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 r3092003 = beta;
        double r3092004 = alpha;
        double r3092005 = r3092003 - r3092004;
        double r3092006 = r3092004 + r3092003;
        double r3092007 = 2.0;
        double r3092008 = /* ERROR: no posit support in C */;
        double r3092009 = r3092006 + r3092008;
        double r3092010 = r3092005 / r3092009;
        double r3092011 = 1.0;
        double r3092012 = /* ERROR: no posit support in C */;
        double r3092013 = r3092010 + r3092012;
        double r3092014 = r3092013 / r3092008;
        return r3092014;
}
↓
double f(double alpha, double beta) {
        double r3092015 = beta;
        double r3092016 = alpha;
        double r3092017 = r3092015 - r3092016;
        double r3092018 = 2.0;
        double r3092019 = r3092015 + r3092018;
        double r3092020 = r3092016 + r3092019;
        double r3092021 = r3092017 / r3092020;
        double r3092022 = 1.0;
        double r3092023 = r3092021 + r3092022;
        double r3092024 = r3092023 / r3092018;
        return r3092024;
}

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 +o rules:numerics
(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)))