Average Error: 0.2 → 0.1
Time: 6.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\]
\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 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}
double f(double a, double rand) {
        double r88735 = a;
        double r88736 = 1.0;
        double r88737 = 3.0;
        double r88738 = r88736 / r88737;
        double r88739 = r88735 - r88738;
        double r88740 = 9.0;
        double r88741 = r88740 * r88739;
        double r88742 = sqrt(r88741);
        double r88743 = r88736 / r88742;
        double r88744 = rand;
        double r88745 = r88743 * r88744;
        double r88746 = r88736 + r88745;
        double r88747 = r88739 * r88746;
        return r88747;
}
↓
double f(double a, double rand) {
        double r88748 = a;
        double r88749 = 1.0;
        double r88750 = 3.0;
        double r88751 = r88749 / r88750;
        double r88752 = r88748 - r88751;
        double r88753 = r88752 * r88749;
        double r88754 = rand;
        double r88755 = r88749 * r88754;
        double r88756 = 9.0;
        double r88757 = sqrt(r88756);
        double r88758 = sqrt(r88752);
        double r88759 = r88757 * r88758;
        double r88760 = r88755 / r88759;
        double r88761 = r88752 * r88760;
        double r88762 = r88753 + r88761;
        return r88762;
}

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 0.2

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

  3. Applied associate-*l/0.1

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

  5. Applied distribute-lft-in0.1

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

  7. Applied sqrt-prod0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 +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))))