Average Error: 0.2 → 0.2
Time: 20.8s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \frac{a - \frac{1.0}{3.0}}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(1 \cdot rand\right)\]
double f(double a, double rand) {
        double r1616608 = a;
        double r1616609 = 1.0;
        double r1616610 = 3.0;
        double r1616611 = r1616609 / r1616610;
        double r1616612 = r1616608 - r1616611;
        double r1616613 = 1.0;
        double r1616614 = 9.0;
        double r1616615 = r1616614 * r1616612;
        double r1616616 = sqrt(r1616615);
        double r1616617 = r1616613 / r1616616;
        double r1616618 = rand;
        double r1616619 = r1616617 * r1616618;
        double r1616620 = r1616613 + r1616619;
        double r1616621 = r1616612 * r1616620;
        return r1616621;
}
↓
double f(double a, double rand) {
        double r1616622 = a;
        double r1616623 = 1.0;
        double r1616624 = 3.0;
        double r1616625 = r1616623 / r1616624;
        double r1616626 = r1616622 - r1616625;
        double r1616627 = 1.0;
        double r1616628 = r1616626 * r1616627;
        double r1616629 = 9.0;
        double r1616630 = r1616626 * r1616629;
        double r1616631 = sqrt(r1616630);
        double r1616632 = r1616626 / r1616631;
        double r1616633 = rand;
        double r1616634 = r1616627 * r1616633;
        double r1616635 = r1616632 * r1616634;
        double r1616636 = r1616628 + r1616635;
        return r1616636;
}


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

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 sub-neg0.2

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

  4. Applied distribute-rgt-in0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Simplified0.2

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

  9. Applied associate-/l*0.2

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

  11. Applied associate-/r/0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019101 
(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))))