Average Error: 0.1 → 0.1
Time: 25.1s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \frac{\frac{rand}{\sqrt{9}} \cdot 1}{\sqrt{a - \frac{1}{3}}} + \left(a - \frac{1}{3}\right) \cdot 1\]
\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 \frac{\frac{rand}{\sqrt{9}} \cdot 1}{\sqrt{a - \frac{1}{3}}} + \left(a - \frac{1}{3}\right) \cdot 1
double f(double a, double rand) {
        double r3655604 = a;
        double r3655605 = 1.0;
        double r3655606 = 3.0;
        double r3655607 = r3655605 / r3655606;
        double r3655608 = r3655604 - r3655607;
        double r3655609 = 9.0;
        double r3655610 = r3655609 * r3655608;
        double r3655611 = sqrt(r3655610);
        double r3655612 = r3655605 / r3655611;
        double r3655613 = rand;
        double r3655614 = r3655612 * r3655613;
        double r3655615 = r3655605 + r3655614;
        double r3655616 = r3655608 * r3655615;
        return r3655616;
}
↓
double f(double a, double rand) {
        double r3655617 = a;
        double r3655618 = 1.0;
        double r3655619 = 3.0;
        double r3655620 = r3655618 / r3655619;
        double r3655621 = r3655617 - r3655620;
        double r3655622 = rand;
        double r3655623 = 9.0;
        double r3655624 = sqrt(r3655623);
        double r3655625 = r3655622 / r3655624;
        double r3655626 = r3655625 * r3655618;
        double r3655627 = sqrt(r3655621);
        double r3655628 = r3655626 / r3655627;
        double r3655629 = r3655621 * r3655628;
        double r3655630 = r3655621 * r3655618;
        double r3655631 = r3655629 + r3655630;
        return r3655631;
}

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-rgt-in0.1

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

  5. Applied *-commutative0.1

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

  7. Applied sqrt-prod0.1

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

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

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

  9. Applied times-frac0.2

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

  10. Applied associate-*r*0.2

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

  11. Simplified0.1

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

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

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

  14. Applied associate-*r*0.1

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

  15. Simplified0.1

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

  16. Final simplification0.1

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

Reproduce

herbie shell --seed 2019192 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))