Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 21.5s

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 r3406419 = a;
        double r3406420 = 1.0;
        double r3406421 = 3.0;
        double r3406422 = r3406420 / r3406421;
        double r3406423 = r3406419 - r3406422;
        double r3406424 = 1.0;
        double r3406425 = 9.0;
        double r3406426 = r3406425 * r3406423;
        double r3406427 = sqrt(r3406426);
        double r3406428 = r3406424 / r3406427;
        double r3406429 = rand;
        double r3406430 = r3406428 * r3406429;
        double r3406431 = r3406424 + r3406430;
        double r3406432 = r3406423 * r3406431;
        return r3406432;
}

↓

double f(double a, double rand) {
        double r3406433 = a;
        double r3406434 = 1.0;
        double r3406435 = 3.0;
        double r3406436 = r3406434 / r3406435;
        double r3406437 = r3406433 - r3406436;
        double r3406438 = sqrt(r3406437);
        double r3406439 = 3.0;
        double r3406440 = r3406438 / r3406439;
        double r3406441 = rand;
        double r3406442 = r3406440 * r3406441;
        double r3406443 = r3406442 + r3406437;
        return r3406443;
}

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 2019162 
(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))))