Average Error: 0.7 → 0.7
Time: 20.9s
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 r3513049 = beta;
        double r3513050 = alpha;
        double r3513051 = r3513049 - r3513050;
        double r3513052 = r3513050 + r3513049;
        double r3513053 = 2.0;
        double r3513054 = /* ERROR: no posit support in C */;
        double r3513055 = r3513052 + r3513054;
        double r3513056 = r3513051 / r3513055;
        double r3513057 = 1.0;
        double r3513058 = /* ERROR: no posit support in C */;
        double r3513059 = r3513056 + r3513058;
        double r3513060 = r3513059 / r3513054;
        return r3513060;
}
↓
double f(double alpha, double beta) {
        double r3513061 = beta;
        double r3513062 = alpha;
        double r3513063 = r3513061 - r3513062;
        double r3513064 = 2.0;
        double r3513065 = r3513061 + r3513064;
        double r3513066 = r3513062 + r3513065;
        double r3513067 = r3513063 / r3513066;
        double r3513068 = 1.0;
        double r3513069 = r3513067 + r3513068;
        double r3513070 = r3513069 / r3513064;
        return r3513070;
}

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 2019125 
(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)))