Average Error: 1.0 → 0.6
Time: 47.5s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right) \land i \gt \left(0\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\left(\frac{\alpha}{\beta}\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot i\right)}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]
↓
\[\frac{\frac{\beta + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \frac{\beta - \alpha}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(2 \cdot i\right)\right), \alpha, 1.0\right)\right), \beta, 1.0\right)\right)} + 1.0}{2.0}\]
\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{\beta + \alpha}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \frac{\beta - \alpha}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(2 \cdot i\right)\right), \alpha, 1.0\right)\right), \beta, 1.0\right)\right)} + 1.0}{2.0}
double f(double alpha, double beta, double i) {
        double r1257226 = alpha;
        double r1257227 = beta;
        double r1257228 = r1257226 + r1257227;
        double r1257229 = r1257227 - r1257226;
        double r1257230 = r1257228 * r1257229;
        double r1257231 = 2.0;
        double r1257232 = /* ERROR: no posit support in C */;
        double r1257233 = i;
        double r1257234 = r1257232 * r1257233;
        double r1257235 = r1257228 + r1257234;
        double r1257236 = r1257230 / r1257235;
        double r1257237 = 2.0;
        double r1257238 = /* ERROR: no posit support in C */;
        double r1257239 = r1257235 + r1257238;
        double r1257240 = r1257236 / r1257239;
        double r1257241 = 1.0;
        double r1257242 = /* ERROR: no posit support in C */;
        double r1257243 = r1257240 + r1257242;
        double r1257244 = r1257243 / r1257238;
        return r1257244;
}
↓
double f(double alpha, double beta, double i) {
        double r1257245 = beta;
        double r1257246 = alpha;
        double r1257247 = r1257245 + r1257246;
        double r1257248 = r1257246 + r1257245;
        double r1257249 = 2.0;
        double r1257250 = i;
        double r1257251 = r1257249 * r1257250;
        double r1257252 = r1257248 + r1257251;
        double r1257253 = 2.0;
        double r1257254 = r1257252 + r1257253;
        double r1257255 = r1257247 / r1257254;
        double r1257256 = r1257245 - r1257246;
        double r1257257 = /*Error: no posit support in C */;
        double r1257258 = 1.0;
        double r1257259 = /*Error: no posit support in C */;
        double r1257260 = /*Error: no posit support in C */;
        double r1257261 = /*Error: no posit support in C */;
        double r1257262 = r1257256 / r1257261;
        double r1257263 = r1257255 * r1257262;
        double r1257264 = r1257263 + r1257258;
        double r1257265 = r1257264 / r1257253;
        return r1257265;
}

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 p16-*-un-lft-identity1.0

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

  4. Applied p16-times-frac0.6

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

  5. Simplified0.6

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

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

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

  8. Applied p16-times-frac0.6

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

  9. Simplified0.6

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

  11. Applied introduce-quire0.6

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

  12. Applied insert-quire-add0.6

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

  13. Applied insert-quire-add0.6

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

  14. Final simplification0.6

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

Reproduce

herbie shell --seed 2019156 +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)))