Average Error: 0.7 → 0.7
Time: 28.2s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{1.0}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]
\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{1.0}{\frac{\left(\alpha + \beta\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r4613502 = beta;
        double r4613503 = alpha;
        double r4613504 = r4613502 - r4613503;
        double r4613505 = r4613503 + r4613502;
        double r4613506 = 2.0;
        double r4613507 = /* ERROR: no posit support in C */;
        double r4613508 = r4613505 + r4613507;
        double r4613509 = r4613504 / r4613508;
        double r4613510 = 1.0;
        double r4613511 = /* ERROR: no posit support in C */;
        double r4613512 = r4613509 + r4613511;
        double r4613513 = r4613512 / r4613507;
        return r4613513;
}
↓
double f(double alpha, double beta) {
        double r4613514 = 1.0;
        double r4613515 = alpha;
        double r4613516 = beta;
        double r4613517 = r4613515 + r4613516;
        double r4613518 = 2.0;
        double r4613519 = r4613517 + r4613518;
        double r4613520 = r4613516 - r4613515;
        double r4613521 = r4613519 / r4613520;
        double r4613522 = r4613514 / r4613521;
        double r4613523 = r4613522 + r4613514;
        double r4613524 = r4613523 / r4613518;
        return r4613524;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019158 +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)))