Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 21.2s

Precision: 64

Math

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

↓

\[\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

C

double f(double a, double rand) {
        double r4647031 = a;
        double r4647032 = 1.0;
        double r4647033 = 3.0;
        double r4647034 = r4647032 / r4647033;
        double r4647035 = r4647031 - r4647034;
        double r4647036 = 1.0;
        double r4647037 = 9.0;
        double r4647038 = r4647037 * r4647035;
        double r4647039 = sqrt(r4647038);
        double r4647040 = r4647036 / r4647039;
        double r4647041 = rand;
        double r4647042 = r4647040 * r4647041;
        double r4647043 = r4647036 + r4647042;
        double r4647044 = r4647035 * r4647043;
        return r4647044;
}

↓

double f(double a, double rand) {
        double r4647045 = a;
        double r4647046 = 1.0;
        double r4647047 = 3.0;
        double r4647048 = r4647046 / r4647047;
        double r4647049 = r4647045 - r4647048;
        double r4647050 = sqrt(r4647049);
        double r4647051 = 3.0;
        double r4647052 = r4647050 / r4647051;
        double r4647053 = rand;
        double r4647054 = r4647052 * r4647053;
        double r4647055 = r4647054 + r4647049;
        return r4647055;
}

Error

Bits error versus a

Bits error versus rand

Derivation

1. Initial program 0.1

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

3. Applied distribute-lft-in 0.1

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

4. Simplified 0.1

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

5. Simplified 0.1

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

7. Applied div-inv 0.1

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

8. Applied associate-*l* 0.1

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

9. Simplified 0.1

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

11. Applied sqrt-prod 0.1

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

12. Applied add-sqr-sqrt 0.1

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

13. Applied times-frac 0.1

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

14. Applied associate-*r* 0.1

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

15. Simplified 0.1

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

16. Final simplification 0.1

    \[\leadsto \frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot rand + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019163 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))