Average Error: 0.2 → 0.2
Time: 5.4m
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 r3422765 = a;
        double r3422766 = 1.0;
        double r3422767 = /* ERROR: no posit support in C */;
        double r3422768 = 3.0;
        double r3422769 = /* ERROR: no posit support in C */;
        double r3422770 = r3422767 / r3422769;
        double r3422771 = r3422765 - r3422770;
        double r3422772 = 1.0;
        double r3422773 = /* ERROR: no posit support in C */;
        double r3422774 = 9.0;
        double r3422775 = /* ERROR: no posit support in C */;
        double r3422776 = r3422775 * r3422771;
        double r3422777 = sqrt(r3422776);
        double r3422778 = r3422773 / r3422777;
        double r3422779 = rand;
        double r3422780 = r3422778 * r3422779;
        double r3422781 = r3422773 + r3422780;
        double r3422782 = r3422771 * r3422781;
        return r3422782;
}
↓
double f(double a, double rand) {
        double r3422783 = a;
        double r3422784 = 1.0;
        double r3422785 = 3.0;
        double r3422786 = r3422784 / r3422785;
        double r3422787 = r3422783 - r3422786;
        double r3422788 = 1.0;
        double r3422789 = r3422787 * r3422788;
        double r3422790 = rand;
        double r3422791 = r3422790 * r3422788;
        double r3422792 = 9.0;
        double r3422793 = r3422787 * r3422792;
        double r3422794 = sqrt(r3422793);
        double r3422795 = r3422791 / r3422794;
        double r3422796 = r3422787 * r3422795;
        double r3422797 = r3422789 + r3422796;
        return r3422797;
}

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 2019158 +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))))