Average Error: 0.1 → 0.2
Time: 24.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r75969 = a;
        double r75970 = 1.0;
        double r75971 = 3.0;
        double r75972 = r75970 / r75971;
        double r75973 = r75969 - r75972;
        double r75974 = 9.0;
        double r75975 = r75974 * r75973;
        double r75976 = sqrt(r75975);
        double r75977 = r75970 / r75976;
        double r75978 = rand;
        double r75979 = r75977 * r75978;
        double r75980 = r75970 + r75979;
        double r75981 = r75973 * r75980;
        return r75981;
}
↓
double f(double a, double rand) {
        double r75982 = 1.0;
        double r75983 = a;
        double r75984 = 3.0;
        double r75985 = r75982 / r75984;
        double r75986 = r75983 - r75985;
        double r75987 = sqrt(r75986);
        double r75988 = r75982 / r75987;
        double r75989 = 9.0;
        double r75990 = sqrt(r75989);
        double r75991 = r75988 / r75990;
        double r75992 = rand;
        double r75993 = fma(r75991, r75992, r75982);
        double r75994 = r75993 * r75986;
        return r75994;
}

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

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

  4. Applied sqrt-prod0.2

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))