Average Error: 0.1 → 0.1
Time: 6.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{\frac{1 \cdot rand}{\sqrt{9}}}{\sqrt{a - \frac{1}{3}}}\right)
double f(double a, double rand) {
        double r70839 = a;
        double r70840 = 1.0;
        double r70841 = 3.0;
        double r70842 = r70840 / r70841;
        double r70843 = r70839 - r70842;
        double r70844 = 9.0;
        double r70845 = r70844 * r70843;
        double r70846 = sqrt(r70845);
        double r70847 = r70840 / r70846;
        double r70848 = rand;
        double r70849 = r70847 * r70848;
        double r70850 = r70840 + r70849;
        double r70851 = r70843 * r70850;
        return r70851;
}
↓
double f(double a, double rand) {
        double r70852 = a;
        double r70853 = 1.0;
        double r70854 = 3.0;
        double r70855 = r70853 / r70854;
        double r70856 = r70852 - r70855;
        double r70857 = rand;
        double r70858 = r70853 * r70857;
        double r70859 = 9.0;
        double r70860 = sqrt(r70859);
        double r70861 = r70858 / r70860;
        double r70862 = sqrt(r70856);
        double r70863 = r70861 / r70862;
        double r70864 = r70853 + r70863;
        double r70865 = r70856 * r70864;
        return r70865;
}

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 sqrt-prod0.1

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

  6. Applied associate-/r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2020057 +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))))