Average Error: 0.1 → 0.1
Time: 7.4s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(\left(a - \frac{1}{3}\right) \cdot \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot rand + \left(a - \frac{1}{3}\right) \cdot 1\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\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 1
double f(double a, double rand) {
        double r127411 = a;
        double r127412 = 1.0;
        double r127413 = 3.0;
        double r127414 = r127412 / r127413;
        double r127415 = r127411 - r127414;
        double r127416 = 9.0;
        double r127417 = r127416 * r127415;
        double r127418 = sqrt(r127417);
        double r127419 = r127412 / r127418;
        double r127420 = rand;
        double r127421 = r127419 * r127420;
        double r127422 = r127412 + r127421;
        double r127423 = r127415 * r127422;
        return r127423;
}
↓
double f(double a, double rand) {
        double r127424 = a;
        double r127425 = 1.0;
        double r127426 = 3.0;
        double r127427 = r127425 / r127426;
        double r127428 = r127424 - r127427;
        double r127429 = 9.0;
        double r127430 = r127429 * r127428;
        double r127431 = sqrt(r127430);
        double r127432 = r127425 / r127431;
        double r127433 = r127428 * r127432;
        double r127434 = rand;
        double r127435 = r127433 * r127434;
        double r127436 = r127428 * r127425;
        double r127437 = r127435 + r127436;
        return r127437;
}

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. Using strategy rm

  7. Applied *-un-lft-identity0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2020025 +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))))