Average Error: 0.1 → 0.1
Time: 21.8s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\mathsf{fma}\left(\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)\]
\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(\left(\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}}\right), \left(a - \frac{1.0}{3.0}\right), \left(a - \frac{1.0}{3.0}\right)\right)
double f(double a, double rand) {
        double r1217224 = a;
        double r1217225 = 1.0;
        double r1217226 = 3.0;
        double r1217227 = r1217225 / r1217226;
        double r1217228 = r1217224 - r1217227;
        double r1217229 = 1.0;
        double r1217230 = 9.0;
        double r1217231 = r1217230 * r1217228;
        double r1217232 = sqrt(r1217231);
        double r1217233 = r1217229 / r1217232;
        double r1217234 = rand;
        double r1217235 = r1217233 * r1217234;
        double r1217236 = r1217229 + r1217235;
        double r1217237 = r1217228 * r1217236;
        return r1217237;
}
↓
double f(double a, double rand) {
        double r1217238 = rand;
        double r1217239 = a;
        double r1217240 = 1.0;
        double r1217241 = 3.0;
        double r1217242 = r1217240 / r1217241;
        double r1217243 = r1217239 - r1217242;
        double r1217244 = 9.0;
        double r1217245 = r1217243 * r1217244;
        double r1217246 = sqrt(r1217245);
        double r1217247 = r1217238 / r1217246;
        double r1217248 = fma(r1217247, r1217243, r1217243);
        return r1217248;
}

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

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 +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))))