Average Error: 0.1 → 0.1
Time: 8.9s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(1 + \frac{\frac{rand \cdot 1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(1 + \frac{\frac{rand \cdot 1}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r94516 = a;
        double r94517 = 1.0;
        double r94518 = 3.0;
        double r94519 = r94517 / r94518;
        double r94520 = r94516 - r94519;
        double r94521 = 9.0;
        double r94522 = r94521 * r94520;
        double r94523 = sqrt(r94522);
        double r94524 = r94517 / r94523;
        double r94525 = rand;
        double r94526 = r94524 * r94525;
        double r94527 = r94517 + r94526;
        double r94528 = r94520 * r94527;
        return r94528;
}
↓
double f(double a, double rand) {
        double r94529 = 1.0;
        double r94530 = rand;
        double r94531 = r94530 * r94529;
        double r94532 = 9.0;
        double r94533 = sqrt(r94532);
        double r94534 = r94531 / r94533;
        double r94535 = a;
        double r94536 = 3.0;
        double r94537 = r94529 / r94536;
        double r94538 = r94535 - r94537;
        double r94539 = sqrt(r94538);
        double r94540 = r94534 / r94539;
        double r94541 = r94529 + r94540;
        double r94542 = r94541 * r94538;
        return r94542;
}

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 sqrt-prod0.2

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

  4. Applied add-sqr-sqrt0.2

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

  5. Applied times-frac0.2

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

  7. Applied frac-times0.2

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

  8. Applied associate-*l/0.1

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

  9. Simplified0.1

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

  11. Applied pow10.1

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

  13. Applied associate-/r*0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019354 +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))))