Average Error: 0.1 → 0.1
Time: 2.9m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + rand \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}\]
double f(double a, double rand) {
        double r7157876 = a;
        double r7157877 = 1.0;
        double r7157878 = 3.0;
        double r7157879 = r7157877 / r7157878;
        double r7157880 = r7157876 - r7157879;
        double r7157881 = 1.0;
        double r7157882 = 9.0;
        double r7157883 = r7157882 * r7157880;
        double r7157884 = sqrt(r7157883);
        double r7157885 = r7157881 / r7157884;
        double r7157886 = rand;
        double r7157887 = r7157885 * r7157886;
        double r7157888 = r7157881 + r7157887;
        double r7157889 = r7157880 * r7157888;
        return r7157889;
}
↓
double f(double a, double rand) {
        double r7157890 = a;
        double r7157891 = 1.0;
        double r7157892 = 3.0;
        double r7157893 = r7157891 / r7157892;
        double r7157894 = r7157890 - r7157893;
        double r7157895 = rand;
        double r7157896 = sqrt(r7157894);
        double r7157897 = 3.0;
        double r7157898 = r7157896 / r7157897;
        double r7157899 = r7157895 * r7157898;
        double r7157900 = r7157894 + r7157899;
        return r7157900;
}


\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) + rand \cdot \frac{\sqrt{a - \frac{1.0}{3.0}}}{3}

Error

Bits error versus a

Bits error versus rand

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. Simplified0.1

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

  8. Simplified0.1

    \[\leadsto rand \cdot \left(\frac{\sqrt{a - \frac{1.0}{3.0}}}{3} \cdot \color{blue}{1}\right) + \left(a - \frac{1.0}{3.0}\right)\]

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019102 
(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))))