Average Error: 0.4 → 0.4
Time: 31.3s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}
↓
\frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}
double f(double alpha, double beta) {
        double r1430256 = alpha;
        double r1430257 = beta;
        double r1430258 = r1430256 + r1430257;
        double r1430259 = r1430257 * r1430256;
        double r1430260 = r1430258 + r1430259;
        double r1430261 = 1.0;
        double r1430262 = /* ERROR: no posit support in C */;
        double r1430263 = r1430260 + r1430262;
        double r1430264 = 2.0;
        double r1430265 = /* ERROR: no posit support in C */;
        double r1430266 = 1.0;
        double r1430267 = /* ERROR: no posit support in C */;
        double r1430268 = r1430265 * r1430267;
        double r1430269 = r1430258 + r1430268;
        double r1430270 = r1430263 / r1430269;
        double r1430271 = r1430270 / r1430269;
        double r1430272 = r1430269 + r1430262;
        double r1430273 = r1430271 / r1430272;
        return r1430273;
}
↓
double f(double alpha, double beta) {
        double r1430274 = 1.0;
        double r1430275 = alpha;
        double r1430276 = beta;
        double r1430277 = r1430275 + r1430276;
        double r1430278 = 2.0;
        double r1430279 = 1.0;
        double r1430280 = r1430278 * r1430279;
        double r1430281 = r1430277 + r1430280;
        double r1430282 = r1430274 / r1430281;
        double r1430283 = /*Error: no posit support in C */;
        double r1430284 = /*Error: no posit support in C */;
        double r1430285 = /*Error: no posit support in C */;
        double r1430286 = /*Error: no posit support in C */;
        double r1430287 = r1430282 * r1430286;
        double r1430288 = r1430287 / r1430281;
        double r1430289 = r1430281 + r1430274;
        double r1430290 = r1430288 / r1430289;
        return r1430290;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

  1. Initial program 0.4

    \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  2. Using strategy rm

  3. Applied introduce-quire0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\left(\frac{\alpha}{\beta}\right)\right)\right)}}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  4. Applied insert-quire-fdp-add0.4

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right)\right)}}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  5. Applied insert-quire-add0.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
  6. Using strategy rm

  7. Applied *p16-rgt-identity-expand0.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  8. Applied p16-*-un-lft-identity0.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  9. Applied p16-times-frac0.4

    \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\left(1.0\right)}\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  10. Simplified0.4

    \[\leadsto \frac{\left(\frac{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \color{blue}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

  11. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1} \cdot \left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019154 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))