Average Error: 0.8 → 0.7
Time: 17.8s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]
\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{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r1610222 = beta;
        double r1610223 = alpha;
        double r1610224 = r1610222 - r1610223;
        double r1610225 = r1610223 + r1610222;
        double r1610226 = 2.0;
        double r1610227 = /* ERROR: no posit support in C */;
        double r1610228 = r1610225 + r1610227;
        double r1610229 = r1610224 / r1610228;
        double r1610230 = 1.0;
        double r1610231 = /* ERROR: no posit support in C */;
        double r1610232 = r1610229 + r1610231;
        double r1610233 = r1610232 / r1610227;
        return r1610233;
}
↓
double f(double alpha, double beta) {
        double r1610234 = beta;
        double r1610235 = alpha;
        double r1610236 = r1610234 - r1610235;
        double r1610237 = 2.0;
        double r1610238 = r1610234 + r1610237;
        double r1610239 = r1610235 + r1610238;
        double r1610240 = r1610236 / r1610239;
        double r1610241 = 1.0;
        double r1610242 = r1610240 + r1610241;
        double r1610243 = r1610242 / r1610237;
        return r1610243;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.8

    \[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019121 +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)))