Average Error: 0.1 → 0.1
Time: 23.3s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + 1 \cdot \frac{a - \frac{1}{3}}{\frac{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}{rand}}\]
\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 + 1 \cdot \frac{a - \frac{1}{3}}{\frac{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}{rand}}
double f(double a, double rand) {
        double r76789 = a;
        double r76790 = 1.0;
        double r76791 = 3.0;
        double r76792 = r76790 / r76791;
        double r76793 = r76789 - r76792;
        double r76794 = 9.0;
        double r76795 = r76794 * r76793;
        double r76796 = sqrt(r76795);
        double r76797 = r76790 / r76796;
        double r76798 = rand;
        double r76799 = r76797 * r76798;
        double r76800 = r76790 + r76799;
        double r76801 = r76793 * r76800;
        return r76801;
}
↓
double f(double a, double rand) {
        double r76802 = a;
        double r76803 = 1.0;
        double r76804 = 3.0;
        double r76805 = r76803 / r76804;
        double r76806 = r76802 - r76805;
        double r76807 = r76806 * r76803;
        double r76808 = 9.0;
        double r76809 = sqrt(r76808);
        double r76810 = sqrt(r76806);
        double r76811 = r76809 * r76810;
        double r76812 = rand;
        double r76813 = r76811 / r76812;
        double r76814 = r76806 / r76813;
        double r76815 = r76803 * r76814;
        double r76816 = r76807 + r76815;
        return r76816;
}

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-prod0.1

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

  5. Applied div-inv0.1

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

  6. Applied associate-*l*0.1

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

  7. Simplified0.1

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

  9. Applied clear-num0.1

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

  11. Applied distribute-lft-in0.1

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

  12. Simplified0.1

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

  13. Final simplification0.1

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

Reproduce

herbie shell --seed 2019209 +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))))