Octave 3.8, jcobi/4

Average Error: 3.4 → 1.6

Time: 1.2m

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(1\right)\]

Math

\[\frac{\left(\frac{\left(\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right) \cdot \left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

↓

\[\frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\frac{\alpha + \left(\beta + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(i \cdot \left(\alpha + \left(\beta + i\right)\right)\right)\right) + \frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\frac{\alpha + \left(\beta + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\beta \cdot \alpha\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

C

double f(double alpha, double beta, double i) {
        double r3659501 = i;
        double r3659502 = alpha;
        double r3659503 = beta;
        double r3659504 = r3659502 + r3659503;
        double r3659505 = r3659504 + r3659501;
        double r3659506 = r3659501 * r3659505;
        double r3659507 = r3659503 * r3659502;
        double r3659508 = r3659507 + r3659506;
        double r3659509 = r3659506 * r3659508;
        double r3659510 = 2.0;
        double r3659511 = /* ERROR: no posit support in C */;
        double r3659512 = r3659511 * r3659501;
        double r3659513 = r3659504 + r3659512;
        double r3659514 = r3659513 * r3659513;
        double r3659515 = r3659509 / r3659514;
        double r3659516 = 1.0;
        double r3659517 = /* ERROR: no posit support in C */;
        double r3659518 = r3659514 - r3659517;
        double r3659519 = r3659515 / r3659518;
        return r3659519;
}

↓

double f(double alpha, double beta, double i) {
        double r3659520 = i;
        double r3659521 = alpha;
        double r3659522 = beta;
        double r3659523 = r3659521 + r3659522;
        double r3659524 = 2.0;
        double r3659525 = r3659524 * r3659520;
        double r3659526 = r3659523 + r3659525;
        double r3659527 = r3659520 / r3659526;
        double r3659528 = r3659522 + r3659520;
        double r3659529 = r3659521 + r3659528;
        double r3659530 = r3659529 / r3659526;
        double r3659531 = r3659520 * r3659529;
        double r3659532 = r3659530 * r3659531;
        double r3659533 = r3659527 * r3659532;
        double r3659534 = r3659522 * r3659521;
        double r3659535 = r3659530 * r3659534;
        double r3659536 = r3659527 * r3659535;
        double r3659537 = r3659533 + r3659536;
        double r3659538 = r3659526 * r3659526;
        double r3659539 = 1.0;
        double r3659540 = r3659538 - r3659539;
        double r3659541 = r3659537 / r3659540;
        return r3659541;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Initial program 3.4

   \[\frac{\left(\frac{\left(\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right) \cdot \left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]
2. Using strategy rm

3. Applied p16-times-frac 1.8

   \[\leadsto \frac{\color{blue}{\left(\left(\frac{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]
4. Using strategy rm

5. Applied *p16-rgt-identity-expand 1.8

   \[\leadsto \frac{\left(\left(\frac{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(1.0\right)\right)}}\right) \cdot \left(\frac{\left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

6. Applied p16-times-frac 1.6

   \[\leadsto \frac{\left(\color{blue}{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)}{\left(1.0\right)}\right)\right)} \cdot \left(\frac{\left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

7. Applied associate-*l* 1.7

   \[\leadsto \frac{\color{blue}{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(\frac{\left(\beta \cdot \alpha\right)}{\left(i \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{i}\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)\right)\right)}}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

8. Simplified 1.7

   \[\leadsto \frac{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\frac{\left(i \cdot \left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)\right)}{\left(\beta \cdot \alpha\right)}\right)\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]
9. Using strategy rm

10. Applied distribute-lft-in 1.7

    \[\leadsto \frac{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \color{blue}{\left(\frac{\left(\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(i \cdot \left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)\right)\right)}{\left(\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\beta \cdot \alpha\right)\right)}\right)}\right)}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

11. Applied distribute-lft-in 1.6

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(i \cdot \left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)\right)\right)\right)}{\left(\left(\frac{i}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{i}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right) \cdot \left(\beta \cdot \alpha\right)\right)\right)}\right)}}{\left(\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right) \cdot \left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)\right) - \left(1.0\right)\right)}\]

12. Final simplification 1.6

    \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\frac{\alpha + \left(\beta + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(i \cdot \left(\alpha + \left(\beta + i\right)\right)\right)\right) + \frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\frac{\alpha + \left(\beta + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\beta \cdot \alpha\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 1)))
  (/.p16 (/.p16 (*.p16 (*.p16 i (+.p16 (+.p16 alpha beta) i)) (+.p16 (*.p16 beta alpha) (*.p16 i (+.p16 (+.p16 alpha beta) i)))) (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)))) (-.p16 (*.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (real->posit16 1.0))))