Average Error: 0.1 → 0.1
Time: 14.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\frac{1}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)\]
\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{\frac{1}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}}, rand, 1\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r135692 = a;
        double r135693 = 1.0;
        double r135694 = 3.0;
        double r135695 = r135693 / r135694;
        double r135696 = r135692 - r135695;
        double r135697 = 9.0;
        double r135698 = r135697 * r135696;
        double r135699 = sqrt(r135698);
        double r135700 = r135693 / r135699;
        double r135701 = rand;
        double r135702 = r135700 * r135701;
        double r135703 = r135693 + r135702;
        double r135704 = r135696 * r135703;
        return r135704;
}
↓
double f(double a, double rand) {
        double r135705 = 1.0;
        double r135706 = a;
        double r135707 = 3.0;
        double r135708 = r135705 / r135707;
        double r135709 = r135706 - r135708;
        double r135710 = sqrt(r135709);
        double r135711 = r135705 / r135710;
        double r135712 = 9.0;
        double r135713 = sqrt(r135712);
        double r135714 = r135711 / r135713;
        double r135715 = rand;
        double r135716 = fma(r135714, r135715, r135705);
        double r135717 = r135716 * r135709;
        return r135717;
}

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 sqrt-prod0.2

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

  5. Applied *-un-lft-identity0.2

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

  6. Applied times-frac0.2

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020046 +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))))