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 r59004 = a;
        double r59005 = 1.0;
        double r59006 = 3.0;
        double r59007 = r59005 / r59006;
        double r59008 = r59004 - r59007;
        double r59009 = 9.0;
        double r59010 = r59009 * r59008;
        double r59011 = sqrt(r59010);
        double r59012 = r59005 / r59011;
        double r59013 = rand;
        double r59014 = r59012 * r59013;
        double r59015 = r59005 + r59014;
        double r59016 = r59008 * r59015;
        return r59016;
}
↓
double f(double a, double rand) {
        double r59017 = 1.0;
        double r59018 = a;
        double r59019 = 3.0;
        double r59020 = r59017 / r59019;
        double r59021 = r59018 - r59020;
        double r59022 = r59017 * r59021;
        double r59023 = rand;
        double r59024 = 9.0;
        double r59025 = r59024 * r59021;
        double r59026 = sqrt(r59025);
        double r59027 = r59023 / r59026;
        double r59028 = r59017 * r59027;
        double r59029 = r59028 * r59021;
        double r59030 = r59022 + r59029;
        return r59030;
}

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 +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))))