Average Error: 0.2 → 0.2
Time: 46.4s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)\]
\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)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right)
double f(double a, double rand) {
        double r5738219 = a;
        double r5738220 = 1.0;
        double r5738221 = /* ERROR: no posit support in C */;
        double r5738222 = 3.0;
        double r5738223 = /* ERROR: no posit support in C */;
        double r5738224 = r5738221 / r5738223;
        double r5738225 = r5738219 - r5738224;
        double r5738226 = 1.0;
        double r5738227 = /* ERROR: no posit support in C */;
        double r5738228 = 9.0;
        double r5738229 = /* ERROR: no posit support in C */;
        double r5738230 = r5738229 * r5738225;
        double r5738231 = sqrt(r5738230);
        double r5738232 = r5738227 / r5738231;
        double r5738233 = rand;
        double r5738234 = r5738232 * r5738233;
        double r5738235 = r5738227 + r5738234;
        double r5738236 = r5738225 * r5738235;
        return r5738236;
}
↓
double f(double a, double rand) {
        double r5738237 = a;
        double r5738238 = 1.0;
        double r5738239 = 3.0;
        double r5738240 = r5738238 / r5738239;
        double r5738241 = r5738237 - r5738240;
        double r5738242 = 1.0;
        double r5738243 = rand;
        double r5738244 = r5738243 * r5738242;
        double r5738245 = 9.0;
        double r5738246 = r5738245 * r5738241;
        double r5738247 = sqrt(r5738246);
        double r5738248 = r5738244 / r5738247;
        double r5738249 = r5738242 + r5738248;
        double r5738250 = r5738241 * r5738249;
        return r5738250;
}

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. Simplified0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019134 
(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))))