Average Error: 0.2 → 0.2
Time: 32.4s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot \left(9 \cdot \left(a + \frac{1.0}{3.0}\right)\right)}{a + \frac{1.0}{3.0}}}}\right)\]
\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{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot \left(9 \cdot \left(a + \frac{1.0}{3.0}\right)\right)}{a + \frac{1.0}{3.0}}}}\right)
double f(double a, double rand) {
        double r2893829 = a;
        double r2893830 = 1.0;
        double r2893831 = /* ERROR: no posit support in C */;
        double r2893832 = 3.0;
        double r2893833 = /* ERROR: no posit support in C */;
        double r2893834 = r2893831 / r2893833;
        double r2893835 = r2893829 - r2893834;
        double r2893836 = 1.0;
        double r2893837 = /* ERROR: no posit support in C */;
        double r2893838 = 9.0;
        double r2893839 = /* ERROR: no posit support in C */;
        double r2893840 = r2893839 * r2893835;
        double r2893841 = sqrt(r2893840);
        double r2893842 = r2893837 / r2893841;
        double r2893843 = rand;
        double r2893844 = r2893842 * r2893843;
        double r2893845 = r2893837 + r2893844;
        double r2893846 = r2893835 * r2893845;
        return r2893846;
}
↓
double f(double a, double rand) {
        double r2893847 = a;
        double r2893848 = 1.0;
        double r2893849 = 3.0;
        double r2893850 = r2893848 / r2893849;
        double r2893851 = r2893847 - r2893850;
        double r2893852 = 1.0;
        double r2893853 = rand;
        double r2893854 = r2893853 * r2893852;
        double r2893855 = 9.0;
        double r2893856 = r2893847 + r2893850;
        double r2893857 = r2893855 * r2893856;
        double r2893858 = r2893851 * r2893857;
        double r2893859 = r2893858 / r2893856;
        double r2893860 = sqrt(r2893859);
        double r2893861 = r2893854 / r2893860;
        double r2893862 = r2893852 + r2893861;
        double r2893863 = r2893851 * r2893862;
        return r2893863;
}

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 p16-flip--0.2

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

  8. Applied associate-*l/0.2

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

  9. Simplified0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019162 +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))))