Average Error: 0.1 → 0.1
Time: 1.7m
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 r2462960 = a;
        double r2462961 = 1.0;
        double r2462962 = 3.0;
        double r2462963 = r2462961 / r2462962;
        double r2462964 = r2462960 - r2462963;
        double r2462965 = 1.0;
        double r2462966 = 9.0;
        double r2462967 = r2462966 * r2462964;
        double r2462968 = sqrt(r2462967);
        double r2462969 = r2462965 / r2462968;
        double r2462970 = rand;
        double r2462971 = r2462969 * r2462970;
        double r2462972 = r2462965 + r2462971;
        double r2462973 = r2462964 * r2462972;
        return r2462973;
}
↓
double f(double a, double rand) {
        double r2462974 = rand;
        double r2462975 = a;
        double r2462976 = 1.0;
        double r2462977 = 3.0;
        double r2462978 = r2462976 / r2462977;
        double r2462979 = r2462975 - r2462978;
        double r2462980 = 9.0;
        double r2462981 = r2462979 * r2462980;
        double r2462982 = sqrt(r2462981);
        double r2462983 = r2462974 / r2462982;
        double r2462984 = fma(r2462983, r2462979, r2462979);
        return r2462984;
}

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 2019124 +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))))