Average Error: 0.2 → 0.2
Time: 30.8s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{\frac{9 \cdot \left(a \cdot a - \frac{1.0}{3.0} \cdot \frac{1.0}{3.0}\right)}{a + \frac{1.0}{3.0}}}} \cdot rand\right)\]
\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{1}{\sqrt{\frac{9 \cdot \left(a \cdot a - \frac{1.0}{3.0} \cdot \frac{1.0}{3.0}\right)}{a + \frac{1.0}{3.0}}}} \cdot rand\right)
double f(double a, double rand) {
        double r2822045 = a;
        double r2822046 = 1.0;
        double r2822047 = /* ERROR: no posit support in C */;
        double r2822048 = 3.0;
        double r2822049 = /* ERROR: no posit support in C */;
        double r2822050 = r2822047 / r2822049;
        double r2822051 = r2822045 - r2822050;
        double r2822052 = 1.0;
        double r2822053 = /* ERROR: no posit support in C */;
        double r2822054 = 9.0;
        double r2822055 = /* ERROR: no posit support in C */;
        double r2822056 = r2822055 * r2822051;
        double r2822057 = sqrt(r2822056);
        double r2822058 = r2822053 / r2822057;
        double r2822059 = rand;
        double r2822060 = r2822058 * r2822059;
        double r2822061 = r2822053 + r2822060;
        double r2822062 = r2822051 * r2822061;
        return r2822062;
}
↓
double f(double a, double rand) {
        double r2822063 = a;
        double r2822064 = 1.0;
        double r2822065 = 3.0;
        double r2822066 = r2822064 / r2822065;
        double r2822067 = r2822063 - r2822066;
        double r2822068 = 1.0;
        double r2822069 = 9.0;
        double r2822070 = r2822063 * r2822063;
        double r2822071 = r2822066 * r2822066;
        double r2822072 = r2822070 - r2822071;
        double r2822073 = r2822069 * r2822072;
        double r2822074 = r2822063 + r2822066;
        double r2822075 = r2822073 / r2822074;
        double r2822076 = sqrt(r2822075);
        double r2822077 = r2822068 / r2822076;
        double r2822078 = rand;
        double r2822079 = r2822077 * r2822078;
        double r2822080 = r2822068 + r2822079;
        double r2822081 = r2822067 * r2822080;
        return r2822081;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019107 
(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))))