Average Error: 0.7 → 0.7
Time: 57.8s
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 r5772043 = beta;
        double r5772044 = alpha;
        double r5772045 = r5772043 - r5772044;
        double r5772046 = r5772044 + r5772043;
        double r5772047 = 2.0;
        double r5772048 = /* ERROR: no posit support in C */;
        double r5772049 = r5772046 + r5772048;
        double r5772050 = r5772045 / r5772049;
        double r5772051 = 1.0;
        double r5772052 = /* ERROR: no posit support in C */;
        double r5772053 = r5772050 + r5772052;
        double r5772054 = r5772053 / r5772048;
        return r5772054;
}
↓
double f(double alpha, double beta) {
        double r5772055 = beta;
        double r5772056 = alpha;
        double r5772057 = r5772055 - r5772056;
        double r5772058 = r5772056 + r5772055;
        double r5772059 = 2.0;
        double r5772060 = r5772058 + r5772059;
        double r5772061 = r5772057 / r5772060;
        double r5772062 = 1.0;
        double r5772063 = r5772061 + r5772062;
        double r5772064 = r5772063 / r5772059;
        return r5772064;
}

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 
(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)))