Average Error: 0.1 → 0.1
Time: 42.6s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)\]
\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)
↓
\frac{rand}{\sqrt{\left(a - \frac{1.0}{3.0}\right) \cdot 9}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2764246 = a;
        double r2764247 = 1.0;
        double r2764248 = 3.0;
        double r2764249 = r2764247 / r2764248;
        double r2764250 = r2764246 - r2764249;
        double r2764251 = 1.0;
        double r2764252 = 9.0;
        double r2764253 = r2764252 * r2764250;
        double r2764254 = sqrt(r2764253);
        double r2764255 = r2764251 / r2764254;
        double r2764256 = rand;
        double r2764257 = r2764255 * r2764256;
        double r2764258 = r2764251 + r2764257;
        double r2764259 = r2764250 * r2764258;
        return r2764259;
}
↓
double f(double a, double rand) {
        double r2764260 = rand;
        double r2764261 = a;
        double r2764262 = 1.0;
        double r2764263 = 3.0;
        double r2764264 = r2764262 / r2764263;
        double r2764265 = r2764261 - r2764264;
        double r2764266 = 9.0;
        double r2764267 = r2764265 * r2764266;
        double r2764268 = sqrt(r2764267);
        double r2764269 = r2764260 / r2764268;
        double r2764270 = r2764269 * r2764265;
        double r2764271 = r2764270 + r2764265;
        return r2764271;
}

Error

Bits error versus a

Bits error versus rand

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  4. Applied fma-udef0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019129 +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))))