Average Error: 0.9 → 0.6
Time: 54.7s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\left(\frac{\left(\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(2\right) \cdot i\right)}\right)}{\alpha}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
\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{\left(\frac{\left(\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(\frac{\left(i \cdot \left(2\right)\right)}{\left(\frac{\left(\frac{\beta}{\alpha}\right)}{\left(2.0\right)}\right)}\right)}\right) \cdot \left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\beta}{\left(\left(2\right) \cdot i\right)}\right)}{\alpha}\right)}\right)\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}
double f(double alpha, double beta, double i) {
        double r2613125 = alpha;
        double r2613126 = beta;
        double r2613127 = r2613125 + r2613126;
        double r2613128 = r2613126 - r2613125;
        double r2613129 = r2613127 * r2613128;
        double r2613130 = 2.0;
        double r2613131 = /* ERROR: no posit support in C */;
        double r2613132 = i;
        double r2613133 = r2613131 * r2613132;
        double r2613134 = r2613127 + r2613133;
        double r2613135 = r2613129 / r2613134;
        double r2613136 = 2.0;
        double r2613137 = /* ERROR: no posit support in C */;
        double r2613138 = r2613134 + r2613137;
        double r2613139 = r2613135 / r2613138;
        double r2613140 = 1.0;
        double r2613141 = /* ERROR: no posit support in C */;
        double r2613142 = r2613139 + r2613141;
        double r2613143 = r2613142 / r2613137;
        return r2613143;
}
↓
double f(double alpha, double beta, double i) {
        double r2613144 = beta;
        double r2613145 = alpha;
        double r2613146 = r2613144 + r2613145;
        double r2613147 = i;
        double r2613148 = 2.0;
        double r2613149 = /* ERROR: no posit support in C */;
        double r2613150 = r2613147 * r2613149;
        double r2613151 = 2.0;
        double r2613152 = /* ERROR: no posit support in C */;
        double r2613153 = r2613146 + r2613152;
        double r2613154 = r2613150 + r2613153;
        double r2613155 = r2613146 / r2613154;
        double r2613156 = r2613144 - r2613145;
        double r2613157 = r2613149 * r2613147;
        double r2613158 = r2613144 + r2613157;
        double r2613159 = r2613158 + r2613145;
        double r2613160 = r2613156 / r2613159;
        double r2613161 = r2613155 * r2613160;
        double r2613162 = 1.0;
        double r2613163 = /* ERROR: no posit support in C */;
        double r2613164 = r2613161 + r2613163;
        double r2613165 = r2613164 / r2613152;
        return r2613165;
}

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 *p16-rgt-identity-expand0.9

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

  4. Applied p16-*-un-lft-identity0.9

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

  5. Applied p16-times-frac0.6

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

  6. Applied p16-times-frac0.6

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

  7. Simplified0.6

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

  8. Simplified0.6

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

  10. Applied *p16-rgt-identity-expand0.6

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

  11. Applied associate-/l*0.6

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

  12. Simplified0.6

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

  13. Final simplification0.6

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

Reproduce

herbie shell --seed 2019163 +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)))