Average Error: 0.2 → 0.1
Time: 35.5s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[1 \cdot \left(a - \frac{1}{3}\right) + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)\]
\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)
↓
1 \cdot \left(a - \frac{1}{3}\right) + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right) \cdot \left(a - \frac{1}{3}\right)
double f(double a, double rand) {
        double r95103 = a;
        double r95104 = 1.0;
        double r95105 = 3.0;
        double r95106 = r95104 / r95105;
        double r95107 = r95103 - r95106;
        double r95108 = 9.0;
        double r95109 = r95108 * r95107;
        double r95110 = sqrt(r95109);
        double r95111 = r95104 / r95110;
        double r95112 = rand;
        double r95113 = r95111 * r95112;
        double r95114 = r95104 + r95113;
        double r95115 = r95107 * r95114;
        return r95115;
}
↓
double f(double a, double rand) {
        double r95116 = 1.0;
        double r95117 = a;
        double r95118 = 3.0;
        double r95119 = r95116 / r95118;
        double r95120 = r95117 - r95119;
        double r95121 = r95116 * r95120;
        double r95122 = 9.0;
        double r95123 = r95122 * r95120;
        double r95124 = sqrt(r95123);
        double r95125 = r95116 / r95124;
        double r95126 = rand;
        double r95127 = r95125 * r95126;
        double r95128 = r95127 * r95120;
        double r95129 = r95121 + r95128;
        return r95129;
}

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 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. Simplified0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019306 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  :precision binary64
  (* (- a (/ 1 3)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1 3))))) rand))))