Average Error: 0.2 → 0.2
Time: 46.8s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)\]
\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 \left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)
double f(double a, double rand) {
        double r5997629 = a;
        double r5997630 = 1.0;
        double r5997631 = /* ERROR: no posit support in C */;
        double r5997632 = 3.0;
        double r5997633 = /* ERROR: no posit support in C */;
        double r5997634 = r5997631 / r5997633;
        double r5997635 = r5997629 - r5997634;
        double r5997636 = 1.0;
        double r5997637 = /* ERROR: no posit support in C */;
        double r5997638 = 9.0;
        double r5997639 = /* ERROR: no posit support in C */;
        double r5997640 = r5997639 * r5997635;
        double r5997641 = sqrt(r5997640);
        double r5997642 = r5997637 / r5997641;
        double r5997643 = rand;
        double r5997644 = r5997642 * r5997643;
        double r5997645 = r5997637 + r5997644;
        double r5997646 = r5997635 * r5997645;
        return r5997646;
}
↓
double f(double a, double rand) {
        double r5997647 = a;
        double r5997648 = 1.0;
        double r5997649 = 3.0;
        double r5997650 = r5997648 / r5997649;
        double r5997651 = r5997647 - r5997650;
        double r5997652 = 1.0;
        double r5997653 = rand;
        double r5997654 = r5997653 * r5997652;
        double r5997655 = 9.0;
        double r5997656 = r5997655 * r5997651;
        double r5997657 = sqrt(r5997656);
        double r5997658 = r5997654 / r5997657;
        double r5997659 = r5997652 + r5997658;
        double r5997660 = r5997651 * r5997659;
        return r5997660;
}

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 p16-flip--0.2

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

  4. Applied associate-*r/0.2

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

  5. Simplified0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019134 +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))))