Octave 3.8, jcobi/4

Average Error: 53.4 → 36.5

Time: 1.1m

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 r187499 = i;
        double r187500 = alpha;
        double r187501 = beta;
        double r187502 = r187500 + r187501;
        double r187503 = r187502 + r187499;
        double r187504 = r187499 * r187503;
        double r187505 = r187501 * r187500;
        double r187506 = r187505 + r187504;
        double r187507 = r187504 * r187506;
        double r187508 = 2.0;
        double r187509 = r187508 * r187499;
        double r187510 = r187502 + r187509;
        double r187511 = r187510 * r187510;
        double r187512 = r187507 / r187511;
        double r187513 = 1.0;
        double r187514 = r187511 - r187513;
        double r187515 = r187512 / r187514;
        return r187515;
}

↓

double f(double alpha, double beta, double i) {
        double r187516 = beta;
        double r187517 = 8.06252390488854e+212;
        bool r187518 = r187516 <= r187517;
        double r187519 = i;
        double r187520 = alpha;
        double r187521 = r187519 + r187520;
        double r187522 = r187521 + r187516;
        double r187523 = r187519 * r187522;
        double r187524 = r187520 * r187516;
        double r187525 = r187523 + r187524;
        double r187526 = sqrt(r187525);
        double r187527 = 2.0;
        double r187528 = r187519 * r187527;
        double r187529 = r187516 + r187520;
        double r187530 = r187528 + r187529;
        double r187531 = sqrt(r187530);
        double r187532 = r187526 / r187531;
        double r187533 = sqrt(r187532);
        double r187534 = 1.0;
        double r187535 = r187522 / r187530;
        double r187536 = r187534 / r187535;
        double r187537 = r187533 / r187536;
        double r187538 = r187525 / r187530;
        double r187539 = sqrt(r187538);
        double r187540 = 1.0;
        double r187541 = sqrt(r187540);
        double r187542 = r187530 - r187541;
        double r187543 = r187542 / r187519;
        double r187544 = r187539 / r187543;
        double r187545 = r187537 * r187544;
        double r187546 = r187516 + r187528;
        double r187547 = r187520 + r187546;
        double r187548 = sqrt(r187547);
        double r187549 = r187526 / r187548;
        double r187550 = sqrt(r187549);
        double r187551 = r187541 + r187520;
        double r187552 = r187551 + r187516;
        double r187553 = r187552 + r187528;
        double r187554 = r187550 / r187553;
        double r187555 = r187545 * r187554;
        double r187556 = r187547 * r187547;
        double r187557 = r187556 - r187540;
        double r187558 = r187519 + r187529;
        double r187559 = r187558 / r187547;
        double r187560 = r187519 * r187559;
        double r187561 = r187557 / r187560;
        double r187562 = r187519 / r187561;
        double r187563 = r187518 ? r187555 : r187562;
        return r187563;
}

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)))