Average Error: 0.7 → 0.7
Time: 1.1m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{1.0}{2.0} \cdot \left(1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)\]
\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{1.0}{2.0} \cdot \left(1.0 + \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0}\right)
double f(double alpha, double beta) {
        double r308545 = beta;
        double r308546 = alpha;
        double r308547 = r308545 - r308546;
        double r308548 = r308546 + r308545;
        double r308549 = 2.0;
        double r308550 = /* ERROR: no posit support in C */;
        double r308551 = r308548 + r308550;
        double r308552 = r308547 / r308551;
        double r308553 = 1.0;
        double r308554 = /* ERROR: no posit support in C */;
        double r308555 = r308552 + r308554;
        double r308556 = r308555 / r308550;
        return r308556;
}
↓
double f(double alpha, double beta) {
        double r308557 = 1.0;
        double r308558 = 2.0;
        double r308559 = r308557 / r308558;
        double r308560 = beta;
        double r308561 = alpha;
        double r308562 = r308560 - r308561;
        double r308563 = r308561 + r308560;
        double r308564 = r308563 + r308558;
        double r308565 = r308562 / r308564;
        double r308566 = r308557 + r308565;
        double r308567 = r308559 * r308566;
        return r308567;
}

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-rgt-identity-expand0.7

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

  4. Applied associate-/l/0.7

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

  5. Simplified0.7

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

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

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

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

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

  9. Applied p16-times-frac0.7

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

  10. Simplified0.7

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

  11. Final simplification0.7

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

Reproduce

herbie shell --seed 2019152 +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)))