Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 41.2s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r1412139 = a;
        double r1412140 = 1.0;
        double r1412141 = /* ERROR: no posit support in C */;
        double r1412142 = 3.0;
        double r1412143 = /* ERROR: no posit support in C */;
        double r1412144 = r1412141 / r1412143;
        double r1412145 = r1412139 - r1412144;
        double r1412146 = 1.0;
        double r1412147 = /* ERROR: no posit support in C */;
        double r1412148 = 9.0;
        double r1412149 = /* ERROR: no posit support in C */;
        double r1412150 = r1412149 * r1412145;
        double r1412151 = sqrt(r1412150);
        double r1412152 = r1412147 / r1412151;
        double r1412153 = rand;
        double r1412154 = r1412152 * r1412153;
        double r1412155 = r1412147 + r1412154;
        double r1412156 = r1412145 * r1412155;
        return r1412156;
}

↓

double f(double a, double rand) {
        double r1412157 = a;
        double r1412158 = 1.0;
        double r1412159 = 3.0;
        double r1412160 = r1412158 / r1412159;
        double r1412161 = r1412157 - r1412160;
        double r1412162 = 1.0;
        double r1412163 = r1412161 * r1412162;
        double r1412164 = rand;
        double r1412165 = r1412164 * r1412161;
        double r1412166 = 9.0;
        double r1412167 = r1412166 * r1412161;
        double r1412168 = sqrt(r1412167);
        double r1412169 = r1412162 / r1412168;
        double r1412170 = r1412165 * r1412169;
        double r1412171 = r1412163 + r1412170;
        return r1412171;
}

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-rgt-in 0.2

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

6. Applied distribute-lft-in 0.2

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

7. Simplified 0.2

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

9. Applied *p16-lft-identity-expand 0.2

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

10. Applied p16-times-frac 0.2

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

11. Simplified 0.2

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

13. Applied p16-*-un-lft-identity 0.2

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

14. Applied p16-times-frac 0.2

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

15. Applied associate-*r* 0.2

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

16. Simplified 0.2

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

17. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019152 +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))))