Average Error: 0.1 → 0.1
Time: 2.3m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \left(\frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}\right) \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}\]
\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)
↓
\left(a - \frac{1.0}{3.0}\right) + \left(\frac{rand}{3} \cdot \sqrt{a - \frac{1.0}{3.0}}\right) \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{\sqrt{a - \frac{1.0}{3.0}}}
double f(double a, double rand) {
        double r13145869 = a;
        double r13145870 = 1.0;
        double r13145871 = 3.0;
        double r13145872 = r13145870 / r13145871;
        double r13145873 = r13145869 - r13145872;
        double r13145874 = 1.0;
        double r13145875 = 9.0;
        double r13145876 = r13145875 * r13145873;
        double r13145877 = sqrt(r13145876);
        double r13145878 = r13145874 / r13145877;
        double r13145879 = rand;
        double r13145880 = r13145878 * r13145879;
        double r13145881 = r13145874 + r13145880;
        double r13145882 = r13145873 * r13145881;
        return r13145882;
}
↓
double f(double a, double rand) {
        double r13145883 = a;
        double r13145884 = 1.0;
        double r13145885 = 3.0;
        double r13145886 = r13145884 / r13145885;
        double r13145887 = r13145883 - r13145886;
        double r13145888 = rand;
        double r13145889 = 3.0;
        double r13145890 = r13145888 / r13145889;
        double r13145891 = sqrt(r13145887);
        double r13145892 = r13145890 * r13145891;
        double r13145893 = r13145891 / r13145891;
        double r13145894 = r13145892 * r13145893;
        double r13145895 = r13145887 + r13145894;
        return r13145895;
}

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

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

  4. Applied sqrt-prod0.1

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

  5. Applied add-sqr-sqrt0.1

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

  6. Applied times-frac0.1

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

  7. Applied associate-*r*0.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019125 
(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))))