Average Error: 0.2 → 0.2
Time: 25.0s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot 9}{1.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]
\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(1 + \frac{rand \cdot 1}{\sqrt{\frac{\left(a - \frac{1.0}{3.0}\right) \cdot 9}{1.0}}}\right) \cdot \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r3734802 = a;
        double r3734803 = 1.0;
        double r3734804 = /* ERROR: no posit support in C */;
        double r3734805 = 3.0;
        double r3734806 = /* ERROR: no posit support in C */;
        double r3734807 = r3734804 / r3734806;
        double r3734808 = r3734802 - r3734807;
        double r3734809 = 1.0;
        double r3734810 = /* ERROR: no posit support in C */;
        double r3734811 = 9.0;
        double r3734812 = /* ERROR: no posit support in C */;
        double r3734813 = r3734812 * r3734808;
        double r3734814 = sqrt(r3734813);
        double r3734815 = r3734810 / r3734814;
        double r3734816 = rand;
        double r3734817 = r3734815 * r3734816;
        double r3734818 = r3734810 + r3734817;
        double r3734819 = r3734808 * r3734818;
        return r3734819;
}
↓
double f(double a, double rand) {
        double r3734820 = 1.0;
        double r3734821 = rand;
        double r3734822 = r3734821 * r3734820;
        double r3734823 = a;
        double r3734824 = 1.0;
        double r3734825 = 3.0;
        double r3734826 = r3734824 / r3734825;
        double r3734827 = r3734823 - r3734826;
        double r3734828 = 9.0;
        double r3734829 = r3734827 * r3734828;
        double r3734830 = r3734829 / r3734824;
        double r3734831 = sqrt(r3734830);
        double r3734832 = r3734822 / r3734831;
        double r3734833 = r3734820 + r3734832;
        double r3734834 = r3734833 * r3734827;
        return r3734834;
}

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 p16-flip--0.2

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

  4. Applied associate-*r/0.2

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

  5. Simplified0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019125 
(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))))