Average Error: 0.7 → 0.7
Time: 20.2s
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 r1620474 = beta;
        double r1620475 = alpha;
        double r1620476 = r1620474 - r1620475;
        double r1620477 = r1620475 + r1620474;
        double r1620478 = 2.0;
        double r1620479 = /* ERROR: no posit support in C */;
        double r1620480 = r1620477 + r1620479;
        double r1620481 = r1620476 / r1620480;
        double r1620482 = 1.0;
        double r1620483 = /* ERROR: no posit support in C */;
        double r1620484 = r1620481 + r1620483;
        double r1620485 = r1620484 / r1620479;
        return r1620485;
}
↓
double f(double alpha, double beta) {
        double r1620486 = beta;
        double r1620487 = alpha;
        double r1620488 = r1620486 - r1620487;
        double r1620489 = 2.0;
        double r1620490 = r1620486 + r1620489;
        double r1620491 = r1620487 + r1620490;
        double r1620492 = r1620488 / r1620491;
        double r1620493 = 1.0;
        double r1620494 = r1620492 + r1620493;
        double r1620495 = r1620494 / r1620489;
        return r1620495;
}

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 2019107 +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)))