Average Error: 0.1 → 0.1
Time: 3.6m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \left(\left(a - \frac{1.0}{3.0}\right) \cdot \frac{1}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right) \cdot rand\]
double f(double a, double rand) {
        double r12731353 = a;
        double r12731354 = 1.0;
        double r12731355 = 3.0;
        double r12731356 = r12731354 / r12731355;
        double r12731357 = r12731353 - r12731356;
        double r12731358 = 1.0;
        double r12731359 = 9.0;
        double r12731360 = r12731359 * r12731357;
        double r12731361 = sqrt(r12731360);
        double r12731362 = r12731358 / r12731361;
        double r12731363 = rand;
        double r12731364 = r12731362 * r12731363;
        double r12731365 = r12731358 + r12731364;
        double r12731366 = r12731357 * r12731365;
        return r12731366;
}
↓
double f(double a, double rand) {
        double r12731367 = a;
        double r12731368 = 1.0;
        double r12731369 = 3.0;
        double r12731370 = r12731368 / r12731369;
        double r12731371 = r12731367 - r12731370;
        double r12731372 = 1.0;
        double r12731373 = 9.0;
        double r12731374 = r12731371 * r12731373;
        double r12731375 = sqrt(r12731374);
        double r12731376 = r12731372 / r12731375;
        double r12731377 = r12731371 * r12731376;
        double r12731378 = rand;
        double r12731379 = r12731377 * r12731378;
        double r12731380 = r12731371 + r12731379;
        return r12731380;
}


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

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{(\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*}\]
  3. Using strategy rm

  4. Applied fma-udef0.1

    \[\leadsto \color{blue}{\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)}\]
  5. Using strategy rm

  6. Applied div-inv0.1

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

  7. Applied associate-*l*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019102 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))