Average Error: 0.7 → 0.7
Time: 56.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\alpha + \left(\beta + 2.0\right)} + 1.0}{2.0}\]
\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 r5374589 = beta;
        double r5374590 = alpha;
        double r5374591 = r5374589 - r5374590;
        double r5374592 = r5374590 + r5374589;
        double r5374593 = 2.0;
        double r5374594 = /* ERROR: no posit support in C */;
        double r5374595 = r5374592 + r5374594;
        double r5374596 = r5374591 / r5374595;
        double r5374597 = 1.0;
        double r5374598 = /* ERROR: no posit support in C */;
        double r5374599 = r5374596 + r5374598;
        double r5374600 = r5374599 / r5374594;
        return r5374600;
}
↓
double f(double alpha, double beta) {
        double r5374601 = beta;
        double r5374602 = alpha;
        double r5374603 = r5374601 - r5374602;
        double r5374604 = 2.0;
        double r5374605 = r5374601 + r5374604;
        double r5374606 = r5374602 + r5374605;
        double r5374607 = r5374603 / r5374606;
        double r5374608 = 1.0;
        double r5374609 = r5374607 + r5374608;
        double r5374610 = r5374609 / r5374604;
        return r5374610;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019133 +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)))