Average Error: 0.7 → 0.7
Time: 1.0m
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 r3261036 = beta;
        double r3261037 = alpha;
        double r3261038 = r3261036 - r3261037;
        double r3261039 = r3261037 + r3261036;
        double r3261040 = 2.0;
        double r3261041 = /* ERROR: no posit support in C */;
        double r3261042 = r3261039 + r3261041;
        double r3261043 = r3261038 / r3261042;
        double r3261044 = 1.0;
        double r3261045 = /* ERROR: no posit support in C */;
        double r3261046 = r3261043 + r3261045;
        double r3261047 = r3261046 / r3261041;
        return r3261047;
}
↓
double f(double alpha, double beta) {
        double r3261048 = beta;
        double r3261049 = alpha;
        double r3261050 = r3261048 - r3261049;
        double r3261051 = r3261049 + r3261048;
        double r3261052 = 2.0;
        double r3261053 = r3261051 + r3261052;
        double r3261054 = r3261050 / r3261053;
        double r3261055 = 1.0;
        double r3261056 = r3261054 + r3261055;
        double r3261057 = r3261056 / r3261052;
        return r3261057;
}

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 
(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)))