Octave 3.8, oct_fill_randg

Average Error: 0.1 → 0.1

Time: 18.8s

Precision: 64

Math

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

↓

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

C

double f(double a, double rand) {
        double r70735 = a;
        double r70736 = 1.0;
        double r70737 = 3.0;
        double r70738 = r70736 / r70737;
        double r70739 = r70735 - r70738;
        double r70740 = 9.0;
        double r70741 = r70740 * r70739;
        double r70742 = sqrt(r70741);
        double r70743 = r70736 / r70742;
        double r70744 = rand;
        double r70745 = r70743 * r70744;
        double r70746 = r70736 + r70745;
        double r70747 = r70739 * r70746;
        return r70747;
}

↓

double f(double a, double rand) {
        double r70748 = 1.0;
        double r70749 = a;
        double r70750 = 3.0;
        double r70751 = r70748 / r70750;
        double r70752 = r70749 - r70751;
        double r70753 = r70748 * r70752;
        double r70754 = rand;
        double r70755 = r70748 * r70754;
        double r70756 = 9.0;
        double r70757 = sqrt(r70756);
        double r70758 = r70755 / r70757;
        double r70759 = sqrt(r70752);
        double r70760 = r70758 * r70759;
        double r70761 = r70753 + r70760;
        return r70761;
}

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 sqrt-prod 0.1

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

4. Applied *-un-lft-identity 0.1

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

5. Applied times-frac 0.2

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

6. Applied associate-*l* 0.2

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

7. Simplified 0.1

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

9. Applied frac-times 0.1

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

10. Simplified 0.1

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

11. Simplified 0.1

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

13. Applied distribute-lft-in 0.1

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

14. Simplified 0.1

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

16. Applied add-sqr-sqrt 0.1

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

17. Applied associate-/l* 0.1

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

18. Simplified 0.1

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

19. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019179 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))