Average Error: 0.1 → 0.1
Time: 58.5s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(rand, \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, a - \frac{1.0}{3.0}\right)\]
\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(rand, \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r4721610 = a;
        double r4721611 = 1.0;
        double r4721612 = 3.0;
        double r4721613 = r4721611 / r4721612;
        double r4721614 = r4721610 - r4721613;
        double r4721615 = 1.0;
        double r4721616 = 9.0;
        double r4721617 = r4721616 * r4721614;
        double r4721618 = sqrt(r4721617);
        double r4721619 = r4721615 / r4721618;
        double r4721620 = rand;
        double r4721621 = r4721619 * r4721620;
        double r4721622 = r4721615 + r4721621;
        double r4721623 = r4721614 * r4721622;
        return r4721623;
}
↓
double f(double a, double rand) {
        double r4721624 = rand;
        double r4721625 = a;
        double r4721626 = 1.0;
        double r4721627 = 3.0;
        double r4721628 = r4721626 / r4721627;
        double r4721629 = r4721625 - r4721628;
        double r4721630 = sqrt(r4721629);
        double r4721631 = 3.0;
        double r4721632 = r4721630 / r4721631;
        double r4721633 = fma(r4721624, r4721632, r4721629);
        return r4721633;
}

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

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

  4. Applied sqrt-prod0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(rand, \color{blue}{\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}} \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}, a - \frac{1.0}{3.0}\right)\]

  8. Simplified0.1

    \[\leadsto \mathsf{fma}\left(rand, \frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot \color{blue}{1}, a - \frac{1.0}{3.0}\right)\]

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(rand, \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019165 +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))))