Average Error: 0.2 → 0.2
Time: 39.9s
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 r5196351 = a;
        double r5196352 = 1.0;
        double r5196353 = /* ERROR: no posit support in C */;
        double r5196354 = 3.0;
        double r5196355 = /* ERROR: no posit support in C */;
        double r5196356 = r5196353 / r5196355;
        double r5196357 = r5196351 - r5196356;
        double r5196358 = 1.0;
        double r5196359 = /* ERROR: no posit support in C */;
        double r5196360 = 9.0;
        double r5196361 = /* ERROR: no posit support in C */;
        double r5196362 = r5196361 * r5196357;
        double r5196363 = sqrt(r5196362);
        double r5196364 = r5196359 / r5196363;
        double r5196365 = rand;
        double r5196366 = r5196364 * r5196365;
        double r5196367 = r5196359 + r5196366;
        double r5196368 = r5196357 * r5196367;
        return r5196368;
}
↓
double f(double a, double rand) {
        double r5196369 = a;
        double r5196370 = 1.0;
        double r5196371 = 3.0;
        double r5196372 = r5196370 / r5196371;
        double r5196373 = r5196369 - r5196372;
        double r5196374 = 1.0;
        double r5196375 = 9.0;
        double r5196376 = r5196375 * r5196369;
        double r5196377 = -r5196372;
        double r5196378 = r5196375 * r5196377;
        double r5196379 = r5196376 + r5196378;
        double r5196380 = sqrt(r5196379);
        double r5196381 = r5196374 / r5196380;
        double r5196382 = rand;
        double r5196383 = r5196381 * r5196382;
        double r5196384 = r5196374 + r5196383;
        double r5196385 = r5196373 * r5196384;
        return r5196385;
}

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 2019133 +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))))