Average Error: 0.2 → 0.2
Time: 24.6s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\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(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)
double f(double a, double rand) {
        double r2787538 = a;
        double r2787539 = 1.0;
        double r2787540 = /* ERROR: no posit support in C */;
        double r2787541 = 3.0;
        double r2787542 = /* ERROR: no posit support in C */;
        double r2787543 = r2787540 / r2787542;
        double r2787544 = r2787538 - r2787543;
        double r2787545 = 1.0;
        double r2787546 = /* ERROR: no posit support in C */;
        double r2787547 = 9.0;
        double r2787548 = /* ERROR: no posit support in C */;
        double r2787549 = r2787548 * r2787544;
        double r2787550 = sqrt(r2787549);
        double r2787551 = r2787546 / r2787550;
        double r2787552 = rand;
        double r2787553 = r2787551 * r2787552;
        double r2787554 = r2787546 + r2787553;
        double r2787555 = r2787544 * r2787554;
        return r2787555;
}
↓
double f(double a, double rand) {
        double r2787556 = a;
        double r2787557 = 1.0;
        double r2787558 = 3.0;
        double r2787559 = r2787557 / r2787558;
        double r2787560 = r2787556 - r2787559;
        double r2787561 = 1.0;
        double r2787562 = r2787560 * r2787561;
        double r2787563 = 9.0;
        double r2787564 = r2787563 * r2787560;
        double r2787565 = sqrt(r2787564);
        double r2787566 = r2787561 / r2787565;
        double r2787567 = rand;
        double r2787568 = r2787566 * r2787567;
        double r2787569 = r2787560 * r2787568;
        double r2787570 = r2787562 + r2787569;
        return r2787570;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 +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))))