Average Error: 0.1 → 0.1
Time: 33.7s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\frac{1}{3} \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)\]
\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{1}{3} \cdot \frac{rand}{\sqrt{a - \frac{1.0}{3.0}}}, a - \frac{1.0}{3.0}, a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2795218 = a;
        double r2795219 = 1.0;
        double r2795220 = 3.0;
        double r2795221 = r2795219 / r2795220;
        double r2795222 = r2795218 - r2795221;
        double r2795223 = 1.0;
        double r2795224 = 9.0;
        double r2795225 = r2795224 * r2795222;
        double r2795226 = sqrt(r2795225);
        double r2795227 = r2795223 / r2795226;
        double r2795228 = rand;
        double r2795229 = r2795227 * r2795228;
        double r2795230 = r2795223 + r2795229;
        double r2795231 = r2795222 * r2795230;
        return r2795231;
}
↓
double f(double a, double rand) {
        double r2795232 = 0.3333333333333333;
        double r2795233 = rand;
        double r2795234 = a;
        double r2795235 = 1.0;
        double r2795236 = 3.0;
        double r2795237 = r2795235 / r2795236;
        double r2795238 = r2795234 - r2795237;
        double r2795239 = sqrt(r2795238);
        double r2795240 = r2795233 / r2795239;
        double r2795241 = r2795232 * r2795240;
        double r2795242 = fma(r2795241, r2795238, r2795238);
        return r2795242;
}

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

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

  4. Applied sqrt-prod0.1

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

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

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019149 +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))))