Octave 3.8, jcobi/4

Average Error: 54.1 → 11.5

Time: 38.4s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.243617314926794442030271342965650187278 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r188577 = i;
        double r188578 = alpha;
        double r188579 = beta;
        double r188580 = r188578 + r188579;
        double r188581 = r188580 + r188577;
        double r188582 = r188577 * r188581;
        double r188583 = r188579 * r188578;
        double r188584 = r188583 + r188582;
        double r188585 = r188582 * r188584;
        double r188586 = 2.0;
        double r188587 = r188586 * r188577;
        double r188588 = r188580 + r188587;
        double r188589 = r188588 * r188588;
        double r188590 = r188585 / r188589;
        double r188591 = 1.0;
        double r188592 = r188589 - r188591;
        double r188593 = r188590 / r188592;
        return r188593;
}

↓

double f(double alpha, double beta, double i) {
        double r188594 = i;
        double r188595 = 1.2436173149267944e+103;
        bool r188596 = r188594 <= r188595;
        double r188597 = alpha;
        double r188598 = beta;
        double r188599 = r188597 + r188598;
        double r188600 = 2.0;
        double r188601 = r188600 * r188594;
        double r188602 = r188599 + r188601;
        double r188603 = r188599 + r188594;
        double r188604 = r188602 / r188603;
        double r188605 = r188594 / r188604;
        double r188606 = 1.0;
        double r188607 = sqrt(r188606);
        double r188608 = r188602 + r188607;
        double r188609 = r188605 / r188608;
        double r188610 = r188598 * r188597;
        double r188611 = r188594 * r188603;
        double r188612 = r188610 + r188611;
        double r188613 = r188612 / r188602;
        double r188614 = r188602 - r188607;
        double r188615 = r188613 / r188614;
        double r188616 = r188609 * r188615;
        double r188617 = 0.25;
        double r188618 = r188617 * r188597;
        double r188619 = 0.5;
        double r188620 = r188619 * r188594;
        double r188621 = r188617 * r188598;
        double r188622 = r188620 + r188621;
        double r188623 = r188618 + r188622;
        double r188624 = r188623 / r188614;
        double r188625 = r188609 * r188624;
        double r188626 = log(r188625);
        double r188627 = exp(r188626);
        double r188628 = r188596 ? r188616 : r188627;
        return r188628;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 1.2436173149267944e+103

   1. Initial program 33.8

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

   3. Applied add-sqr-sqrt 33.8

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

   4. Applied difference-of-squares 33.8

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

   5. Applied times-frac 13.7

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

   6. Applied times-frac 9.0

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

   8. Applied associate-/l* 8.9

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

   if 1.2436173149267944e+103 < 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. Using strategy rm

   3. Applied add-sqr-sqrt 64.0

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

   4. Applied difference-of-squares 64.0

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

   5. Applied times-frac 52.8

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

   6. Applied times-frac 52.2

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

   8. Applied associate-/l* 52.2

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

   9. Taylor expanded around 0 12.8

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

   11. Applied add-exp-log 17.9

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

   12. Applied add-exp-log 16.9

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

   13. Applied div-exp 16.9

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

   14. Applied add-exp-log 18.0

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

   15. Applied add-exp-log 18.4

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

   16. Applied add-exp-log 18.8

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

   17. Applied div-exp 18.8

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

   18. Applied add-exp-log 18.9

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

   19. Applied div-exp 18.9

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

   20. Applied div-exp 18.9

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

   21. Applied prod-exp 18.9

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

   22. Simplified 12.8

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

4. Final simplification 11.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.243617314926794442030271342965650187278 \cdot 10^{103}:\\ \;\;\;\;\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\left(\alpha + \beta\right) + i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \frac{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1)))