Average Error: 0.1 → 0.1
Time: 22.6s
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 r2515096 = a;
        double r2515097 = 1.0;
        double r2515098 = 3.0;
        double r2515099 = r2515097 / r2515098;
        double r2515100 = r2515096 - r2515099;
        double r2515101 = 1.0;
        double r2515102 = 9.0;
        double r2515103 = r2515102 * r2515100;
        double r2515104 = sqrt(r2515103);
        double r2515105 = r2515101 / r2515104;
        double r2515106 = rand;
        double r2515107 = r2515105 * r2515106;
        double r2515108 = r2515101 + r2515107;
        double r2515109 = r2515100 * r2515108;
        return r2515109;
}
↓
double f(double a, double rand) {
        double r2515110 = rand;
        double r2515111 = a;
        double r2515112 = 1.0;
        double r2515113 = 3.0;
        double r2515114 = r2515112 / r2515113;
        double r2515115 = r2515111 - r2515114;
        double r2515116 = 9.0;
        double r2515117 = r2515115 * r2515116;
        double r2515118 = sqrt(r2515117);
        double r2515119 = r2515110 / r2515118;
        double r2515120 = fma(r2515119, r2515115, r2515115);
        return r2515120;
}

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 2019132 +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))))