Average Error: 0.1 → 0.1
Time: 12.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \frac{a - \frac{1}{3}}{\sqrt{9}} \cdot \frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}\]
\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 + \frac{a - \frac{1}{3}}{\sqrt{9}} \cdot \frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}
double f(double a, double rand) {
        double r81610 = a;
        double r81611 = 1.0;
        double r81612 = 3.0;
        double r81613 = r81611 / r81612;
        double r81614 = r81610 - r81613;
        double r81615 = 9.0;
        double r81616 = r81615 * r81614;
        double r81617 = sqrt(r81616);
        double r81618 = r81611 / r81617;
        double r81619 = rand;
        double r81620 = r81618 * r81619;
        double r81621 = r81611 + r81620;
        double r81622 = r81614 * r81621;
        return r81622;
}
↓
double f(double a, double rand) {
        double r81623 = a;
        double r81624 = 1.0;
        double r81625 = 3.0;
        double r81626 = r81624 / r81625;
        double r81627 = r81623 - r81626;
        double r81628 = r81627 * r81624;
        double r81629 = 9.0;
        double r81630 = sqrt(r81629);
        double r81631 = r81627 / r81630;
        double r81632 = rand;
        double r81633 = r81624 * r81632;
        double r81634 = sqrt(r81627);
        double r81635 = r81633 / r81634;
        double r81636 = r81631 * r81635;
        double r81637 = r81628 + r81636;
        return r81637;
}

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. Using strategy rm

  7. Applied sqrt-prod0.2

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

  8. Applied *-un-lft-identity0.2

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

  9. Applied times-frac0.2

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

  10. Applied associate-*r*0.2

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

  11. Simplified0.1

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

  13. Applied associate-*l*0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))))