Average Error: 0.1 → 0.1
Time: 7.1s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{rand}}\]
\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}{\frac{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}{rand}}
double f(double a, double rand) {
        double r75801 = a;
        double r75802 = 1.0;
        double r75803 = 3.0;
        double r75804 = r75802 / r75803;
        double r75805 = r75801 - r75804;
        double r75806 = 9.0;
        double r75807 = r75806 * r75805;
        double r75808 = sqrt(r75807);
        double r75809 = r75802 / r75808;
        double r75810 = rand;
        double r75811 = r75809 * r75810;
        double r75812 = r75802 + r75811;
        double r75813 = r75805 * r75812;
        return r75813;
}
↓
double f(double a, double rand) {
        double r75814 = a;
        double r75815 = 1.0;
        double r75816 = 3.0;
        double r75817 = r75815 / r75816;
        double r75818 = r75814 - r75817;
        double r75819 = r75818 * r75815;
        double r75820 = 9.0;
        double r75821 = r75820 * r75818;
        double r75822 = sqrt(r75821);
        double r75823 = rand;
        double r75824 = r75822 / r75823;
        double r75825 = r75815 / r75824;
        double r75826 = r75818 * r75825;
        double r75827 = r75819 + r75826;
        return r75827;
}

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 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 associate-/l*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020049 +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))))