Average Error: 0.1 → 0.1
Time: 29.1s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left({\left(\left(a - \frac{1.0}{3.0}\right) \cdot 9\right)}^{\frac{-1}{2}} \cdot rand, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
\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({\left(\left(a - \frac{1.0}{3.0}\right) \cdot 9\right)}^{\frac{-1}{2}} \cdot rand, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3056736 = a;
        double r3056737 = 1.0;
        double r3056738 = 3.0;
        double r3056739 = r3056737 / r3056738;
        double r3056740 = r3056736 - r3056739;
        double r3056741 = 1.0;
        double r3056742 = 9.0;
        double r3056743 = r3056742 * r3056740;
        double r3056744 = sqrt(r3056743);
        double r3056745 = r3056741 / r3056744;
        double r3056746 = rand;
        double r3056747 = r3056745 * r3056746;
        double r3056748 = r3056741 + r3056747;
        double r3056749 = r3056740 * r3056748;
        return r3056749;
}
↓
double f(double a, double rand) {
        double r3056750 = a;
        double r3056751 = 1.0;
        double r3056752 = 3.0;
        double r3056753 = r3056751 / r3056752;
        double r3056754 = r3056750 - r3056753;
        double r3056755 = 9.0;
        double r3056756 = r3056754 * r3056755;
        double r3056757 = -0.5;
        double r3056758 = pow(r3056756, r3056757);
        double r3056759 = rand;
        double r3056760 = r3056758 * r3056759;
        double r3056761 = fma(r3056760, r3056754, r3056754);
        return r3056761;
}

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 div-inv0.1

    \[\leadsto \mathsf{fma}\left(\color{blue}{rand \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
  5. Using strategy rm

  6. Applied pow10.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \frac{1}{\sqrt{\color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{1}}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  7. Applied sqrt-pow10.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \frac{1}{\color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\left(\frac{1}{2}\right)}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  8. Applied pow-flip0.1

    \[\leadsto \mathsf{fma}\left(rand \cdot \color{blue}{{\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\left(-\frac{1}{2}\right)}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  9. Simplified0.1

    \[\leadsto \mathsf{fma}\left(rand \cdot {\left(9 \cdot \left(a - \frac{1.0}{3.0}\right)\right)}^{\color{blue}{\frac{-1}{2}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

  10. Final simplification0.1

    \[\leadsto \mathsf{fma}\left({\left(\left(a - \frac{1.0}{3.0}\right) \cdot 9\right)}^{\frac{-1}{2}} \cdot rand, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019151 +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))))