Average Error: 0.7 → 0.7
Time: 1.0m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \beta + \frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \left(-\alpha\right)\right) + 1.0}{2.0}\]
\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{\left(\frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \beta + \frac{1.0}{\left(\alpha + \beta\right) + 2.0} \cdot \left(-\alpha\right)\right) + 1.0}{2.0}
double f(double alpha, double beta) {
        double r1772736 = beta;
        double r1772737 = alpha;
        double r1772738 = r1772736 - r1772737;
        double r1772739 = r1772737 + r1772736;
        double r1772740 = 2.0;
        double r1772741 = /* ERROR: no posit support in C */;
        double r1772742 = r1772739 + r1772741;
        double r1772743 = r1772738 / r1772742;
        double r1772744 = 1.0;
        double r1772745 = /* ERROR: no posit support in C */;
        double r1772746 = r1772743 + r1772745;
        double r1772747 = r1772746 / r1772741;
        return r1772747;
}
↓
double f(double alpha, double beta) {
        double r1772748 = 1.0;
        double r1772749 = alpha;
        double r1772750 = beta;
        double r1772751 = r1772749 + r1772750;
        double r1772752 = 2.0;
        double r1772753 = r1772751 + r1772752;
        double r1772754 = r1772748 / r1772753;
        double r1772755 = r1772754 * r1772750;
        double r1772756 = -r1772749;
        double r1772757 = r1772754 * r1772756;
        double r1772758 = r1772755 + r1772757;
        double r1772759 = r1772758 + r1772748;
        double r1772760 = r1772759 / r1772752;
        return r1772760;
}

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 p16-*-un-lft-identity0.7

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

  4. Applied associate-/l*0.7

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  5. Using strategy rm

  6. Applied associate-/r/0.8

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right) \cdot \left(\beta - \alpha\right)\right)}}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  7. Using strategy rm

  8. Applied sub-neg0.8

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

  9. Applied distribute-lft-in0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019153 +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)))