Average Error: 0.2 → 0.2
Time: 30.1s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot a + 9 \cdot \left(-\frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\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{9 \cdot a + 9 \cdot \left(-\frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r3023302 = a;
        double r3023303 = 1.0;
        double r3023304 = /* ERROR: no posit support in C */;
        double r3023305 = 3.0;
        double r3023306 = /* ERROR: no posit support in C */;
        double r3023307 = r3023304 / r3023306;
        double r3023308 = r3023302 - r3023307;
        double r3023309 = 1.0;
        double r3023310 = /* ERROR: no posit support in C */;
        double r3023311 = 9.0;
        double r3023312 = /* ERROR: no posit support in C */;
        double r3023313 = r3023312 * r3023308;
        double r3023314 = sqrt(r3023313);
        double r3023315 = r3023310 / r3023314;
        double r3023316 = rand;
        double r3023317 = r3023315 * r3023316;
        double r3023318 = r3023310 + r3023317;
        double r3023319 = r3023308 * r3023318;
        return r3023319;
}
↓
double f(double a, double rand) {
        double r3023320 = a;
        double r3023321 = 1.0;
        double r3023322 = 3.0;
        double r3023323 = r3023321 / r3023322;
        double r3023324 = r3023320 - r3023323;
        double r3023325 = 1.0;
        double r3023326 = 9.0;
        double r3023327 = r3023326 * r3023320;
        double r3023328 = -r3023323;
        double r3023329 = r3023326 * r3023328;
        double r3023330 = r3023327 + r3023329;
        double r3023331 = sqrt(r3023330);
        double r3023332 = r3023325 / r3023331;
        double r3023333 = rand;
        double r3023334 = r3023332 * r3023333;
        double r3023335 = r3023325 + r3023334;
        double r3023336 = r3023324 * r3023335;
        return r3023336;
}

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 sub-neg0.2

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

  4. Applied distribute-lft-in0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 
(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))))