Average Error: 0.7 → 0.7
Time: 20.1s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]
\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}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r2355049 = beta;
        double r2355050 = alpha;
        double r2355051 = r2355049 - r2355050;
        double r2355052 = r2355050 + r2355049;
        double r2355053 = 2.0;
        double r2355054 = /* ERROR: no posit support in C */;
        double r2355055 = r2355052 + r2355054;
        double r2355056 = r2355051 / r2355055;
        double r2355057 = 1.0;
        double r2355058 = /* ERROR: no posit support in C */;
        double r2355059 = r2355056 + r2355058;
        double r2355060 = r2355059 / r2355054;
        return r2355060;
}
↓
double f(double alpha, double beta) {
        double r2355061 = beta;
        double r2355062 = alpha;
        double r2355063 = r2355061 - r2355062;
        double r2355064 = 2.0;
        double r2355065 = r2355061 + r2355064;
        double r2355066 = r2355062 + r2355065;
        double r2355067 = r2355063 / r2355066;
        double r2355068 = 1.0;
        double r2355069 = r2355067 + r2355068;
        double r2355070 = r2355069 / r2355064;
        return r2355070;
}

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. Using strategy rm

  3. Applied associate-+l+0.7

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019119 
(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)))