Average Error: 0.1 → 0.2
Time: 43.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right)
double f(double a, double rand) {
        double r195326 = a;
        double r195327 = 1.0;
        double r195328 = 3.0;
        double r195329 = r195327 / r195328;
        double r195330 = r195326 - r195329;
        double r195331 = 9.0;
        double r195332 = r195331 * r195330;
        double r195333 = sqrt(r195332);
        double r195334 = r195327 / r195333;
        double r195335 = rand;
        double r195336 = r195334 * r195335;
        double r195337 = r195327 + r195336;
        double r195338 = r195330 * r195337;
        return r195338;
}
↓
double f(double a, double rand) {
        double r195339 = a;
        double r195340 = 1.0;
        double r195341 = 3.0;
        double r195342 = r195340 / r195341;
        double r195343 = r195339 - r195342;
        double r195344 = 9.0;
        double r195345 = sqrt(r195344);
        double r195346 = r195340 / r195345;
        double r195347 = sqrt(r195343);
        double r195348 = r195346 / r195347;
        double r195349 = rand;
        double r195350 = fma(r195348, r195349, r195340);
        double r195351 = r195343 * r195350;
        return r195351;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{\left(1 \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)} \cdot \left(a - \frac{1}{3}\right)\]

  5. Applied associate-*l*0.1

    \[\leadsto \color{blue}{1 \cdot \left(\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\right)}\]

  6. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\left(\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right)\right)}\]
  7. Using strategy rm

  8. Applied sqrt-prod0.2

    \[\leadsto 1 \cdot \left(\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{1}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}}, rand, 1\right)\right)\]

  9. Applied associate-/r*0.2

    \[\leadsto 1 \cdot \left(\left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\color{blue}{\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}}, rand, 1\right)\right)\]

  10. Final simplification0.2

    \[\leadsto \left(a - \frac{1}{3}\right) \cdot \mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right)\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))