Average Error: 0.2 → 0.2
Time: 18.7s
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 r1783658 = a;
        double r1783659 = 1.0;
        double r1783660 = /* ERROR: no posit support in C */;
        double r1783661 = 3.0;
        double r1783662 = /* ERROR: no posit support in C */;
        double r1783663 = r1783660 / r1783662;
        double r1783664 = r1783658 - r1783663;
        double r1783665 = 1.0;
        double r1783666 = /* ERROR: no posit support in C */;
        double r1783667 = 9.0;
        double r1783668 = /* ERROR: no posit support in C */;
        double r1783669 = r1783668 * r1783664;
        double r1783670 = sqrt(r1783669);
        double r1783671 = r1783666 / r1783670;
        double r1783672 = rand;
        double r1783673 = r1783671 * r1783672;
        double r1783674 = r1783666 + r1783673;
        double r1783675 = r1783664 * r1783674;
        return r1783675;
}
↓
double f(double a, double rand) {
        double r1783676 = a;
        double r1783677 = 1.0;
        double r1783678 = 3.0;
        double r1783679 = r1783677 / r1783678;
        double r1783680 = r1783676 - r1783679;
        double r1783681 = 1.0;
        double r1783682 = 9.0;
        double r1783683 = r1783682 * r1783680;
        double r1783684 = sqrt(r1783683);
        double r1783685 = r1783681 / r1783684;
        double r1783686 = rand;
        double r1783687 = r1783685 * r1783686;
        double r1783688 = r1783681 + r1783687;
        double r1783689 = r1783680 * r1783688;
        return r1783689;
}

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 2019124 +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))))