Average Error: 0.1 → 0.1
Time: 30.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}}, a - \frac{1}{3}, a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
1 \cdot \mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}}, a - \frac{1}{3}, a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r3485035 = a;
        double r3485036 = 1.0;
        double r3485037 = 3.0;
        double r3485038 = r3485036 / r3485037;
        double r3485039 = r3485035 - r3485038;
        double r3485040 = 9.0;
        double r3485041 = r3485040 * r3485039;
        double r3485042 = sqrt(r3485041);
        double r3485043 = r3485036 / r3485042;
        double r3485044 = rand;
        double r3485045 = r3485043 * r3485044;
        double r3485046 = r3485036 + r3485045;
        double r3485047 = r3485039 * r3485046;
        return r3485047;
}
↓
double f(double a, double rand) {
        double r3485048 = 1.0;
        double r3485049 = rand;
        double r3485050 = a;
        double r3485051 = 3.0;
        double r3485052 = r3485048 / r3485051;
        double r3485053 = r3485050 - r3485052;
        double r3485054 = 9.0;
        double r3485055 = r3485053 * r3485054;
        double r3485056 = sqrt(r3485055);
        double r3485057 = r3485049 / r3485056;
        double r3485058 = fma(r3485057, r3485053, r3485053);
        double r3485059 = r3485048 * r3485058;
        return r3485059;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +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))))