Average Error: 0.1 → 0.1
Time: 30.2s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r57769 = a;
        double r57770 = 1.0;
        double r57771 = 3.0;
        double r57772 = r57770 / r57771;
        double r57773 = r57769 - r57772;
        double r57774 = 9.0;
        double r57775 = r57774 * r57773;
        double r57776 = sqrt(r57775);
        double r57777 = r57770 / r57776;
        double r57778 = rand;
        double r57779 = r57777 * r57778;
        double r57780 = r57770 + r57779;
        double r57781 = r57773 * r57780;
        return r57781;
}
↓
double f(double a, double rand) {
        double r57782 = 1.0;
        double r57783 = a;
        double r57784 = 3.0;
        double r57785 = r57782 / r57784;
        double r57786 = r57783 - r57785;
        double r57787 = r57782 * r57786;
        double r57788 = rand;
        double r57789 = 9.0;
        double r57790 = r57789 * r57786;
        double r57791 = sqrt(r57790);
        double r57792 = r57788 / r57791;
        double r57793 = r57782 * r57792;
        double r57794 = r57793 * r57786;
        double r57795 = r57787 + r57794;
        return r57795;
}

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. Simplified0.1

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

  5. Simplified0.1

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

  7. Applied div-inv0.1

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

  8. Applied associate-*l*0.1

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

  9. Simplified0.1

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

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

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))