Average Error: 0.1 → 0.1
Time: 6.0s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9 \cdot a + 9 \cdot \left(-\frac{1}{3}\right)}}\]
\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 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9 \cdot a + 9 \cdot \left(-\frac{1}{3}\right)}}
double f(double a, double rand) {
        double r74642 = a;
        double r74643 = 1.0;
        double r74644 = 3.0;
        double r74645 = r74643 / r74644;
        double r74646 = r74642 - r74645;
        double r74647 = 9.0;
        double r74648 = r74647 * r74646;
        double r74649 = sqrt(r74648);
        double r74650 = r74643 / r74649;
        double r74651 = rand;
        double r74652 = r74650 * r74651;
        double r74653 = r74643 + r74652;
        double r74654 = r74646 * r74653;
        return r74654;
}
↓
double f(double a, double rand) {
        double r74655 = a;
        double r74656 = 1.0;
        double r74657 = 3.0;
        double r74658 = r74656 / r74657;
        double r74659 = r74655 - r74658;
        double r74660 = r74659 * r74656;
        double r74661 = rand;
        double r74662 = r74656 * r74661;
        double r74663 = 9.0;
        double r74664 = r74663 * r74655;
        double r74665 = -r74658;
        double r74666 = r74663 * r74665;
        double r74667 = r74664 + r74666;
        double r74668 = sqrt(r74667);
        double r74669 = r74662 / r74668;
        double r74670 = r74659 * r74669;
        double r74671 = r74660 + r74670;
        return r74671;
}

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 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 sub-neg0.1

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

  8. Applied distribute-lft-in0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2020059 +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))))