Average Error: 0.1 → 0.1
Time: 25.6s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)\]
\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{\sqrt{a - \frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r1730698 = a;
        double r1730699 = 1.0;
        double r1730700 = 3.0;
        double r1730701 = r1730699 / r1730700;
        double r1730702 = r1730698 - r1730701;
        double r1730703 = 1.0;
        double r1730704 = 9.0;
        double r1730705 = r1730704 * r1730702;
        double r1730706 = sqrt(r1730705);
        double r1730707 = r1730703 / r1730706;
        double r1730708 = rand;
        double r1730709 = r1730707 * r1730708;
        double r1730710 = r1730703 + r1730709;
        double r1730711 = r1730702 * r1730710;
        return r1730711;
}
↓
double f(double a, double rand) {
        double r1730712 = a;
        double r1730713 = 1.0;
        double r1730714 = 3.0;
        double r1730715 = r1730713 / r1730714;
        double r1730716 = r1730712 - r1730715;
        double r1730717 = sqrt(r1730716);
        double r1730718 = 3.0;
        double r1730719 = r1730717 / r1730718;
        double r1730720 = rand;
        double r1730721 = fma(r1730719, r1730720, r1730716);
        return r1730721;
}

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

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

  4. Applied sqrt-prod0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

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))))