Average Error: 0.1 → 0.1
Time: 27.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \mathsf{fma}\left(\frac{rand}{\sqrt{\left(a - \frac{1}{3}\right) \cdot 9}}, a - \frac{1}{3}, a - \frac{1}{3}\right)\]
\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 r2892385 = a;
        double r2892386 = 1.0;
        double r2892387 = 3.0;
        double r2892388 = r2892386 / r2892387;
        double r2892389 = r2892385 - r2892388;
        double r2892390 = 9.0;
        double r2892391 = r2892390 * r2892389;
        double r2892392 = sqrt(r2892391);
        double r2892393 = r2892386 / r2892392;
        double r2892394 = rand;
        double r2892395 = r2892393 * r2892394;
        double r2892396 = r2892386 + r2892395;
        double r2892397 = r2892389 * r2892396;
        return r2892397;
}
↓
double f(double a, double rand) {
        double r2892398 = 1.0;
        double r2892399 = rand;
        double r2892400 = a;
        double r2892401 = 3.0;
        double r2892402 = r2892398 / r2892401;
        double r2892403 = r2892400 - r2892402;
        double r2892404 = 9.0;
        double r2892405 = r2892403 * r2892404;
        double r2892406 = sqrt(r2892405);
        double r2892407 = r2892399 / r2892406;
        double r2892408 = fma(r2892407, r2892403, r2892403);
        double r2892409 = r2892398 * r2892408;
        return r2892409;
}

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 2019169 +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))))