Average Error: 0.7 → 0.7
Time: 19.4s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r4181233 = beta;
        double r4181234 = alpha;
        double r4181235 = r4181233 - r4181234;
        double r4181236 = r4181234 + r4181233;
        double r4181237 = 2.0;
        double r4181238 = /* ERROR: no posit support in C */;
        double r4181239 = r4181236 + r4181238;
        double r4181240 = r4181235 / r4181239;
        double r4181241 = 1.0;
        double r4181242 = /* ERROR: no posit support in C */;
        double r4181243 = r4181240 + r4181242;
        double r4181244 = r4181243 / r4181238;
        return r4181244;
}
↓
double f(double alpha, double beta) {
        double r4181245 = beta;
        double r4181246 = alpha;
        double r4181247 = r4181245 - r4181246;
        double r4181248 = 2.0;
        double r4181249 = r4181245 + r4181248;
        double r4181250 = r4181246 + r4181249;
        double r4181251 = r4181247 / r4181250;
        double r4181252 = 1.0;
        double r4181253 = r4181251 + r4181252;
        double r4181254 = r4181253 / r4181248;
        return r4181254;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.7

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied associate-+l+0.7

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\color{blue}{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  4. Final simplification0.7

    \[\leadsto \frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))