Average Error: 0.8 → 0.7
Time: 19.1s
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 r1833369 = beta;
        double r1833370 = alpha;
        double r1833371 = r1833369 - r1833370;
        double r1833372 = r1833370 + r1833369;
        double r1833373 = 2.0;
        double r1833374 = /* ERROR: no posit support in C */;
        double r1833375 = r1833372 + r1833374;
        double r1833376 = r1833371 / r1833375;
        double r1833377 = 1.0;
        double r1833378 = /* ERROR: no posit support in C */;
        double r1833379 = r1833376 + r1833378;
        double r1833380 = r1833379 / r1833374;
        return r1833380;
}
↓
double f(double alpha, double beta) {
        double r1833381 = beta;
        double r1833382 = alpha;
        double r1833383 = r1833381 - r1833382;
        double r1833384 = 2.0;
        double r1833385 = r1833381 + r1833384;
        double r1833386 = r1833382 + r1833385;
        double r1833387 = r1833383 / r1833386;
        double r1833388 = 1.0;
        double r1833389 = r1833387 + r1833388;
        double r1833390 = r1833389 / r1833384;
        return r1833390;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.8

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  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 2019121 
(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)))