Average Error: 0.1 → 0.1
Time: 28.8s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{1}{\frac{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}{rand}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\mathsf{fma}\left(\frac{1}{\frac{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}{rand}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2013809 = a;
        double r2013810 = 1.0;
        double r2013811 = 3.0;
        double r2013812 = r2013810 / r2013811;
        double r2013813 = r2013809 - r2013812;
        double r2013814 = 1.0;
        double r2013815 = 9.0;
        double r2013816 = r2013815 * r2013813;
        double r2013817 = sqrt(r2013816);
        double r2013818 = r2013814 / r2013817;
        double r2013819 = rand;
        double r2013820 = r2013818 * r2013819;
        double r2013821 = r2013814 + r2013820;
        double r2013822 = r2013813 * r2013821;
        return r2013822;
}
↓
double f(double a, double rand) {
        double r2013823 = 1.0;
        double r2013824 = 3.0;
        double r2013825 = a;
        double r2013826 = 1.0;
        double r2013827 = 3.0;
        double r2013828 = r2013826 / r2013827;
        double r2013829 = r2013825 - r2013828;
        double r2013830 = sqrt(r2013829);
        double r2013831 = r2013824 * r2013830;
        double r2013832 = rand;
        double r2013833 = r2013831 / r2013832;
        double r2013834 = r2013823 / r2013833;
        double r2013835 = fma(r2013834, r2013829, r2013829);
        return r2013835;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.1

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

    \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  5. Simplified0.1

    \[\leadsto \mathsf{fma}\left(\frac{rand}{\color{blue}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
  6. Using strategy rm

  7. Applied *-un-lft-identity0.1

    \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{1 \cdot rand}}{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  8. Applied associate-/l*0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}{rand}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  9. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{3 \cdot \sqrt{a - \frac{1.0}{3.0}}}{rand}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019139 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))