Average Error: 0.2 → 0.2
Time: 17.1s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[1 \cdot a + \left(1 \cdot \left(-\frac{1.0}{3.0}\right) + \left(\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot a + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(-\frac{1.0}{3.0}\right)\right)\right)\]
\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)
↓
1 \cdot a + \left(1 \cdot \left(-\frac{1.0}{3.0}\right) + \left(\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot a + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(-\frac{1.0}{3.0}\right)\right)\right)
double f(double a, double rand) {
        double r1633667 = a;
        double r1633668 = 1.0;
        double r1633669 = /* ERROR: no posit support in C */;
        double r1633670 = 3.0;
        double r1633671 = /* ERROR: no posit support in C */;
        double r1633672 = r1633669 / r1633671;
        double r1633673 = r1633667 - r1633672;
        double r1633674 = 1.0;
        double r1633675 = /* ERROR: no posit support in C */;
        double r1633676 = 9.0;
        double r1633677 = /* ERROR: no posit support in C */;
        double r1633678 = r1633677 * r1633673;
        double r1633679 = sqrt(r1633678);
        double r1633680 = r1633675 / r1633679;
        double r1633681 = rand;
        double r1633682 = r1633680 * r1633681;
        double r1633683 = r1633675 + r1633682;
        double r1633684 = r1633673 * r1633683;
        return r1633684;
}
↓
double f(double a, double rand) {
        double r1633685 = 1.0;
        double r1633686 = a;
        double r1633687 = r1633685 * r1633686;
        double r1633688 = 1.0;
        double r1633689 = 3.0;
        double r1633690 = r1633688 / r1633689;
        double r1633691 = -r1633690;
        double r1633692 = r1633685 * r1633691;
        double r1633693 = 9.0;
        double r1633694 = r1633686 - r1633690;
        double r1633695 = r1633693 * r1633694;
        double r1633696 = sqrt(r1633695);
        double r1633697 = r1633685 / r1633696;
        double r1633698 = rand;
        double r1633699 = r1633697 * r1633698;
        double r1633700 = r1633699 * r1633686;
        double r1633701 = r1633699 * r1633691;
        double r1633702 = r1633700 + r1633701;
        double r1633703 = r1633692 + r1633702;
        double r1633704 = r1633687 + r1633703;
        return r1633704;
}

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-rgt-in0.2

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

  5. Applied sub-neg0.2

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

  6. Applied distribute-lft-in0.2

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

  7. Applied associate-+l+0.2

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

  9. Applied sub-neg0.2

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

  10. Applied distribute-lft-in0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019128 
(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))))