Average Error: 0.2 → 0.2
Time: 28.8s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \frac{rand \cdot 1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\]
\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 \frac{rand \cdot 1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}
double f(double a, double rand) {
        double r1650468 = a;
        double r1650469 = 1.0;
        double r1650470 = /* ERROR: no posit support in C */;
        double r1650471 = 3.0;
        double r1650472 = /* ERROR: no posit support in C */;
        double r1650473 = r1650470 / r1650472;
        double r1650474 = r1650468 - r1650473;
        double r1650475 = 1.0;
        double r1650476 = /* ERROR: no posit support in C */;
        double r1650477 = 9.0;
        double r1650478 = /* ERROR: no posit support in C */;
        double r1650479 = r1650478 * r1650474;
        double r1650480 = sqrt(r1650479);
        double r1650481 = r1650476 / r1650480;
        double r1650482 = rand;
        double r1650483 = r1650481 * r1650482;
        double r1650484 = r1650476 + r1650483;
        double r1650485 = r1650474 * r1650484;
        return r1650485;
}
↓
double f(double a, double rand) {
        double r1650486 = a;
        double r1650487 = 1.0;
        double r1650488 = 3.0;
        double r1650489 = r1650487 / r1650488;
        double r1650490 = r1650486 - r1650489;
        double r1650491 = 1.0;
        double r1650492 = r1650490 * r1650491;
        double r1650493 = rand;
        double r1650494 = r1650493 * r1650491;
        double r1650495 = 9.0;
        double r1650496 = r1650490 * r1650495;
        double r1650497 = sqrt(r1650496);
        double r1650498 = r1650494 / r1650497;
        double r1650499 = r1650490 * r1650498;
        double r1650500 = r1650492 + r1650499;
        return r1650500;
}

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-rgt-identity-expand0.2

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

  4. Applied associate-*l*0.2

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

  5. Simplified0.2

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

  7. Applied distribute-lft-in0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019165 
(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))))