Average Error: 0.7 → 0.7
Time: 21.2s
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 r3914537 = beta;
        double r3914538 = alpha;
        double r3914539 = r3914537 - r3914538;
        double r3914540 = r3914538 + r3914537;
        double r3914541 = 2.0;
        double r3914542 = /* ERROR: no posit support in C */;
        double r3914543 = r3914540 + r3914542;
        double r3914544 = r3914539 / r3914543;
        double r3914545 = 1.0;
        double r3914546 = /* ERROR: no posit support in C */;
        double r3914547 = r3914544 + r3914546;
        double r3914548 = r3914547 / r3914542;
        return r3914548;
}
↓
double f(double alpha, double beta) {
        double r3914549 = beta;
        double r3914550 = alpha;
        double r3914551 = r3914549 - r3914550;
        double r3914552 = 2.0;
        double r3914553 = r3914549 + r3914552;
        double r3914554 = r3914550 + r3914553;
        double r3914555 = r3914551 / r3914554;
        double r3914556 = 1.0;
        double r3914557 = r3914555 + r3914556;
        double r3914558 = r3914557 / r3914552;
        return r3914558;
}

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 2019128 +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)))