Average Error: 0.1 → 0.1
Time: 7.6s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand\]
\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 1 + \left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand
double f(double a, double rand) {
        double r83175 = a;
        double r83176 = 1.0;
        double r83177 = 3.0;
        double r83178 = r83176 / r83177;
        double r83179 = r83175 - r83178;
        double r83180 = 9.0;
        double r83181 = r83180 * r83179;
        double r83182 = sqrt(r83181);
        double r83183 = r83176 / r83182;
        double r83184 = rand;
        double r83185 = r83183 * r83184;
        double r83186 = r83176 + r83185;
        double r83187 = r83179 * r83186;
        return r83187;
}
↓
double f(double a, double rand) {
        double r83188 = a;
        double r83189 = 1.0;
        double r83190 = 3.0;
        double r83191 = r83189 / r83190;
        double r83192 = r83188 - r83191;
        double r83193 = r83192 * r83189;
        double r83194 = 9.0;
        double r83195 = r83194 * r83192;
        double r83196 = sqrt(r83195);
        double r83197 = r83189 / r83196;
        double r83198 = r83192 * r83197;
        double r83199 = rand;
        double r83200 = r83198 * r83199;
        double r83201 = r83193 + r83200;
        return r83201;
}

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 distribute-lft-in0.1

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

  5. Applied associate-*r*0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020047 +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))))