Average Error: 0.1 → 0.1
Time: 27.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{a - \frac{1}{3}}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{1 \cdot rand}}\]
\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 + \frac{a - \frac{1}{3}}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{1 \cdot rand}}
double f(double a, double rand) {
        double r66473 = a;
        double r66474 = 1.0;
        double r66475 = 3.0;
        double r66476 = r66474 / r66475;
        double r66477 = r66473 - r66476;
        double r66478 = 9.0;
        double r66479 = r66478 * r66477;
        double r66480 = sqrt(r66479);
        double r66481 = r66474 / r66480;
        double r66482 = rand;
        double r66483 = r66481 * r66482;
        double r66484 = r66474 + r66483;
        double r66485 = r66477 * r66484;
        return r66485;
}
↓
double f(double a, double rand) {
        double r66486 = a;
        double r66487 = 1.0;
        double r66488 = 3.0;
        double r66489 = r66487 / r66488;
        double r66490 = r66486 - r66489;
        double r66491 = r66490 * r66487;
        double r66492 = 9.0;
        double r66493 = r66492 * r66490;
        double r66494 = sqrt(r66493);
        double r66495 = rand;
        double r66496 = r66487 * r66495;
        double r66497 = r66494 / r66496;
        double r66498 = r66490 / r66497;
        double r66499 = r66491 + r66498;
        return r66499;
}

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 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 clear-num0.1

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

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

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

  10. Applied associate-*l*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))