Average Error: 0.2 → 0.2
Time: 26.8s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\]
\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 + \frac{\left(a - \frac{1.0}{3.0}\right) \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand
double f(double a, double rand) {
        double r1261562 = a;
        double r1261563 = 1.0;
        double r1261564 = /* ERROR: no posit support in C */;
        double r1261565 = 3.0;
        double r1261566 = /* ERROR: no posit support in C */;
        double r1261567 = r1261564 / r1261566;
        double r1261568 = r1261562 - r1261567;
        double r1261569 = 1.0;
        double r1261570 = /* ERROR: no posit support in C */;
        double r1261571 = 9.0;
        double r1261572 = /* ERROR: no posit support in C */;
        double r1261573 = r1261572 * r1261568;
        double r1261574 = sqrt(r1261573);
        double r1261575 = r1261570 / r1261574;
        double r1261576 = rand;
        double r1261577 = r1261575 * r1261576;
        double r1261578 = r1261570 + r1261577;
        double r1261579 = r1261568 * r1261578;
        return r1261579;
}
↓
double f(double a, double rand) {
        double r1261580 = a;
        double r1261581 = 1.0;
        double r1261582 = 3.0;
        double r1261583 = r1261581 / r1261582;
        double r1261584 = r1261580 - r1261583;
        double r1261585 = 1.0;
        double r1261586 = r1261584 * r1261585;
        double r1261587 = 9.0;
        double r1261588 = r1261587 * r1261584;
        double r1261589 = sqrt(r1261588);
        double r1261590 = r1261586 / r1261589;
        double r1261591 = rand;
        double r1261592 = r1261590 * r1261591;
        double r1261593 = r1261586 + r1261592;
        return r1261593;
}

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. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019155 
(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))))