Average Error: 0.1 → 0.2
Time: 26.0s
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 r78484 = a;
        double r78485 = 1.0;
        double r78486 = 3.0;
        double r78487 = r78485 / r78486;
        double r78488 = r78484 - r78487;
        double r78489 = 9.0;
        double r78490 = r78489 * r78488;
        double r78491 = sqrt(r78490);
        double r78492 = r78485 / r78491;
        double r78493 = rand;
        double r78494 = r78492 * r78493;
        double r78495 = r78485 + r78494;
        double r78496 = r78488 * r78495;
        return r78496;
}
↓
double f(double a, double rand) {
        double r78497 = 1.0;
        double r78498 = a;
        double r78499 = 3.0;
        double r78500 = r78497 / r78499;
        double r78501 = r78498 - r78500;
        double r78502 = sqrt(r78501);
        double r78503 = r78497 / r78502;
        double r78504 = 9.0;
        double r78505 = sqrt(r78504);
        double r78506 = r78503 / r78505;
        double r78507 = rand;
        double r78508 = fma(r78506, r78507, r78497);
        double r78509 = r78508 * r78501;
        return r78509;
}

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

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

  4. Applied sqrt-prod0.2

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

  5. Applied associate-/r*0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))