Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 36.5s

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 r1447542 = a;
        double r1447543 = 1.0;
        double r1447544 = /* ERROR: no posit support in C */;
        double r1447545 = 3.0;
        double r1447546 = /* ERROR: no posit support in C */;
        double r1447547 = r1447544 / r1447546;
        double r1447548 = r1447542 - r1447547;
        double r1447549 = 1.0;
        double r1447550 = /* ERROR: no posit support in C */;
        double r1447551 = 9.0;
        double r1447552 = /* ERROR: no posit support in C */;
        double r1447553 = r1447552 * r1447548;
        double r1447554 = sqrt(r1447553);
        double r1447555 = r1447550 / r1447554;
        double r1447556 = rand;
        double r1447557 = r1447555 * r1447556;
        double r1447558 = r1447550 + r1447557;
        double r1447559 = r1447548 * r1447558;
        return r1447559;
}

↓

double f(double a, double rand) {
        double r1447560 = rand;
        double r1447561 = 1.0;
        double r1447562 = r1447560 * r1447561;
        double r1447563 = 9.0;
        double r1447564 = a;
        double r1447565 = 1.0;
        double r1447566 = 3.0;
        double r1447567 = r1447565 / r1447566;
        double r1447568 = r1447564 - r1447567;
        double r1447569 = r1447563 * r1447568;
        double r1447570 = sqrt(r1447569);
        double r1447571 = r1447562 / r1447570;
        double r1447572 = r1447571 * r1447568;
        double r1447573 = r1447568 * r1447561;
        double r1447574 = r1447572 + r1447573;
        return r1447574;
}

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-lft-in 0.2

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

7. Simplified 0.2

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

9. Applied p16-flip-- 0.2

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

10. Simplified 0.2

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

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 +o rules:numerics
(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))))