Average Error: 0.7 → 0.7
Time: 1.1m
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}\]
\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}{\left(\alpha + \beta\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta) {
        double r4501715 = beta;
        double r4501716 = alpha;
        double r4501717 = r4501715 - r4501716;
        double r4501718 = r4501716 + r4501715;
        double r4501719 = 2.0;
        double r4501720 = /* ERROR: no posit support in C */;
        double r4501721 = r4501718 + r4501720;
        double r4501722 = r4501717 / r4501721;
        double r4501723 = 1.0;
        double r4501724 = /* ERROR: no posit support in C */;
        double r4501725 = r4501722 + r4501724;
        double r4501726 = r4501725 / r4501720;
        return r4501726;
}
↓
double f(double alpha, double beta) {
        double r4501727 = beta;
        double r4501728 = alpha;
        double r4501729 = r4501727 - r4501728;
        double r4501730 = r4501728 + r4501727;
        double r4501731 = 2.0;
        double r4501732 = r4501730 + r4501731;
        double r4501733 = r4501729 / r4501732;
        double r4501734 = 1.0;
        double r4501735 = r4501733 + r4501734;
        double r4501736 = r4501735 / r4501731;
        return r4501736;
}

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. Final simplification0.7

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

Reproduce

herbie shell --seed 2019132 +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)))