Average Error: 0.2 → 0.2
Time: 31.0s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[\left(a - \frac{1.0}{3.0}\right) \cdot \left(1 + \frac{1}{\sqrt{9 \cdot a + 9 \cdot \left(-\frac{1.0}{3.0}\right)}} \cdot rand\right)\]
\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 a + 9 \cdot \left(-\frac{1.0}{3.0}\right)}} \cdot rand\right)
double f(double a, double rand) {
        double r3019838 = a;
        double r3019839 = 1.0;
        double r3019840 = /* ERROR: no posit support in C */;
        double r3019841 = 3.0;
        double r3019842 = /* ERROR: no posit support in C */;
        double r3019843 = r3019840 / r3019842;
        double r3019844 = r3019838 - r3019843;
        double r3019845 = 1.0;
        double r3019846 = /* ERROR: no posit support in C */;
        double r3019847 = 9.0;
        double r3019848 = /* ERROR: no posit support in C */;
        double r3019849 = r3019848 * r3019844;
        double r3019850 = sqrt(r3019849);
        double r3019851 = r3019846 / r3019850;
        double r3019852 = rand;
        double r3019853 = r3019851 * r3019852;
        double r3019854 = r3019846 + r3019853;
        double r3019855 = r3019844 * r3019854;
        return r3019855;
}
↓
double f(double a, double rand) {
        double r3019856 = a;
        double r3019857 = 1.0;
        double r3019858 = 3.0;
        double r3019859 = r3019857 / r3019858;
        double r3019860 = r3019856 - r3019859;
        double r3019861 = 1.0;
        double r3019862 = 9.0;
        double r3019863 = r3019862 * r3019856;
        double r3019864 = -r3019859;
        double r3019865 = r3019862 * r3019864;
        double r3019866 = r3019863 + r3019865;
        double r3019867 = sqrt(r3019866);
        double r3019868 = r3019861 / r3019867;
        double r3019869 = rand;
        double r3019870 = r3019868 * r3019869;
        double r3019871 = r3019861 + r3019870;
        double r3019872 = r3019860 * r3019871;
        return r3019872;
}

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 sub-neg0.2

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

  4. Applied distribute-lft-in0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +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))))