Average Error: 0.2 → 0.1
Time: 3.2m
Precision: 64
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\]
\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
↓
\left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r6025127 = a;
        double r6025128 = 1.0;
        double r6025129 = 3.0;
        double r6025130 = r6025128 / r6025129;
        double r6025131 = r6025127 - r6025130;
        double r6025132 = 1.0;
        double r6025133 = 9.0;
        double r6025134 = r6025133 * r6025131;
        double r6025135 = sqrt(r6025134);
        double r6025136 = r6025132 / r6025135;
        double r6025137 = rand;
        double r6025138 = r6025136 * r6025137;
        double r6025139 = r6025132 + r6025138;
        double r6025140 = r6025131 * r6025139;
        return r6025140;
}
↓
double f(double a, double rand) {
        double r6025141 = a;
        double r6025142 = 1.0;
        double r6025143 = 3.0;
        double r6025144 = r6025142 / r6025143;
        double r6025145 = r6025141 - r6025144;
        double r6025146 = rand;
        double r6025147 = 3.0;
        double r6025148 = r6025145 * r6025147;
        double r6025149 = r6025147 * r6025148;
        double r6025150 = sqrt(r6025149);
        double r6025151 = r6025146 / r6025150;
        double r6025152 = r6025151 * r6025145;
        double r6025153 = r6025145 + r6025152;
        return r6025153;
}

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

    \[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{(\left(\frac{rand}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt0.1

    \[\leadsto (\left(\frac{rand}{\sqrt{\color{blue}{\left(\sqrt{9} \cdot \sqrt{9}\right)} \cdot \left(a - \frac{1.0}{3.0}\right)}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]

  5. Applied associate-*l*0.1

    \[\leadsto (\left(\frac{rand}{\sqrt{\color{blue}{\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right))_*\]
  6. Using strategy rm

  7. Applied fma-udef0.1

    \[\leadsto \color{blue}{\frac{rand}{\sqrt{\sqrt{9} \cdot \left(\sqrt{9} \cdot \left(a - \frac{1.0}{3.0}\right)\right)}} \cdot \left(a - \frac{1.0}{3.0}\right) + \left(a - \frac{1.0}{3.0}\right)}\]

  8. Final simplification0.1

    \[\leadsto \left(a - \frac{1.0}{3.0}\right) + \frac{rand}{\sqrt{3 \cdot \left(\left(a - \frac{1.0}{3.0}\right) \cdot 3\right)}} \cdot \left(a - \frac{1.0}{3.0}\right)\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (* (- a (/ 1.0 3.0)) (+ 1 (* (/ 1 (sqrt (* 9 (- a (/ 1.0 3.0))))) rand))))