Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 51.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r2778894 = a;
        double r2778895 = 1.0;
        double r2778896 = /* ERROR: no posit support in C */;
        double r2778897 = 3.0;
        double r2778898 = /* ERROR: no posit support in C */;
        double r2778899 = r2778896 / r2778898;
        double r2778900 = r2778894 - r2778899;
        double r2778901 = 1.0;
        double r2778902 = /* ERROR: no posit support in C */;
        double r2778903 = 9.0;
        double r2778904 = /* ERROR: no posit support in C */;
        double r2778905 = r2778904 * r2778900;
        double r2778906 = sqrt(r2778905);
        double r2778907 = r2778902 / r2778906;
        double r2778908 = rand;
        double r2778909 = r2778907 * r2778908;
        double r2778910 = r2778902 + r2778909;
        double r2778911 = r2778900 * r2778910;
        return r2778911;
}

↓

double f(double a, double rand) {
        double r2778912 = 1.0;
        double r2778913 = a;
        double r2778914 = 1.0;
        double r2778915 = 3.0;
        double r2778916 = r2778914 / r2778915;
        double r2778917 = r2778913 - r2778916;
        double r2778918 = r2778912 * r2778917;
        double r2778919 = rand;
        double r2778920 = 9.0;
        double r2778921 = r2778917 * r2778920;
        double r2778922 = sqrt(r2778921);
        double r2778923 = r2778919 / r2778922;
        double r2778924 = r2778923 * r2778918;
        double r2778925 = r2778918 + r2778924;
        return r2778925;
}

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

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

7. Applied +p16-lft-identity-expand 0.2

   \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\color{blue}{\left(\frac{\left(0.0\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right)}\right)}}\right)\]

8. Applied associate-+r+ 0.2

   \[\leadsto \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \color{blue}{\left(\frac{\left(\frac{\left(1\right)}{\left(0.0\right)}\right)}{\left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right)}\right)}\]

9. Applied distribute-lft-in 0.2

   \[\leadsto \color{blue}{\frac{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(0.0\right)}\right)\right)}{\left(\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(rand \cdot \left(1\right)\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right)\right)}}\]

10. Simplified 0.2

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

11. Simplified 0.2

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

12. Final simplification 0.2

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

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))))