Average Error: 0.1 → 0.1
Time: 24.4s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(a - \frac{1.0}{3.0}, \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, a - \frac{1.0}{3.0}\right)\]
\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(a - \frac{1.0}{3.0}, \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3142466 = a;
        double r3142467 = 1.0;
        double r3142468 = 3.0;
        double r3142469 = r3142467 / r3142468;
        double r3142470 = r3142466 - r3142469;
        double r3142471 = 1.0;
        double r3142472 = 9.0;
        double r3142473 = r3142472 * r3142470;
        double r3142474 = sqrt(r3142473);
        double r3142475 = r3142471 / r3142474;
        double r3142476 = rand;
        double r3142477 = r3142475 * r3142476;
        double r3142478 = r3142471 + r3142477;
        double r3142479 = r3142470 * r3142478;
        return r3142479;
}
↓
double f(double a, double rand) {
        double r3142480 = a;
        double r3142481 = 1.0;
        double r3142482 = 3.0;
        double r3142483 = r3142481 / r3142482;
        double r3142484 = r3142480 - r3142483;
        double r3142485 = rand;
        double r3142486 = 9.0;
        double r3142487 = r3142486 * r3142484;
        double r3142488 = sqrt(r3142487);
        double r3142489 = r3142485 / r3142488;
        double r3142490 = fma(r3142484, r3142489, r3142484);
        return r3142490;
}

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. Using strategy rm

  3. Applied *-un-lft-identity0.1

    \[\leadsto \color{blue}{\left(1 \cdot \left(a - \frac{1.0}{3.0}\right)\right)} \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  4. Applied associate-*l*0.1

    \[\leadsto \color{blue}{1 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\right)}\]

  5. Simplified0.1

    \[\leadsto 1 \cdot \color{blue}{\mathsf{fma}\left(a - \frac{1.0}{3.0}, \frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}, a - \frac{1.0}{3.0}\right)}\]

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019168 +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))))