Octave 3.8, jcobi/4

Average Error: 53.4 → 36.5

Time: 1.2m

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 8.062523904888540452301986752113315617235 \cdot 10^{212}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}}{\sqrt{i \cdot 2 + \left(\beta + \alpha\right)}}}}{\frac{1}{\frac{\left(i + \alpha\right) + \beta}{i \cdot 2 + \left(\beta + \alpha\right)}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\frac{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) - \sqrt{1}}{i}}\right) \cdot \frac{\sqrt{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}}{\sqrt{\alpha + \left(\beta + i \cdot 2\right)}}}}{\left(\left(\sqrt{1} + \alpha\right) + \beta\right) + i \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r187662 = i;
        double r187663 = alpha;
        double r187664 = beta;
        double r187665 = r187663 + r187664;
        double r187666 = r187665 + r187662;
        double r187667 = r187662 * r187666;
        double r187668 = r187664 * r187663;
        double r187669 = r187668 + r187667;
        double r187670 = r187667 * r187669;
        double r187671 = 2.0;
        double r187672 = r187671 * r187662;
        double r187673 = r187665 + r187672;
        double r187674 = r187673 * r187673;
        double r187675 = r187670 / r187674;
        double r187676 = 1.0;
        double r187677 = r187674 - r187676;
        double r187678 = r187675 / r187677;
        return r187678;
}

↓

double f(double alpha, double beta, double i) {
        double r187679 = beta;
        double r187680 = 8.06252390488854e+212;
        bool r187681 = r187679 <= r187680;
        double r187682 = i;
        double r187683 = alpha;
        double r187684 = r187682 + r187683;
        double r187685 = r187684 + r187679;
        double r187686 = r187682 * r187685;
        double r187687 = r187683 * r187679;
        double r187688 = r187686 + r187687;
        double r187689 = sqrt(r187688);
        double r187690 = 2.0;
        double r187691 = r187682 * r187690;
        double r187692 = r187679 + r187683;
        double r187693 = r187691 + r187692;
        double r187694 = sqrt(r187693);
        double r187695 = r187689 / r187694;
        double r187696 = sqrt(r187695);
        double r187697 = 1.0;
        double r187698 = r187685 / r187693;
        double r187699 = r187697 / r187698;
        double r187700 = r187696 / r187699;
        double r187701 = r187688 / r187693;
        double r187702 = sqrt(r187701);
        double r187703 = 1.0;
        double r187704 = sqrt(r187703);
        double r187705 = r187693 - r187704;
        double r187706 = r187705 / r187682;
        double r187707 = r187702 / r187706;
        double r187708 = r187700 * r187707;
        double r187709 = r187679 + r187691;
        double r187710 = r187683 + r187709;
        double r187711 = sqrt(r187710);
        double r187712 = r187689 / r187711;
        double r187713 = sqrt(r187712);
        double r187714 = r187704 + r187683;
        double r187715 = r187714 + r187679;
        double r187716 = r187715 + r187691;
        double r187717 = r187713 / r187716;
        double r187718 = r187708 * r187717;
        double r187719 = r187710 * r187710;
        double r187720 = r187719 - r187703;
        double r187721 = r187682 + r187692;
        double r187722 = r187721 / r187710;
        double r187723 = r187682 * r187722;
        double r187724 = r187720 / r187723;
        double r187725 = r187682 / r187724;
        double r187726 = r187681 ? r187718 : r187725;
        return r187726;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if beta < 8.06252390488854e+212

   1. Initial program 52.2

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

   2. Simplified 37.6

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

   4. Applied add-sqr-sqrt 37.6

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

   5. Applied difference-of-squares 37.6

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

   6. Applied times-frac 36.1

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

   7. Applied add-sqr-sqrt 36.3

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

   8. Applied times-frac 35.6

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

   9. Simplified 35.6

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

   10. Simplified 35.6

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

   12. Applied div-inv 35.6

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

   13. Applied add-sqr-sqrt 35.5

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

   14. Applied add-sqr-sqrt 35.5

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

   15. Applied times-frac 35.5

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

   16. Applied sqrt-prod 35.6

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

   17. Applied times-frac 35.6

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

   18. Applied associate-*l* 35.6

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

   19. Simplified 35.6

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

   if 8.06252390488854e+212 < beta

   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 57.5

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

   3. Taylor expanded around inf 44.0

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

4. Final simplification 36.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 8.062523904888540452301986752113315617235 \cdot 10^{212}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}}{\sqrt{i \cdot 2 + \left(\beta + \alpha\right)}}}}{\frac{1}{\frac{\left(i + \alpha\right) + \beta}{i \cdot 2 + \left(\beta + \alpha\right)}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}{i \cdot 2 + \left(\beta + \alpha\right)}}}{\frac{\left(i \cdot 2 + \left(\beta + \alpha\right)\right) - \sqrt{1}}{i}}\right) \cdot \frac{\sqrt{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \alpha \cdot \beta}}{\sqrt{\alpha + \left(\beta + i \cdot 2\right)}}}}{\left(\left(\sqrt{1} + \alpha\right) + \beta\right) + i \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{i}{\frac{\left(\alpha + \left(\beta + i \cdot 2\right)\right) \cdot \left(\alpha + \left(\beta + i \cdot 2\right)\right) - 1}{i \cdot \frac{i + \left(\beta + \alpha\right)}{\alpha + \left(\beta + i \cdot 2\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))