Average Error: 0.1 → 0.1
Time: 28.9s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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 \left(a - \frac{1}{3}\right)}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r145663 = a;
        double r145664 = 1.0;
        double r145665 = 3.0;
        double r145666 = r145664 / r145665;
        double r145667 = r145663 - r145666;
        double r145668 = 9.0;
        double r145669 = r145668 * r145667;
        double r145670 = sqrt(r145669);
        double r145671 = r145664 / r145670;
        double r145672 = rand;
        double r145673 = r145671 * r145672;
        double r145674 = r145664 + r145673;
        double r145675 = r145667 * r145674;
        return r145675;
}
↓
double f(double a, double rand) {
        double r145676 = 1.0;
        double r145677 = 9.0;
        double r145678 = a;
        double r145679 = 3.0;
        double r145680 = r145676 / r145679;
        double r145681 = r145678 - r145680;
        double r145682 = r145677 * r145681;
        double r145683 = sqrt(r145682);
        double r145684 = r145676 / r145683;
        double r145685 = rand;
        double r145686 = fma(r145684, r145685, r145676);
        double r145687 = r145686 * r145681;
        return r145687;
}

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 *-un-lft-identity0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))