Average Error: 0.1 → 0.2
Time: 7.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9}} \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}\right)\]
\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{1}{\sqrt{9}} \cdot \frac{rand}{\sqrt{a - \frac{1}{3}}}\right)
double f(double a, double rand) {
        double r73848 = a;
        double r73849 = 1.0;
        double r73850 = 3.0;
        double r73851 = r73849 / r73850;
        double r73852 = r73848 - r73851;
        double r73853 = 9.0;
        double r73854 = r73853 * r73852;
        double r73855 = sqrt(r73854);
        double r73856 = r73849 / r73855;
        double r73857 = rand;
        double r73858 = r73856 * r73857;
        double r73859 = r73849 + r73858;
        double r73860 = r73852 * r73859;
        return r73860;
}
↓
double f(double a, double rand) {
        double r73861 = a;
        double r73862 = 1.0;
        double r73863 = 3.0;
        double r73864 = r73862 / r73863;
        double r73865 = r73861 - r73864;
        double r73866 = 9.0;
        double r73867 = sqrt(r73866);
        double r73868 = r73862 / r73867;
        double r73869 = rand;
        double r73870 = sqrt(r73865);
        double r73871 = r73869 / r73870;
        double r73872 = r73868 * r73871;
        double r73873 = r73862 + r73872;
        double r73874 = r73865 * r73873;
        return r73874;
}

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 times-frac0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 +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))))