Average Error: 0.7 → 0.8
Time: 1.1m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
\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{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\left(\frac{\beta}{\left(2.0\right)}\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
double f(double alpha, double beta) {
        double r4060019 = beta;
        double r4060020 = alpha;
        double r4060021 = r4060019 - r4060020;
        double r4060022 = r4060020 + r4060019;
        double r4060023 = 2.0;
        double r4060024 = /* ERROR: no posit support in C */;
        double r4060025 = r4060022 + r4060024;
        double r4060026 = r4060021 / r4060025;
        double r4060027 = 1.0;
        double r4060028 = /* ERROR: no posit support in C */;
        double r4060029 = r4060026 + r4060028;
        double r4060030 = r4060029 / r4060024;
        return r4060030;
}
↓
double f(double alpha, double beta) {
        double r4060031 = 1.0;
        double r4060032 = /* ERROR: no posit support in C */;
        double r4060033 = alpha;
        double r4060034 = beta;
        double r4060035 = 2.0;
        double r4060036 = /* ERROR: no posit support in C */;
        double r4060037 = r4060034 + r4060036;
        double r4060038 = r4060033 + r4060037;
        double r4060039 = r4060034 - r4060033;
        double r4060040 = r4060038 / r4060039;
        double r4060041 = r4060032 / r4060040;
        double r4060042 = r4060041 + r4060032;
        double r4060043 = r4060042 / r4060036;
        return r4060043;
}

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. Using strategy rm

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

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

  6. Applied associate-/l*0.8

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

  7. Final simplification0.8

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

Reproduce

herbie shell --seed 2019164 +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)))