Average Error: 0.2 → 0.1
Time: 6.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}
double f(double a, double rand) {
        double r83246 = a;
        double r83247 = 1.0;
        double r83248 = 3.0;
        double r83249 = r83247 / r83248;
        double r83250 = r83246 - r83249;
        double r83251 = 9.0;
        double r83252 = r83251 * r83250;
        double r83253 = sqrt(r83252);
        double r83254 = r83247 / r83253;
        double r83255 = rand;
        double r83256 = r83254 * r83255;
        double r83257 = r83247 + r83256;
        double r83258 = r83250 * r83257;
        return r83258;
}
↓
double f(double a, double rand) {
        double r83259 = a;
        double r83260 = 1.0;
        double r83261 = 3.0;
        double r83262 = r83260 / r83261;
        double r83263 = r83259 - r83262;
        double r83264 = r83263 * r83260;
        double r83265 = rand;
        double r83266 = r83260 * r83265;
        double r83267 = 9.0;
        double r83268 = sqrt(r83267);
        double r83269 = sqrt(r83263);
        double r83270 = r83268 * r83269;
        double r83271 = r83266 / r83270;
        double r83272 = r83263 * r83271;
        double r83273 = r83264 + r83272;
        return r83273;
}

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 0.2

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
  2. Using strategy rm

  3. Applied associate-*l/0.1

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

  5. Applied distribute-lft-in0.1

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

  7. Applied sqrt-prod0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))