Average Error: 0.7 → 0.7
Time: 25.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\beta \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0} + \left(-\alpha\right) \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}{2.0}\]
\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{\left(\beta \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0} + \left(-\alpha\right) \cdot \frac{1.0}{\left(\alpha + \beta\right) + 2.0}\right) + 1.0}{2.0}
double f(double alpha, double beta) {
        double r861370 = beta;
        double r861371 = alpha;
        double r861372 = r861370 - r861371;
        double r861373 = r861371 + r861370;
        double r861374 = 2.0;
        double r861375 = /* ERROR: no posit support in C */;
        double r861376 = r861373 + r861375;
        double r861377 = r861372 / r861376;
        double r861378 = 1.0;
        double r861379 = /* ERROR: no posit support in C */;
        double r861380 = r861377 + r861379;
        double r861381 = r861380 / r861375;
        return r861381;
}
↓
double f(double alpha, double beta) {
        double r861382 = beta;
        double r861383 = 1.0;
        double r861384 = alpha;
        double r861385 = r861384 + r861382;
        double r861386 = 2.0;
        double r861387 = r861385 + r861386;
        double r861388 = r861383 / r861387;
        double r861389 = r861382 * r861388;
        double r861390 = -r861384;
        double r861391 = r861390 * r861388;
        double r861392 = r861389 + r861391;
        double r861393 = r861392 + r861383;
        double r861394 = r861393 / r861386;
        return r861394;
}

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-rgt-in0.7

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

  10. Final simplification0.7

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

Reproduce

herbie shell --seed 2019154 +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)))