Average Error: 0.1 → 0.1
Time: 28.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \left(1 \cdot \frac{rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)\]
\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 r76809 = a;
        double r76810 = 1.0;
        double r76811 = 3.0;
        double r76812 = r76810 / r76811;
        double r76813 = r76809 - r76812;
        double r76814 = 9.0;
        double r76815 = r76814 * r76813;
        double r76816 = sqrt(r76815);
        double r76817 = r76810 / r76816;
        double r76818 = rand;
        double r76819 = r76817 * r76818;
        double r76820 = r76810 + r76819;
        double r76821 = r76813 * r76820;
        return r76821;
}
↓
double f(double a, double rand) {
        double r76822 = 1.0;
        double r76823 = a;
        double r76824 = 3.0;
        double r76825 = r76822 / r76824;
        double r76826 = r76823 - r76825;
        double r76827 = r76822 * r76826;
        double r76828 = rand;
        double r76829 = 9.0;
        double r76830 = r76829 * r76826;
        double r76831 = sqrt(r76830);
        double r76832 = r76828 / r76831;
        double r76833 = r76822 * r76832;
        double r76834 = r76833 * r76826;
        double r76835 = r76827 + r76834;
        return r76835;
}

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 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))