Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 29.4s

Precision: 64

Math

\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

↓

\[1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)\]

C

double f(double a, double rand) {
        double r77975 = a;
        double r77976 = 1.0;
        double r77977 = 3.0;
        double r77978 = r77976 / r77977;
        double r77979 = r77975 - r77978;
        double r77980 = 9.0;
        double r77981 = r77980 * r77979;
        double r77982 = sqrt(r77981);
        double r77983 = r77976 / r77982;
        double r77984 = rand;
        double r77985 = r77983 * r77984;
        double r77986 = r77976 + r77985;
        double r77987 = r77979 * r77986;
        return r77987;
}

↓

double f(double a, double rand) {
        double r77988 = 1.0;
        double r77989 = a;
        double r77990 = 3.0;
        double r77991 = r77988 / r77990;
        double r77992 = r77989 - r77991;
        double r77993 = r77988 * r77992;
        double r77994 = rand;
        double r77995 = 9.0;
        double r77996 = r77995 * r77992;
        double r77997 = sqrt(r77996);
        double r77998 = r77994 / r77997;
        double r77999 = r77988 * r77998;
        double r78000 = r77999 * r77992;
        double r78001 = r77993 + r78000;
        return r78001;
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

   \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
2. Using strategy rm

3. Applied distribute-lft-in 0.1

   \[\leadsto \color{blue}{\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)}\]

4. Simplified 0.1

   \[\leadsto \color{blue}{1 \cdot \left(a - \frac{1}{3}\right)} + \left(a - \frac{1}{3}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

5. Simplified 0.1

   \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \color{blue}{\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)}\]
6. Using strategy rm

7. Applied div-inv 0.1

   \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(\color{blue}{\left(1 \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right)} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)\]

8. Applied associate-*l* 0.1

   \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \color{blue}{\left(1 \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\right)} \cdot \left(a - \frac{1}{3}\right)\]

9. Simplified 0.1

   \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \color{blue}{\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}}\right) \cdot \left(a - \frac{1}{3}\right)\]
10. Using strategy rm

11. Applied pow1 0.1

    \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \color{blue}{{\left(a - \frac{1}{3}\right)}^{1}}}}\right) \cdot \left(a - \frac{1}{3}\right)\]

12. Applied pow1 0.1

    \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{\color{blue}{{9}^{1}} \cdot {\left(a - \frac{1}{3}\right)}^{1}}}\right) \cdot \left(a - \frac{1}{3}\right)\]

13. Applied pow-prod-down 0.1

    \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{\color{blue}{{\left(9 \cdot \left(a - \frac{1}{3}\right)\right)}^{1}}}}\right) \cdot \left(a - \frac{1}{3}\right)\]

14. Final simplification 0.1

    \[\leadsto 1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))