Average Error: 0.1 → 0.1
Time: 7.0s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand\]
\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(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand
double f(double a, double rand) {
        double r77358 = a;
        double r77359 = 1.0;
        double r77360 = 3.0;
        double r77361 = r77359 / r77360;
        double r77362 = r77358 - r77361;
        double r77363 = 9.0;
        double r77364 = r77363 * r77362;
        double r77365 = sqrt(r77364);
        double r77366 = r77359 / r77365;
        double r77367 = rand;
        double r77368 = r77366 * r77367;
        double r77369 = r77359 + r77368;
        double r77370 = r77362 * r77369;
        return r77370;
}
↓
double f(double a, double rand) {
        double r77371 = a;
        double r77372 = 1.0;
        double r77373 = 3.0;
        double r77374 = r77372 / r77373;
        double r77375 = r77371 - r77374;
        double r77376 = r77375 * r77372;
        double r77377 = 9.0;
        double r77378 = r77377 * r77375;
        double r77379 = sqrt(r77378);
        double r77380 = r77372 / r77379;
        double r77381 = r77375 * r77380;
        double r77382 = rand;
        double r77383 = r77381 * r77382;
        double r77384 = r77376 + r77383;
        return r77384;
}

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

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

  3. Applied distribute-lft-in0.1

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

  5. Applied associate-*r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))