Average Error: 0.1 → 0.1
Time: 54.9s
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 r4038336 = a;
        double r4038337 = 1.0;
        double r4038338 = 3.0;
        double r4038339 = r4038337 / r4038338;
        double r4038340 = r4038336 - r4038339;
        double r4038341 = 9.0;
        double r4038342 = r4038341 * r4038340;
        double r4038343 = sqrt(r4038342);
        double r4038344 = r4038337 / r4038343;
        double r4038345 = rand;
        double r4038346 = r4038344 * r4038345;
        double r4038347 = r4038337 + r4038346;
        double r4038348 = r4038340 * r4038347;
        return r4038348;
}
↓
double f(double a, double rand) {
        double r4038349 = 1.0;
        double r4038350 = rand;
        double r4038351 = a;
        double r4038352 = 3.0;
        double r4038353 = r4038349 / r4038352;
        double r4038354 = r4038351 - r4038353;
        double r4038355 = 9.0;
        double r4038356 = r4038354 * r4038355;
        double r4038357 = sqrt(r4038356);
        double r4038358 = r4038350 / r4038357;
        double r4038359 = fma(r4038358, r4038354, r4038354);
        double r4038360 = r4038349 * r4038359;
        return r4038360;
}

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 2019168 +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))))