Average Error: 0.1 → 0.1
Time: 8.1s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot 1 + \left(a - \frac{1}{3}\right) \cdot \frac{1 \cdot rand}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}}\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
\left(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)}}
double f(double a, double rand) {
        double r75127 = a;
        double r75128 = 1.0;
        double r75129 = 3.0;
        double r75130 = r75128 / r75129;
        double r75131 = r75127 - r75130;
        double r75132 = 9.0;
        double r75133 = r75132 * r75131;
        double r75134 = sqrt(r75133);
        double r75135 = r75128 / r75134;
        double r75136 = rand;
        double r75137 = r75135 * r75136;
        double r75138 = r75128 + r75137;
        double r75139 = r75131 * r75138;
        return r75139;
}
↓
double f(double a, double rand) {
        double r75140 = a;
        double r75141 = 1.0;
        double r75142 = 3.0;
        double r75143 = r75141 / r75142;
        double r75144 = r75140 - r75143;
        double r75145 = r75144 * r75141;
        double r75146 = rand;
        double r75147 = r75141 * r75146;
        double r75148 = 9.0;
        double r75149 = r75148 * r75144;
        double r75150 = sqrt(r75149);
        double r75151 = r75147 / r75150;
        double r75152 = r75144 * r75151;
        double r75153 = r75145 + r75152;
        return r75153;
}

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. Final simplification0.1

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

Reproduce

herbie shell --seed 2020100 +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))))