Average Error: 0.2 → 0.2
Time: 25.1s
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 r3734946 = a;
        double r3734947 = 1.0;
        double r3734948 = /* ERROR: no posit support in C */;
        double r3734949 = 3.0;
        double r3734950 = /* ERROR: no posit support in C */;
        double r3734951 = r3734948 / r3734950;
        double r3734952 = r3734946 - r3734951;
        double r3734953 = 1.0;
        double r3734954 = /* ERROR: no posit support in C */;
        double r3734955 = 9.0;
        double r3734956 = /* ERROR: no posit support in C */;
        double r3734957 = r3734956 * r3734952;
        double r3734958 = sqrt(r3734957);
        double r3734959 = r3734954 / r3734958;
        double r3734960 = rand;
        double r3734961 = r3734959 * r3734960;
        double r3734962 = r3734954 + r3734961;
        double r3734963 = r3734952 * r3734962;
        return r3734963;
}
↓
double f(double a, double rand) {
        double r3734964 = 1.0;
        double r3734965 = rand;
        double r3734966 = r3734965 * r3734964;
        double r3734967 = a;
        double r3734968 = 1.0;
        double r3734969 = 3.0;
        double r3734970 = r3734968 / r3734969;
        double r3734971 = r3734967 - r3734970;
        double r3734972 = 9.0;
        double r3734973 = r3734971 * r3734972;
        double r3734974 = r3734973 / r3734968;
        double r3734975 = sqrt(r3734974);
        double r3734976 = r3734966 / r3734975;
        double r3734977 = r3734964 + r3734976;
        double r3734978 = r3734977 * r3734971;
        return r3734978;
}

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 +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))))