Average Error: 0.1 → 0.1
Time: 18.9s
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 r1259392 = a;
        double r1259393 = 1.0;
        double r1259394 = 3.0;
        double r1259395 = r1259393 / r1259394;
        double r1259396 = r1259392 - r1259395;
        double r1259397 = 1.0;
        double r1259398 = 9.0;
        double r1259399 = r1259398 * r1259396;
        double r1259400 = sqrt(r1259399);
        double r1259401 = r1259397 / r1259400;
        double r1259402 = rand;
        double r1259403 = r1259401 * r1259402;
        double r1259404 = r1259397 + r1259403;
        double r1259405 = r1259396 * r1259404;
        return r1259405;
}
↓
double f(double a, double rand) {
        double r1259406 = a;
        double r1259407 = 1.0;
        double r1259408 = 3.0;
        double r1259409 = r1259407 / r1259408;
        double r1259410 = r1259406 - r1259409;
        double r1259411 = sqrt(r1259410);
        double r1259412 = 3.0;
        double r1259413 = r1259411 / r1259412;
        double r1259414 = rand;
        double r1259415 = fma(r1259413, r1259414, r1259410);
        return r1259415;
}

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-sqrt0.2

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

  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 2019153 +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))))