Average Error: 0.7 → 0.7
Time: 18.8s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
\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}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r2232382 = beta;
        double r2232383 = alpha;
        double r2232384 = r2232382 - r2232383;
        double r2232385 = r2232383 + r2232382;
        double r2232386 = 2.0;
        double r2232387 = /* ERROR: no posit support in C */;
        double r2232388 = r2232385 + r2232387;
        double r2232389 = r2232384 / r2232388;
        double r2232390 = 1.0;
        double r2232391 = /* ERROR: no posit support in C */;
        double r2232392 = r2232389 + r2232391;
        double r2232393 = r2232392 / r2232387;
        return r2232393;
}
↓
double f(double alpha, double beta) {
        double r2232394 = beta;
        double r2232395 = alpha;
        double r2232396 = r2232394 - r2232395;
        double r2232397 = r2232395 + r2232394;
        double r2232398 = 2.0;
        double r2232399 = r2232397 + r2232398;
        double r2232400 = r2232396 / r2232399;
        double r2232401 = 1.0;
        double r2232402 = r2232400 + r2232401;
        double r2232403 = r2232402 / r2232398;
        return r2232403;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019112 +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)))