Average Error: 0.1 → 0.1
Time: 26.7s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{\sqrt{a} - \sqrt{\frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{\sqrt{a} + \sqrt{\frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)\]
\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} - \sqrt{\frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{\sqrt{a} + \sqrt{\frac{1.0}{3.0}}}{3}, rand, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r4559440 = a;
        double r4559441 = 1.0;
        double r4559442 = 3.0;
        double r4559443 = r4559441 / r4559442;
        double r4559444 = r4559440 - r4559443;
        double r4559445 = 1.0;
        double r4559446 = 9.0;
        double r4559447 = r4559446 * r4559444;
        double r4559448 = sqrt(r4559447);
        double r4559449 = r4559445 / r4559448;
        double r4559450 = rand;
        double r4559451 = r4559449 * r4559450;
        double r4559452 = r4559445 + r4559451;
        double r4559453 = r4559444 * r4559452;
        return r4559453;
}
↓
double f(double a, double rand) {
        double r4559454 = a;
        double r4559455 = sqrt(r4559454);
        double r4559456 = 1.0;
        double r4559457 = 3.0;
        double r4559458 = r4559456 / r4559457;
        double r4559459 = sqrt(r4559458);
        double r4559460 = r4559455 - r4559459;
        double r4559461 = r4559454 - r4559458;
        double r4559462 = sqrt(r4559461);
        double r4559463 = r4559460 / r4559462;
        double r4559464 = r4559455 + r4559459;
        double r4559465 = 3.0;
        double r4559466 = r4559464 / r4559465;
        double r4559467 = r4559463 * r4559466;
        double r4559468 = rand;
        double r4559469 = fma(r4559467, r4559468, r4559461);
        return r4559469;
}

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

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

  6. Applied add-sqr-sqrt0.2

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

  7. Applied difference-of-squares0.1

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

  8. Applied times-frac0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019162 +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))))