Average Error: 0.2 → 0.2
Time: 47.0s
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 r4981808 = a;
        double r4981809 = 1.0;
        double r4981810 = /* ERROR: no posit support in C */;
        double r4981811 = 3.0;
        double r4981812 = /* ERROR: no posit support in C */;
        double r4981813 = r4981810 / r4981812;
        double r4981814 = r4981808 - r4981813;
        double r4981815 = 1.0;
        double r4981816 = /* ERROR: no posit support in C */;
        double r4981817 = 9.0;
        double r4981818 = /* ERROR: no posit support in C */;
        double r4981819 = r4981818 * r4981814;
        double r4981820 = sqrt(r4981819);
        double r4981821 = r4981816 / r4981820;
        double r4981822 = rand;
        double r4981823 = r4981821 * r4981822;
        double r4981824 = r4981816 + r4981823;
        double r4981825 = r4981814 * r4981824;
        return r4981825;
}
↓
double f(double a, double rand) {
        double r4981826 = a;
        double r4981827 = 1.0;
        double r4981828 = 3.0;
        double r4981829 = r4981827 / r4981828;
        double r4981830 = r4981826 - r4981829;
        double r4981831 = 1.0;
        double r4981832 = 9.0;
        double r4981833 = r4981832 * r4981830;
        double r4981834 = sqrt(r4981833);
        double r4981835 = r4981831 / r4981834;
        double r4981836 = rand;
        double r4981837 = r4981835 * r4981836;
        double r4981838 = r4981831 + r4981837;
        double r4981839 = r4981830 * r4981838;
        return r4981839;
}

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 +o rules:numerics
(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))))