Average Error: 0.7 → 0.7
Time: 17.7s
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 r2449636 = beta;
        double r2449637 = alpha;
        double r2449638 = r2449636 - r2449637;
        double r2449639 = r2449637 + r2449636;
        double r2449640 = 2.0;
        double r2449641 = /* ERROR: no posit support in C */;
        double r2449642 = r2449639 + r2449641;
        double r2449643 = r2449638 / r2449642;
        double r2449644 = 1.0;
        double r2449645 = /* ERROR: no posit support in C */;
        double r2449646 = r2449643 + r2449645;
        double r2449647 = r2449646 / r2449641;
        return r2449647;
}
↓
double f(double alpha, double beta) {
        double r2449648 = beta;
        double r2449649 = alpha;
        double r2449650 = r2449648 - r2449649;
        double r2449651 = 2.0;
        double r2449652 = r2449648 + r2449651;
        double r2449653 = r2449649 + r2449652;
        double r2449654 = r2449650 / r2449653;
        double r2449655 = 1.0;
        double r2449656 = r2449654 + r2449655;
        double r2449657 = r2449656 / r2449651;
        return r2449657;
}

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 2019124 +o rules:numerics
(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)))