Average Error: 0.1 → 0.1
Time: 29.2s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
\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{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2947924 = a;
        double r2947925 = 1.0;
        double r2947926 = 3.0;
        double r2947927 = r2947925 / r2947926;
        double r2947928 = r2947924 - r2947927;
        double r2947929 = 1.0;
        double r2947930 = 9.0;
        double r2947931 = r2947930 * r2947928;
        double r2947932 = sqrt(r2947931);
        double r2947933 = r2947929 / r2947932;
        double r2947934 = rand;
        double r2947935 = r2947933 * r2947934;
        double r2947936 = r2947929 + r2947935;
        double r2947937 = r2947928 * r2947936;
        return r2947937;
}
↓
double f(double a, double rand) {
        double r2947938 = rand;
        double r2947939 = a;
        double r2947940 = 1.0;
        double r2947941 = 3.0;
        double r2947942 = r2947940 / r2947941;
        double r2947943 = r2947939 - r2947942;
        double r2947944 = 9.0;
        double r2947945 = r2947943 * r2947944;
        double r2947946 = sqrt(r2947945);
        double r2947947 = r2947938 / r2947946;
        double r2947948 = fma(r2947947, r2947943, r2947943);
        return r2947948;
}

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

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

  4. Applied *-un-lft-identity0.1

    \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{9 \cdot \color{blue}{\left(1 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  5. Applied associate-*r*0.1

    \[\leadsto \mathsf{fma}\left(\frac{rand}{\sqrt{\color{blue}{\left(9 \cdot 1\right) \cdot \left(a - \frac{1.0}{3.0}\right)}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019144 +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))))