Average Error: 0.1 → 0.1
Time: 31.6s
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 r3401025 = a;
        double r3401026 = 1.0;
        double r3401027 = 3.0;
        double r3401028 = r3401026 / r3401027;
        double r3401029 = r3401025 - r3401028;
        double r3401030 = 9.0;
        double r3401031 = r3401030 * r3401029;
        double r3401032 = sqrt(r3401031);
        double r3401033 = r3401026 / r3401032;
        double r3401034 = rand;
        double r3401035 = r3401033 * r3401034;
        double r3401036 = r3401026 + r3401035;
        double r3401037 = r3401029 * r3401036;
        return r3401037;
}
↓
double f(double a, double rand) {
        double r3401038 = 1.0;
        double r3401039 = rand;
        double r3401040 = a;
        double r3401041 = 3.0;
        double r3401042 = r3401038 / r3401041;
        double r3401043 = r3401040 - r3401042;
        double r3401044 = 9.0;
        double r3401045 = r3401043 * r3401044;
        double r3401046 = sqrt(r3401045);
        double r3401047 = r3401039 / r3401046;
        double r3401048 = fma(r3401047, r3401043, r3401043);
        double r3401049 = r3401038 * r3401048;
        return r3401049;
}

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))))