Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 31.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r2646167 = a;
        double r2646168 = 1.0;
        double r2646169 = /* ERROR: no posit support in C */;
        double r2646170 = 3.0;
        double r2646171 = /* ERROR: no posit support in C */;
        double r2646172 = r2646169 / r2646171;
        double r2646173 = r2646167 - r2646172;
        double r2646174 = 1.0;
        double r2646175 = /* ERROR: no posit support in C */;
        double r2646176 = 9.0;
        double r2646177 = /* ERROR: no posit support in C */;
        double r2646178 = r2646177 * r2646173;
        double r2646179 = sqrt(r2646178);
        double r2646180 = r2646175 / r2646179;
        double r2646181 = rand;
        double r2646182 = r2646180 * r2646181;
        double r2646183 = r2646175 + r2646182;
        double r2646184 = r2646173 * r2646183;
        return r2646184;
}

↓

double f(double a, double rand) {
        double r2646185 = 1.0;
        double r2646186 = a;
        double r2646187 = 1.0;
        double r2646188 = 3.0;
        double r2646189 = r2646187 / r2646188;
        double r2646190 = r2646186 - r2646189;
        double r2646191 = r2646185 * r2646190;
        double r2646192 = rand;
        double r2646193 = r2646192 / r2646187;
        double r2646194 = 9.0;
        double r2646195 = r2646194 * r2646186;
        double r2646196 = -r2646189;
        double r2646197 = r2646194 * r2646196;
        double r2646198 = r2646195 + r2646197;
        double r2646199 = sqrt(r2646198);
        double r2646200 = r2646191 / r2646199;
        double r2646201 = r2646193 * r2646200;
        double r2646202 = r2646191 + r2646201;
        return r2646202;
}

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 *p16-rgt-identity-expand 0.2

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

4. Applied associate-*l* 0.2

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

5. Simplified 0.2

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

7. Applied distribute-rgt-in 0.2

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

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

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

10. Applied p16-times-frac 0.2

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

11. Applied associate-*l* 0.2

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

12. Simplified 0.2

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

14. Applied sub-neg 0.2

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

15. Applied distribute-lft-in 0.2

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

16. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019158 
(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))))