Average Error: 0.7 → 0.8
Time: 3.5m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\beta - \alpha\right) \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
\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(\beta - \alpha\right) \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r4122119 = beta;
        double r4122120 = alpha;
        double r4122121 = r4122119 - r4122120;
        double r4122122 = r4122120 + r4122119;
        double r4122123 = 2.0;
        double r4122124 = /* ERROR: no posit support in C */;
        double r4122125 = r4122122 + r4122124;
        double r4122126 = r4122121 / r4122125;
        double r4122127 = 1.0;
        double r4122128 = /* ERROR: no posit support in C */;
        double r4122129 = r4122126 + r4122128;
        double r4122130 = r4122129 / r4122124;
        return r4122130;
}
↓
double f(double alpha, double beta) {
        double r4122131 = beta;
        double r4122132 = alpha;
        double r4122133 = r4122131 - r4122132;
        double r4122134 = 1.0;
        double r4122135 = r4122132 + r4122131;
        double r4122136 = 2.0;
        double r4122137 = r4122135 + r4122136;
        double r4122138 = r4122134 / r4122137;
        double r4122139 = r4122133 * r4122138;
        double r4122140 = r4122139 + r4122134;
        double r4122141 = r4122140 / r4122136;
        return r4122141;
}

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

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

  4. Applied p16-*-un-lft-identity0.7

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

  5. Applied distribute-lft-out0.7

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

  6. Applied *p16-rgt-identity-expand0.7

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

  7. Applied p16-times-frac0.8

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

  8. Simplified0.8

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

  9. Final simplification0.8

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

Reproduce

herbie shell --seed 2019162 +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)))