Average Error: 0.7 → 0.7
Time: 58.1s
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 r4112209 = beta;
        double r4112210 = alpha;
        double r4112211 = r4112209 - r4112210;
        double r4112212 = r4112210 + r4112209;
        double r4112213 = 2.0;
        double r4112214 = /* ERROR: no posit support in C */;
        double r4112215 = r4112212 + r4112214;
        double r4112216 = r4112211 / r4112215;
        double r4112217 = 1.0;
        double r4112218 = /* ERROR: no posit support in C */;
        double r4112219 = r4112216 + r4112218;
        double r4112220 = r4112219 / r4112214;
        return r4112220;
}
↓
double f(double alpha, double beta) {
        double r4112221 = beta;
        double r4112222 = alpha;
        double r4112223 = r4112221 - r4112222;
        double r4112224 = r4112222 + r4112221;
        double r4112225 = 2.0;
        double r4112226 = r4112224 + r4112225;
        double r4112227 = r4112223 / r4112226;
        double r4112228 = 1.0;
        double r4112229 = r4112227 + r4112228;
        double r4112230 = r4112229 / r4112225;
        return r4112230;
}

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 2019134 +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)))