Average Error: 0.1 → 0.1
Time: 16.1s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\frac{a - \frac{1.0}{3.0}}{3}}{\sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)\]
\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(\frac{\frac{a - \frac{1.0}{3.0}}{3}}{\sqrt{a - \frac{1.0}{3.0}}}, rand, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r1433645 = a;
        double r1433646 = 1.0;
        double r1433647 = 3.0;
        double r1433648 = r1433646 / r1433647;
        double r1433649 = r1433645 - r1433648;
        double r1433650 = 1.0;
        double r1433651 = 9.0;
        double r1433652 = r1433651 * r1433649;
        double r1433653 = sqrt(r1433652);
        double r1433654 = r1433650 / r1433653;
        double r1433655 = rand;
        double r1433656 = r1433654 * r1433655;
        double r1433657 = r1433650 + r1433656;
        double r1433658 = r1433649 * r1433657;
        return r1433658;
}
↓
double f(double a, double rand) {
        double r1433659 = a;
        double r1433660 = 1.0;
        double r1433661 = 3.0;
        double r1433662 = r1433660 / r1433661;
        double r1433663 = r1433659 - r1433662;
        double r1433664 = 3.0;
        double r1433665 = r1433663 / r1433664;
        double r1433666 = sqrt(r1433663);
        double r1433667 = r1433665 / r1433666;
        double r1433668 = rand;
        double r1433669 = fma(r1433667, r1433668, r1433663);
        return r1433669;
}

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

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

  4. Applied sqrt-prod0.1

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

  5. Applied associate-/r*0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019156 +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))))