Average Error: 1.0 → 0.6
Time: 34.8s
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 r1334793 = alpha;
        double r1334794 = beta;
        double r1334795 = r1334793 + r1334794;
        double r1334796 = r1334794 - r1334793;
        double r1334797 = r1334795 * r1334796;
        double r1334798 = 2.0;
        double r1334799 = /* ERROR: no posit support in C */;
        double r1334800 = i;
        double r1334801 = r1334799 * r1334800;
        double r1334802 = r1334795 + r1334801;
        double r1334803 = r1334797 / r1334802;
        double r1334804 = 2.0;
        double r1334805 = /* ERROR: no posit support in C */;
        double r1334806 = r1334802 + r1334805;
        double r1334807 = r1334803 / r1334806;
        double r1334808 = 1.0;
        double r1334809 = /* ERROR: no posit support in C */;
        double r1334810 = r1334807 + r1334809;
        double r1334811 = r1334810 / r1334805;
        return r1334811;
}
↓
double f(double alpha, double beta, double i) {
        double r1334812 = alpha;
        double r1334813 = beta;
        double r1334814 = r1334812 + r1334813;
        double r1334815 = 2.0;
        double r1334816 = i;
        double r1334817 = r1334815 * r1334816;
        double r1334818 = r1334814 + r1334817;
        double r1334819 = r1334813 - r1334812;
        double r1334820 = r1334818 / r1334819;
        double r1334821 = r1334814 / r1334820;
        double r1334822 = 2.0;
        double r1334823 = r1334818 + r1334822;
        double r1334824 = r1334821 / r1334823;
        double r1334825 = 1.0;
        double r1334826 = r1334824 + r1334825;
        double r1334827 = r1334826 / r1334822;
        return r1334827;
}

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. 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 2019120 
(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)))