Average Error: 0.2 → 0.2
Time: 47.5s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\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 \left(1 + \frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r6157136 = a;
        double r6157137 = 1.0;
        double r6157138 = /* ERROR: no posit support in C */;
        double r6157139 = 3.0;
        double r6157140 = /* ERROR: no posit support in C */;
        double r6157141 = r6157138 / r6157140;
        double r6157142 = r6157136 - r6157141;
        double r6157143 = 1.0;
        double r6157144 = /* ERROR: no posit support in C */;
        double r6157145 = 9.0;
        double r6157146 = /* ERROR: no posit support in C */;
        double r6157147 = r6157146 * r6157142;
        double r6157148 = sqrt(r6157147);
        double r6157149 = r6157144 / r6157148;
        double r6157150 = rand;
        double r6157151 = r6157149 * r6157150;
        double r6157152 = r6157144 + r6157151;
        double r6157153 = r6157142 * r6157152;
        return r6157153;
}
↓
double f(double a, double rand) {
        double r6157154 = a;
        double r6157155 = 1.0;
        double r6157156 = 3.0;
        double r6157157 = r6157155 / r6157156;
        double r6157158 = r6157154 - r6157157;
        double r6157159 = 1.0;
        double r6157160 = 9.0;
        double r6157161 = r6157160 * r6157158;
        double r6157162 = sqrt(r6157161);
        double r6157163 = r6157159 / r6157162;
        double r6157164 = rand;
        double r6157165 = r6157163 * r6157164;
        double r6157166 = r6157159 + r6157165;
        double r6157167 = r6157158 * r6157166;
        return r6157167;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019130 
(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))))