Average Error: 0.1 → 0.1
Time: 20.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(\frac{\frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}} + 1\right)\]
\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(\frac{\frac{1 \cdot rand}{\sqrt{a - \frac{1}{3}}}}{\sqrt{9}} + 1\right)
double f(double a, double rand) {
        double r62862 = a;
        double r62863 = 1.0;
        double r62864 = 3.0;
        double r62865 = r62863 / r62864;
        double r62866 = r62862 - r62865;
        double r62867 = 9.0;
        double r62868 = r62867 * r62866;
        double r62869 = sqrt(r62868);
        double r62870 = r62863 / r62869;
        double r62871 = rand;
        double r62872 = r62870 * r62871;
        double r62873 = r62863 + r62872;
        double r62874 = r62866 * r62873;
        return r62874;
}
↓
double f(double a, double rand) {
        double r62875 = a;
        double r62876 = 1.0;
        double r62877 = 3.0;
        double r62878 = r62876 / r62877;
        double r62879 = r62875 - r62878;
        double r62880 = rand;
        double r62881 = r62876 * r62880;
        double r62882 = sqrt(r62879);
        double r62883 = r62881 / r62882;
        double r62884 = 9.0;
        double r62885 = sqrt(r62884);
        double r62886 = r62883 / r62885;
        double r62887 = r62886 + r62876;
        double r62888 = r62879 * r62887;
        return r62888;
}

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-prod0.2

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

  4. Applied *-un-lft-identity0.2

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

  5. Applied times-frac0.2

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

  6. Applied associate-*l*0.2

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

  8. Applied distribute-lft-in0.2

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

  9. Simplified0.2

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

  11. Applied associate-*l/0.2

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2019297 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))