Octave 3.8, oct_fill_randg

Average Error: 0.2 → 0.2

Time: 36.1s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r2730012 = a;
        double r2730013 = 1.0;
        double r2730014 = /* ERROR: no posit support in C */;
        double r2730015 = 3.0;
        double r2730016 = /* ERROR: no posit support in C */;
        double r2730017 = r2730014 / r2730016;
        double r2730018 = r2730012 - r2730017;
        double r2730019 = 1.0;
        double r2730020 = /* ERROR: no posit support in C */;
        double r2730021 = 9.0;
        double r2730022 = /* ERROR: no posit support in C */;
        double r2730023 = r2730022 * r2730018;
        double r2730024 = sqrt(r2730023);
        double r2730025 = r2730020 / r2730024;
        double r2730026 = rand;
        double r2730027 = r2730025 * r2730026;
        double r2730028 = r2730020 + r2730027;
        double r2730029 = r2730018 * r2730028;
        return r2730029;
}

↓

double f(double a, double rand) {
        double r2730030 = a;
        double r2730031 = 1.0;
        double r2730032 = /* ERROR: no posit support in C */;
        double r2730033 = 3.0;
        double r2730034 = /* ERROR: no posit support in C */;
        double r2730035 = r2730032 / r2730034;
        double r2730036 = r2730030 - r2730035;
        double r2730037 = 1.0;
        double r2730038 = /* ERROR: no posit support in C */;
        double r2730039 = r2730036 * r2730038;
        double r2730040 = rand;
        double r2730041 = r2730038 * r2730040;
        double r2730042 = r2730041 * r2730036;
        double r2730043 = 9.0;
        double r2730044 = /* ERROR: no posit support in C */;
        double r2730045 = r2730036 * r2730044;
        double r2730046 = sqrt(r2730045);
        double r2730047 = r2730042 / r2730046;
        double r2730048 = r2730030 + r2730035;
        double r2730049 = r2730047 * r2730048;
        double r2730050 = r2730049 / r2730048;
        double r2730051 = r2730039 + r2730050;
        return r2730051;
}

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 distribute-lft-in 0.2

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

5. Applied sub-neg 0.2

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

6. Applied distribute-lft-in 0.2

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

8. Applied p16-flip-- 0.2

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

9. Applied associate-*l/ 0.3

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

10. Simplified 0.2

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

11. Final simplification 0.2

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

Reproduce

herbie shell --seed 2019164 
(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))))