Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 58.2s

Precision: 64

Math

\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]

↓

\[\left(\mathsf{qma}\left(\left(\left(1 \cdot \left(a - \frac{1.0}{3.0}\right)\right)\right), \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right), \left(a - \frac{1.0}{3.0}\right)\right)\right)\]

C

double f(double a, double rand) {
        double r2235449 = a;
        double r2235450 = 1.0;
        double r2235451 = /* ERROR: no posit support in C */;
        double r2235452 = 3.0;
        double r2235453 = /* ERROR: no posit support in C */;
        double r2235454 = r2235451 / r2235453;
        double r2235455 = r2235449 - r2235454;
        double r2235456 = 1.0;
        double r2235457 = /* ERROR: no posit support in C */;
        double r2235458 = 9.0;
        double r2235459 = /* ERROR: no posit support in C */;
        double r2235460 = r2235459 * r2235455;
        double r2235461 = sqrt(r2235460);
        double r2235462 = r2235457 / r2235461;
        double r2235463 = rand;
        double r2235464 = r2235462 * r2235463;
        double r2235465 = r2235457 + r2235464;
        double r2235466 = r2235455 * r2235465;
        return r2235466;
}

↓

double f(double a, double rand) {
        double r2235467 = 1.0;
        double r2235468 = a;
        double r2235469 = 1.0;
        double r2235470 = 3.0;
        double r2235471 = r2235469 / r2235470;
        double r2235472 = r2235468 - r2235471;
        double r2235473 = r2235467 * r2235472;
        double r2235474 = /*Error: no posit support in C */;
        double r2235475 = 9.0;
        double r2235476 = r2235475 * r2235472;
        double r2235477 = sqrt(r2235476);
        double r2235478 = r2235467 / r2235477;
        double r2235479 = rand;
        double r2235480 = r2235478 * r2235479;
        double r2235481 = /*Error: no posit support in C */;
        double r2235482 = /*Error: no posit support in C */;
        return r2235482;
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.2

   \[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
2. Using strategy rm

3. Applied sub-neg 0.2

   \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \color{blue}{\left(\frac{a}{\left(-\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)}\right)}\right)}\right)}\right) \cdot rand\right)}\right)\]

4. Applied distribute-lft-in 0.2

   \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\color{blue}{\left(\frac{\left(\left(9\right) \cdot a\right)}{\left(\left(9\right) \cdot \left(-\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}}\right)}\right) \cdot rand\right)}\right)\]
5. Using strategy rm

6. Applied distribute-rgt-in 0.2

   \[\leadsto \color{blue}{\frac{\left(\left(1\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}{\left(\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\frac{\left(\left(9\right) \cdot a\right)}{\left(\left(9\right) \cdot \left(-\left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}}\]

7. Simplified 0.2

   \[\leadsto \frac{\left(\left(1\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}{\color{blue}{\left(\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}}\]
8. Using strategy rm

9. Applied introduce-quire 0.2

   \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(1\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)\right)\right)}}{\left(\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\]

10. Applied insert-quire-fdp-add 0.2

    \[\leadsto \color{blue}{\left(\mathsf{qma}\left(\left(\left(\left(1\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)\right), \left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right), \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)\right)}\]

11. Final simplification 0.2

    \[\leadsto \left(\mathsf{qma}\left(\left(\left(1 \cdot \left(a - \frac{1.0}{3.0}\right)\right)\right), \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right), \left(a - \frac{1.0}{3.0}\right)\right)\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))