Average Error: 0.1 → 0.1
Time: 2.4m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]
\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)
↓
\mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)
double f(double a, double rand) {
        double r19397530 = a;
        double r19397531 = 1.0;
        double r19397532 = 3.0;
        double r19397533 = r19397531 / r19397532;
        double r19397534 = r19397530 - r19397533;
        double r19397535 = 1.0;
        double r19397536 = 9.0;
        double r19397537 = r19397536 * r19397534;
        double r19397538 = sqrt(r19397537);
        double r19397539 = r19397535 / r19397538;
        double r19397540 = rand;
        double r19397541 = r19397539 * r19397540;
        double r19397542 = r19397535 + r19397541;
        double r19397543 = r19397534 * r19397542;
        return r19397543;
}
↓
double f(double a, double rand) {
        double r19397544 = rand;
        double r19397545 = a;
        double r19397546 = 1.0;
        double r19397547 = 3.0;
        double r19397548 = r19397546 / r19397547;
        double r19397549 = r19397545 - r19397548;
        double r19397550 = 9.0;
        double r19397551 = r19397549 * r19397550;
        double r19397552 = sqrt(r19397551);
        double r19397553 = r19397544 / r19397552;
        double r19397554 = fma(r19397553, r19397549, r19397549);
        return r19397554;
}

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

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

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]

Reproduce

herbie shell --seed 2019128 +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))))