Average Error: 0.2 → 0.2
Time: 19.7s
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 r2147537 = a;
        double r2147538 = 1.0;
        double r2147539 = /* ERROR: no posit support in C */;
        double r2147540 = 3.0;
        double r2147541 = /* ERROR: no posit support in C */;
        double r2147542 = r2147539 / r2147541;
        double r2147543 = r2147537 - r2147542;
        double r2147544 = 1.0;
        double r2147545 = /* ERROR: no posit support in C */;
        double r2147546 = 9.0;
        double r2147547 = /* ERROR: no posit support in C */;
        double r2147548 = r2147547 * r2147543;
        double r2147549 = sqrt(r2147548);
        double r2147550 = r2147545 / r2147549;
        double r2147551 = rand;
        double r2147552 = r2147550 * r2147551;
        double r2147553 = r2147545 + r2147552;
        double r2147554 = r2147543 * r2147553;
        return r2147554;
}
↓
double f(double a, double rand) {
        double r2147555 = a;
        double r2147556 = 1.0;
        double r2147557 = 3.0;
        double r2147558 = r2147556 / r2147557;
        double r2147559 = r2147555 - r2147558;
        double r2147560 = 1.0;
        double r2147561 = 9.0;
        double r2147562 = r2147561 * r2147559;
        double r2147563 = sqrt(r2147562);
        double r2147564 = r2147560 / r2147563;
        double r2147565 = rand;
        double r2147566 = r2147564 * r2147565;
        double r2147567 = r2147560 + r2147566;
        double r2147568 = r2147559 * r2147567;
        return r2147568;
}

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 
(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))))