Average Error: 0.1 → 0.2
Time: 23.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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{1}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r114924 = a;
        double r114925 = 1.0;
        double r114926 = 3.0;
        double r114927 = r114925 / r114926;
        double r114928 = r114924 - r114927;
        double r114929 = 9.0;
        double r114930 = r114929 * r114928;
        double r114931 = sqrt(r114930);
        double r114932 = r114925 / r114931;
        double r114933 = rand;
        double r114934 = r114932 * r114933;
        double r114935 = r114925 + r114934;
        double r114936 = r114928 * r114935;
        return r114936;
}
↓
double f(double a, double rand) {
        double r114937 = 1.0;
        double r114938 = 9.0;
        double r114939 = sqrt(r114938);
        double r114940 = a;
        double r114941 = 3.0;
        double r114942 = r114937 / r114941;
        double r114943 = r114940 - r114942;
        double r114944 = sqrt(r114943);
        double r114945 = r114939 * r114944;
        double r114946 = r114937 / r114945;
        double r114947 = rand;
        double r114948 = fma(r114946, r114947, r114937);
        double r114949 = r114948 * r114943;
        return r114949;
}

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-prod0.2

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

  5. Final simplification0.2

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

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))))