Average Error: 1.0 → 0.6
Time: 37.4s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \beta}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}\]
\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}{\left(\alpha + \beta\right) + 2 \cdot i}}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}{\beta - \alpha}} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r2966965 = alpha;
        double r2966966 = beta;
        double r2966967 = r2966965 + r2966966;
        double r2966968 = r2966966 - r2966965;
        double r2966969 = r2966967 * r2966968;
        double r2966970 = 2.0;
        double r2966971 = /* ERROR: no posit support in C */;
        double r2966972 = i;
        double r2966973 = r2966971 * r2966972;
        double r2966974 = r2966967 + r2966973;
        double r2966975 = r2966969 / r2966974;
        double r2966976 = 2.0;
        double r2966977 = /* ERROR: no posit support in C */;
        double r2966978 = r2966974 + r2966977;
        double r2966979 = r2966975 / r2966978;
        double r2966980 = 1.0;
        double r2966981 = /* ERROR: no posit support in C */;
        double r2966982 = r2966979 + r2966981;
        double r2966983 = r2966982 / r2966977;
        return r2966983;
}
↓
double f(double alpha, double beta, double i) {
        double r2966984 = alpha;
        double r2966985 = beta;
        double r2966986 = r2966984 + r2966985;
        double r2966987 = 2.0;
        double r2966988 = i;
        double r2966989 = r2966987 * r2966988;
        double r2966990 = r2966986 + r2966989;
        double r2966991 = r2966986 / r2966990;
        double r2966992 = 2.0;
        double r2966993 = r2966990 + r2966992;
        double r2966994 = r2966985 - r2966984;
        double r2966995 = r2966993 / r2966994;
        double r2966996 = r2966991 / r2966995;
        double r2966997 = 1.0;
        double r2966998 = r2966996 + r2966997;
        double r2966999 = r2966998 / r2966992;
        return r2966999;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

  1. Initial program 1.0

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

  5. Applied associate-/r/0.6

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

  6. Applied associate-/l*0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(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)))