Average Error: 0.2 → 0.2
Time: 24.6s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)
↓
\left(a - \frac{1.0}{3.0}\right) \cdot 1 + \left(a - \frac{1.0}{3.0}\right) \cdot \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r3504141 = a;
        double r3504142 = 1.0;
        double r3504143 = /* ERROR: no posit support in C */;
        double r3504144 = 3.0;
        double r3504145 = /* ERROR: no posit support in C */;
        double r3504146 = r3504143 / r3504145;
        double r3504147 = r3504141 - r3504146;
        double r3504148 = 1.0;
        double r3504149 = /* ERROR: no posit support in C */;
        double r3504150 = 9.0;
        double r3504151 = /* ERROR: no posit support in C */;
        double r3504152 = r3504151 * r3504147;
        double r3504153 = sqrt(r3504152);
        double r3504154 = r3504149 / r3504153;
        double r3504155 = rand;
        double r3504156 = r3504154 * r3504155;
        double r3504157 = r3504149 + r3504156;
        double r3504158 = r3504147 * r3504157;
        return r3504158;
}
↓
double f(double a, double rand) {
        double r3504159 = a;
        double r3504160 = 1.0;
        double r3504161 = 3.0;
        double r3504162 = r3504160 / r3504161;
        double r3504163 = r3504159 - r3504162;
        double r3504164 = 1.0;
        double r3504165 = r3504163 * r3504164;
        double r3504166 = 9.0;
        double r3504167 = r3504166 * r3504163;
        double r3504168 = sqrt(r3504167);
        double r3504169 = r3504164 / r3504168;
        double r3504170 = rand;
        double r3504171 = r3504169 * r3504170;
        double r3504172 = r3504163 * r3504171;
        double r3504173 = r3504165 + r3504172;
        return r3504173;
}

Error

Bits error versus a

Bits error versus rand

Derivation

  1. Initial program 0.2

    \[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
  2. Using strategy rm

  3. Applied distribute-lft-in0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019104 
(FPCore (a rand)
  :name "Octave 3.8, oct_fill_randg"
  (*.p16 (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0))) (+.p16 (real->posit16 1) (*.p16 (/.p16 (real->posit16 1) (sqrt.p16 (*.p16 (real->posit16 9) (-.p16 a (/.p16 (real->posit16 1.0) (real->posit16 3.0)))))) rand))))