Average Error: 0.1 → 0.1
Time: 30.7s
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{rand}{3} + \left(a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{rand}{3} + \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3040864 = a;
        double r3040865 = 1.0;
        double r3040866 = 3.0;
        double r3040867 = r3040865 / r3040866;
        double r3040868 = r3040864 - r3040867;
        double r3040869 = 1.0;
        double r3040870 = 9.0;
        double r3040871 = r3040870 * r3040868;
        double r3040872 = sqrt(r3040871);
        double r3040873 = r3040869 / r3040872;
        double r3040874 = rand;
        double r3040875 = r3040873 * r3040874;
        double r3040876 = r3040869 + r3040875;
        double r3040877 = r3040868 * r3040876;
        return r3040877;
}
↓
double f(double a, double rand) {
        double r3040878 = a;
        double r3040879 = 1.0;
        double r3040880 = 3.0;
        double r3040881 = r3040879 / r3040880;
        double r3040882 = r3040878 - r3040881;
        double r3040883 = sqrt(r3040882);
        double r3040884 = r3040882 / r3040883;
        double r3040885 = rand;
        double r3040886 = 3.0;
        double r3040887 = r3040885 / r3040886;
        double r3040888 = r3040884 * r3040887;
        double r3040889 = r3040888 + r3040882;
        return r3040889;
}

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

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}}\]
  3. Using strategy rm

  4. Applied sqrt-prod0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{a - \frac{1.0}{3.0}}{\color{blue}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}}\]

  5. Applied *-un-lft-identity0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{a - \color{blue}{1 \cdot \frac{1.0}{3.0}}}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}\]

  6. Applied *-un-lft-identity0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\color{blue}{1 \cdot a} - 1 \cdot \frac{1.0}{3.0}}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}\]

  7. Applied distribute-lft-out--0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\color{blue}{1 \cdot \left(a - \frac{1.0}{3.0}\right)}}{\sqrt{9} \cdot \sqrt{a - \frac{1.0}{3.0}}}\]

  8. Applied times-frac0.2

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + rand \cdot \color{blue}{\left(\frac{1}{\sqrt{9}} \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}}\right)}\]

  9. Applied associate-*r*0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\left(rand \cdot \frac{1}{\sqrt{9}}\right) \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}}}\]

  10. Simplified0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \color{blue}{\frac{rand}{3}} \cdot \frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}}\]

  11. Final simplification0.1

    \[\leadsto \frac{a - \frac{1.0}{3.0}}{\sqrt{a - \frac{1.0}{3.0}}} \cdot \frac{rand}{3} + \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019143 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))