Average Error: 0.1 → 0.1
Time: 21.8s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(1 + \frac{rand \cdot 1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r75743 = a;
        double r75744 = 1.0;
        double r75745 = 3.0;
        double r75746 = r75744 / r75745;
        double r75747 = r75743 - r75746;
        double r75748 = 9.0;
        double r75749 = r75748 * r75747;
        double r75750 = sqrt(r75749);
        double r75751 = r75744 / r75750;
        double r75752 = rand;
        double r75753 = r75751 * r75752;
        double r75754 = r75744 + r75753;
        double r75755 = r75747 * r75754;
        return r75755;
}
↓
double f(double a, double rand) {
        double r75756 = 1.0;
        double r75757 = rand;
        double r75758 = r75757 * r75756;
        double r75759 = 9.0;
        double r75760 = a;
        double r75761 = 3.0;
        double r75762 = r75756 / r75761;
        double r75763 = r75760 - r75762;
        double r75764 = r75759 * r75763;
        double r75765 = sqrt(r75764);
        double r75766 = r75758 / r75765;
        double r75767 = r75756 + r75766;
        double r75768 = r75767 * r75763;
        return r75768;
}

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. Simplified0.1

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

  4. Applied fma-udef0.1

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

  5. Simplified0.1

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

  7. Applied *-commutative0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1.0 (* (/ 1.0 (sqrt (* 9.0 (- a (/ 1.0 3.0))))) rand))))