Average Error: 0.9 → 0.6
Time: 37.0s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
↓
\frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r2802914 = alpha;
        double r2802915 = beta;
        double r2802916 = r2802914 + r2802915;
        double r2802917 = r2802915 - r2802914;
        double r2802918 = r2802916 * r2802917;
        double r2802919 = 2.0;
        double r2802920 = /* ERROR: no posit support in C */;
        double r2802921 = i;
        double r2802922 = r2802920 * r2802921;
        double r2802923 = r2802916 + r2802922;
        double r2802924 = r2802918 / r2802923;
        double r2802925 = 2.0;
        double r2802926 = /* ERROR: no posit support in C */;
        double r2802927 = r2802923 + r2802926;
        double r2802928 = r2802924 / r2802927;
        double r2802929 = 1.0;
        double r2802930 = /* ERROR: no posit support in C */;
        double r2802931 = r2802928 + r2802930;
        double r2802932 = r2802931 / r2802926;
        return r2802932;
}
↓
double f(double alpha, double beta, double i) {
        double r2802933 = alpha;
        double r2802934 = beta;
        double r2802935 = r2802933 + r2802934;
        double r2802936 = 2.0;
        double r2802937 = i;
        double r2802938 = r2802936 * r2802937;
        double r2802939 = r2802935 + r2802938;
        double r2802940 = r2802934 - r2802933;
        double r2802941 = r2802939 / r2802940;
        double r2802942 = r2802935 / r2802941;
        double r2802943 = 2.0;
        double r2802944 = r2802939 + r2802943;
        double r2802945 = r2802942 / r2802944;
        double r2802946 = 1.0;
        double r2802947 = r2802945 + r2802946;
        double r2802948 = r2802947 / r2802943;
        return r2802948;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 0.9

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
  2. Using strategy rm

  3. Applied associate-/l*0.6

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(\beta - \alpha\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

  4. Final simplification0.6

    \[\leadsto \frac{\frac{\frac{\alpha + \beta}{\frac{\left(\alpha + \beta\right) + 2 \cdot i}{\beta - \alpha}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)) (>.p16 i (real->posit16 0)))
  (/.p16 (+.p16 (/.p16 (/.p16 (*.p16 (+.p16 alpha beta) (-.p16 beta alpha)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) i)) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))