Average Error: 0.2 → 0.2
Time: 19.5s
Precision: 64
\[\left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right) \cdot \left(\frac{\left(1\right)}{\left(\left(\frac{\left(1\right)}{\left(\sqrt{\left(\left(9\right) \cdot \left(a - \left(\frac{\left(1.0\right)}{\left(3.0\right)}\right)\right)\right)}\right)}\right) \cdot rand\right)}\right)\]
↓
\[1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)\]
\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)
↓
1 \cdot \left(a - \frac{1.0}{3.0}\right) + \left(\frac{1}{\sqrt{9 \cdot \left(a - \frac{1.0}{3.0}\right)}} \cdot rand\right) \cdot \left(a - \frac{1.0}{3.0}\right)
double f(double a, double rand) {
        double r2672064 = a;
        double r2672065 = 1.0;
        double r2672066 = /* ERROR: no posit support in C */;
        double r2672067 = 3.0;
        double r2672068 = /* ERROR: no posit support in C */;
        double r2672069 = r2672066 / r2672068;
        double r2672070 = r2672064 - r2672069;
        double r2672071 = 1.0;
        double r2672072 = /* ERROR: no posit support in C */;
        double r2672073 = 9.0;
        double r2672074 = /* ERROR: no posit support in C */;
        double r2672075 = r2672074 * r2672070;
        double r2672076 = sqrt(r2672075);
        double r2672077 = r2672072 / r2672076;
        double r2672078 = rand;
        double r2672079 = r2672077 * r2672078;
        double r2672080 = r2672072 + r2672079;
        double r2672081 = r2672070 * r2672080;
        return r2672081;
}
↓
double f(double a, double rand) {
        double r2672082 = 1.0;
        double r2672083 = a;
        double r2672084 = 1.0;
        double r2672085 = 3.0;
        double r2672086 = r2672084 / r2672085;
        double r2672087 = r2672083 - r2672086;
        double r2672088 = r2672082 * r2672087;
        double r2672089 = 9.0;
        double r2672090 = r2672089 * r2672087;
        double r2672091 = sqrt(r2672090);
        double r2672092 = r2672082 / r2672091;
        double r2672093 = rand;
        double r2672094 = r2672092 * r2672093;
        double r2672095 = r2672094 * r2672087;
        double r2672096 = r2672088 + r2672095;
        return r2672096;
}

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 distribute-rgt-in0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019119 +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))))