Average Error: 0.1 → 0.1
Time: 16.7s
Precision: 64
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1}{3}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1}{3}\right) \cdot \left(1 + \frac{1 \cdot rand}{\sqrt{\left(9 \cdot \sqrt{a - \frac{1}{3}}\right) \cdot \sqrt{a - \frac{1}{3}}}}\right)\]
\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 \left(1 + \frac{1 \cdot rand}{\sqrt{\left(9 \cdot \sqrt{a - \frac{1}{3}}\right) \cdot \sqrt{a - \frac{1}{3}}}}\right)
double f(double a, double rand) {
        double r78123 = a;
        double r78124 = 1.0;
        double r78125 = 3.0;
        double r78126 = r78124 / r78125;
        double r78127 = r78123 - r78126;
        double r78128 = 9.0;
        double r78129 = r78128 * r78127;
        double r78130 = sqrt(r78129);
        double r78131 = r78124 / r78130;
        double r78132 = rand;
        double r78133 = r78131 * r78132;
        double r78134 = r78124 + r78133;
        double r78135 = r78127 * r78134;
        return r78135;
}
↓
double f(double a, double rand) {
        double r78136 = a;
        double r78137 = 1.0;
        double r78138 = 3.0;
        double r78139 = r78137 / r78138;
        double r78140 = r78136 - r78139;
        double r78141 = rand;
        double r78142 = r78137 * r78141;
        double r78143 = 9.0;
        double r78144 = sqrt(r78140);
        double r78145 = r78143 * r78144;
        double r78146 = r78145 * r78144;
        double r78147 = sqrt(r78146);
        double r78148 = r78142 / r78147;
        double r78149 = r78137 + r78148;
        double r78150 = r78140 * r78149;
        return r78150;
}

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 add-sqr-sqrt0.2

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

  6. Applied associate-*r*0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019351 +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))))