Average Error: 0.1 → 0.2
Time: 28.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\mathsf{fma}\left(\frac{\frac{1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r72592 = a;
        double r72593 = 1.0;
        double r72594 = 3.0;
        double r72595 = r72593 / r72594;
        double r72596 = r72592 - r72595;
        double r72597 = 9.0;
        double r72598 = r72597 * r72596;
        double r72599 = sqrt(r72598);
        double r72600 = r72593 / r72599;
        double r72601 = rand;
        double r72602 = r72600 * r72601;
        double r72603 = r72593 + r72602;
        double r72604 = r72596 * r72603;
        return r72604;
}
↓
double f(double a, double rand) {
        double r72605 = 1.0;
        double r72606 = 9.0;
        double r72607 = sqrt(r72606);
        double r72608 = r72605 / r72607;
        double r72609 = a;
        double r72610 = 3.0;
        double r72611 = r72605 / r72610;
        double r72612 = r72609 - r72611;
        double r72613 = sqrt(r72612);
        double r72614 = r72608 / r72613;
        double r72615 = rand;
        double r72616 = fma(r72614, r72615, r72605);
        double r72617 = r72616 * r72612;
        return r72617;
}

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 sqrt-prod0.1

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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))))