Average Error: 0.2 → 0.1
Time: 6.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}\]
\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(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9} \cdot \sqrt{a - \frac{1}{3}}}
double f(double a, double rand) {
        double r80413 = a;
        double r80414 = 1.0;
        double r80415 = 3.0;
        double r80416 = r80414 / r80415;
        double r80417 = r80413 - r80416;
        double r80418 = 9.0;
        double r80419 = r80418 * r80417;
        double r80420 = sqrt(r80419);
        double r80421 = r80414 / r80420;
        double r80422 = rand;
        double r80423 = r80421 * r80422;
        double r80424 = r80414 + r80423;
        double r80425 = r80417 * r80424;
        return r80425;
}
↓
double f(double a, double rand) {
        double r80426 = a;
        double r80427 = 1.0;
        double r80428 = 3.0;
        double r80429 = r80427 / r80428;
        double r80430 = r80426 - r80429;
        double r80431 = r80430 * r80427;
        double r80432 = rand;
        double r80433 = r80427 * r80432;
        double r80434 = 9.0;
        double r80435 = sqrt(r80434);
        double r80436 = sqrt(r80430);
        double r80437 = r80435 * r80436;
        double r80438 = r80433 / r80437;
        double r80439 = r80430 * r80438;
        double r80440 = r80431 + r80439;
        return r80440;
}

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 0.2

    \[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
  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 distribute-lft-in0.1

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

  7. Applied sqrt-prod0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020062 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))