Average Error: 0.2 → 0.2
Time: 22.5s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(\left(a - \frac{1.0}{3.0}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand\]
\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)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(\left(a - \frac{1.0}{3.0}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot rand
double f(double a, double rand) {
        double r3932705 = a;
        double r3932706 = 1.0;
        double r3932707 = /* ERROR: no posit support in C */;
        double r3932708 = 3.0;
        double r3932709 = /* ERROR: no posit support in C */;
        double r3932710 = r3932707 / r3932709;
        double r3932711 = r3932705 - r3932710;
        double r3932712 = 1.0;
        double r3932713 = /* ERROR: no posit support in C */;
        double r3932714 = 9.0;
        double r3932715 = /* ERROR: no posit support in C */;
        double r3932716 = r3932715 * r3932711;
        double r3932717 = sqrt(r3932716);
        double r3932718 = r3932713 / r3932717;
        double r3932719 = rand;
        double r3932720 = r3932718 * r3932719;
        double r3932721 = r3932713 + r3932720;
        double r3932722 = r3932711 * r3932721;
        return r3932722;
}
↓
double f(double a, double rand) {
        double r3932723 = a;
        double r3932724 = 1.0;
        double r3932725 = 3.0;
        double r3932726 = r3932724 / r3932725;
        double r3932727 = r3932723 - r3932726;
        double r3932728 = 1.0;
        double r3932729 = r3932727 * r3932728;
        double r3932730 = 9.0;
        double r3932731 = r3932730 * r3932727;
        double r3932732 = sqrt(r3932731);
        double r3932733 = r3932728 / r3932732;
        double r3932734 = r3932727 * r3932733;
        double r3932735 = rand;
        double r3932736 = r3932734 * r3932735;
        double r3932737 = r3932729 + r3932736;
        return r3932737;
}

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-in0.2

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

  5. Applied associate-*r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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))))