Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 24.8s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r1210565 = a;
        double r1210566 = 1.0;
        double r1210567 = /* ERROR: no posit support in C */;
        double r1210568 = 3.0;
        double r1210569 = /* ERROR: no posit support in C */;
        double r1210570 = r1210567 / r1210569;
        double r1210571 = r1210565 - r1210570;
        double r1210572 = 1.0;
        double r1210573 = /* ERROR: no posit support in C */;
        double r1210574 = 9.0;
        double r1210575 = /* ERROR: no posit support in C */;
        double r1210576 = r1210575 * r1210571;
        double r1210577 = sqrt(r1210576);
        double r1210578 = r1210573 / r1210577;
        double r1210579 = rand;
        double r1210580 = r1210578 * r1210579;
        double r1210581 = r1210573 + r1210580;
        double r1210582 = r1210571 * r1210581;
        return r1210582;
}

↓

double f(double a, double rand) {
        double r1210583 = rand;
        double r1210584 = 1.0;
        double r1210585 = r1210583 * r1210584;
        double r1210586 = 9.0;
        double r1210587 = a;
        double r1210588 = 1.0;
        double r1210589 = 3.0;
        double r1210590 = r1210588 / r1210589;
        double r1210591 = r1210587 - r1210590;
        double r1210592 = r1210586 * r1210591;
        double r1210593 = sqrt(r1210592);
        double r1210594 = r1210585 / r1210593;
        double r1210595 = r1210594 * r1210591;
        double r1210596 = r1210591 * r1210584;
        double r1210597 = r1210595 + r1210596;
        return r1210597;
}

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

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

9. Applied p16-flip-- 0.2

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

10. Simplified 0.2

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

11. Simplified 0.2

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

12. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019153 
(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))))