Average Error: 0.7 → 0.7
Time: 59.0s
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 r4648247 = beta;
        double r4648248 = alpha;
        double r4648249 = r4648247 - r4648248;
        double r4648250 = r4648248 + r4648247;
        double r4648251 = 2.0;
        double r4648252 = /* ERROR: no posit support in C */;
        double r4648253 = r4648250 + r4648252;
        double r4648254 = r4648249 / r4648253;
        double r4648255 = 1.0;
        double r4648256 = /* ERROR: no posit support in C */;
        double r4648257 = r4648254 + r4648256;
        double r4648258 = r4648257 / r4648252;
        return r4648258;
}
↓
double f(double alpha, double beta) {
        double r4648259 = beta;
        double r4648260 = alpha;
        double r4648261 = r4648259 - r4648260;
        double r4648262 = r4648260 + r4648259;
        double r4648263 = 2.0;
        double r4648264 = r4648262 + r4648263;
        double r4648265 = r4648261 / r4648264;
        double r4648266 = 1.0;
        double r4648267 = r4648265 + r4648266;
        double r4648268 = r4648267 / r4648263;
        return r4648268;
}

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 2019135 +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)))