305;
        double r794307 = beta;
        double r794308 = r794301 + r794307;
        double r794309 = 2.0;
        double r794310 = i;
        double r794311 = r794309 * r794310;
        double r794312 = r794308 + r794311;
        double r794313 = r794312 + r794309;
        double r794314 = r794307 - r794301;
        double r794315 = r794314 / r794312;
        double r794316 = r794313 / r794315;
        double r794317 = r794308 / r794305;
        double r794318 = r794316 / r794317;
        double r794319 = r794306 / r794318;
        double r794320 = 3.0;
        double r794321 = pow(r794319, r794320);
        double r794322 = 1.0;
        double r794323 = pow(r794322, r794320);
        double r794324 = r794321 + r794323;
        double r794325 = exp(r794324);
        double r794326 = log(r794325);
        double r794327 = r794319 * r794319;
        double r794328 = r794322 * r794322;
        double r794329 = r794319 * r794322;
        double r794330 = r794328 - r794329;
        double r794331 = r794327 + r794330;
        double r794332 = r794326 / r794331;
        double r794333 = r794332 / r794309;
        double r794334 = 1.262557062960946e+165;
        bool r794335 = r794301 <= r794334;
        double r794336 = r794304 / r794301;
        double r794337 = r794309 * r794336;
        double r794338 = 8.0;
        double r794339 = pow(r794301, r794320);
        double r794340 = r794304 / r794339;
        double r794341 = r794338 * r794340;
        double r794342 = r794337 + r794341;
        double r794343 = 4.0;
        double r794344 = 2.0;
        double r794345 = pow(r794301, r794344);
        double r794346 = r794304 / r794345;
        double r794347 = r794343 * r794346;
        double r794348 = r794342 - r794347;
        double r794349 = r794348 / r794309;
        double r794350 = 1.4682017070687658e+209;
        bool r794351 = r794301 <= r794350;
        double r794352 = sqrt(r794313);
        double r794353 = r794352 / r794304;
        double r794354 = r794353 / r794308;
        double r794355 = r794306 / r794354;
        double r794356 = r794352 / r794315;
        double r794357 = r794356 / r794306;
        double r794358 = r794355 / r794357;
        double r794359 = r794358 + r794322;
        double r794360 = r794359 / r794309;
        double r794361 = r794351 ? r794360 : r794349;
        double r794362 = r794335 ? r794349 : r794361;
        double r794363 = r794303 ? r794333 : r794362;
        return r794363;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 1.9393672272838702e+40

   1. Initial program 11.0

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

   3. Applied *-un-lft-identity 11.0

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

   4. Applied times-frac 1.1

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

   5. Applied associate-/l* 1.1

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

   7. Applied add-sqr-sqrt 1.1

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

   8. Applied *-un-lft-identity 1.1

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

   9. Applied times-frac 1.1

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

   10. Applied associate-/l* 1.1

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

   12. Applied flip3-+ 1.1

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

   14. Applied add-log-exp 1.1

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

   15. Applied add-log-exp 1.1

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

   16. Applied sum-log 1.1

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

   17. Simplified 1.1

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

   if 1.9393672272838702e+40 < alpha < 1.262557062960946e+165 or 1.4682017070687658e+209 < alpha

   1. Initial program 51.9

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

   2. Taylor expanded around inf 41.5

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

   if 1.262557062960946e+165 < alpha < 1.4682017070687658e+209

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied times-frac 41.8

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

   5. Applied associate-/l* 41.8

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

   7. Applied add-sqr-sqrt 41.8

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

   8. Applied *-un-lft-identity 41.8

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

   9. Applied times-frac 41.8

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

   10. Applied associate-/l* 41.8

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

   12. Applied div-inv 41.8

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

   13. Applied *-un-lft-identity 41.8

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

   14. Applied *-un-lft-identity 41.8

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

   15. Applied times-frac 41.8

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

   16. Applied add-sqr-sqrt 42.0

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

   17. Applied times-frac 42.0

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

   18. Applied times-frac 42.1

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

   19. Applied associate-/r* 42.0

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

   20. Simplified 42.0

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 19393672272838702021867389529627543207940:\\ \;\;\;\;\frac{\frac{\log \left(e^{{\left(\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}}\right)}^{3} + {1}^{3}}\right)}{\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} \cdot \frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} + \left(1 \cdot 1 - \frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{\alpha + \beta}{\sqrt{1}}}} \cdot 1\right)}}{2}\\ \mathbf{elif}\;\alpha \le 1.262557062960945982227949818049794246774 \cdot 10^{165}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \mathbf{elif}\;\alpha \le 1.468201707068765789213963828469773650324 \cdot 10^{209}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1}{\sqrt{1}}}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{1}}{\alpha + \beta}}}{\frac{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\frac{1}{\sqrt{1}}}} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019298 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))