Octave 3.8, jcobi/2

Average Error: 23.7 → 11.0

Time: 14.6s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r105429 = alpha;
        double r105430 = beta;
        double r105431 = r105429 + r105430;
        double r105432 = r105430 - r105429;
        double r105433 = r105431 * r105432;
        double r105434 = 2.0;
        double r105435 = i;
        double r105436 = r105434 * r105435;
        double r105437 = r105431 + r105436;
        double r105438 = r105433 / r105437;
        double r105439 = r105437 + r105434;
        double r105440 = r105438 / r105439;
        double r105441 = 1.0;
        double r105442 = r105440 + r105441;
        double r105443 = r105442 / r105434;
        return r105443;
}

↓

double f(double alpha, double beta, double i) {
        double r105444 = alpha;
        double r105445 = 1.5371929371909745e+209;
        bool r105446 = r105444 <= r105445;
        double r105447 = beta;
        double r105448 = r105444 + r105447;
        double r105449 = 2.0;
        double r105450 = i;
        double r105451 = r105449 * r105450;
        double r105452 = r105448 + r105451;
        double r105453 = r105452 + r105449;
        double r105454 = sqrt(r105453);
        double r105455 = r105448 / r105454;
        double r105456 = r105447 - r105444;
        double r105457 = r105456 / r105452;
        double r105458 = r105454 / r105457;
        double r105459 = r105455 / r105458;
        double r105460 = 1.0;
        double r105461 = r105459 + r105460;
        double r105462 = r105461 * r105461;
        double r105463 = r105461 * r105462;
        double r105464 = cbrt(r105463);
        double r105465 = r105464 / r105449;
        double r105466 = r105449 / r105444;
        double r105467 = 8.0;
        double r105468 = 3.0;
        double r105469 = pow(r105444, r105468);
        double r105470 = r105467 / r105469;
        double r105471 = r105466 + r105470;
        double r105472 = 4.0;
        double r105473 = r105444 * r105444;
        double r105474 = r105472 / r105473;
        double r105475 = r105471 - r105474;
        double r105476 = r105475 / r105449;
        double r105477 = r105446 ? r105465 : r105476;
        return r105477;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.5371929371909745e+209

   1. Initial program 18.9

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

   3. Applied add-sqr-sqrt 18.9

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

   4. Applied *-un-lft-identity 18.9

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

   5. Applied times-frac 7.4

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

   6. Applied times-frac 7.4

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

   7. Simplified 7.4

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

   9. Applied add-sqr-sqrt 7.4

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

   10. Applied sqrt-prod 7.5

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

   11. Applied *-un-lft-identity 7.5

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

   12. Applied times-frac 7.5

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

   14. Applied add-cbrt-cube 7.4

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

   15. Simplified 7.4

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

   if 1.5371929371909745e+209 < alpha

   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 add-sqr-sqrt 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 50.9

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

   6. Applied times-frac 50.8

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

   7. Simplified 50.8

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

   9. Applied associate-*r/ 50.9

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

   10. Taylor expanded around inf 40.5

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

   11. Simplified 40.5

       \[\leadsto \frac{\color{blue}{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.5371929371909745 \cdot 10^{209}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right) \cdot \left(\frac{\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}} + 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2}{\alpha} + \frac{8}{{\alpha}^{3}}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))