Octave 3.8, jcobi/1

Average Error: 0.7 → 0.6

Time: 1.1m

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]

Math

\[\frac{\left(\frac{\left(\frac{\left(\beta - \alpha\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(2.0\right)}\right)}\right)}{\left(1.0\right)}\right)}{\left(2.0\right)}\]

↓

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

C

double f(double alpha, double beta) {
        double r3463406 = beta;
        double r3463407 = alpha;
        double r3463408 = r3463406 - r3463407;
        double r3463409 = r3463407 + r3463406;
        double r3463410 = 2.0;
        double r3463411 = /* ERROR: no posit support in C */;
        double r3463412 = r3463409 + r3463411;
        double r3463413 = r3463408 / r3463412;
        double r3463414 = 1.0;
        double r3463415 = /* ERROR: no posit support in C */;
        double r3463416 = r3463413 + r3463415;
        double r3463417 = r3463416 / r3463411;
        return r3463417;
}

↓

double f(double alpha, double beta) {
        double r3463418 = beta;
        double r3463419 = 2.0;
        double r3463420 = alpha;
        double r3463421 = r3463419 + r3463420;
        double r3463422 = r3463418 + r3463421;
        double r3463423 = r3463418 / r3463422;
        double r3463424 = /*Error: no posit support in C */;
        double r3463425 = -r3463420;
        double r3463426 = r3463425 / r3463422;
        double r3463427 = 1.0;
        double r3463428 = /*Error: no posit support in C */;
        double r3463429 = /*Error: no posit support in C */;
        double r3463430 = /*Error: no posit support in C */;
        double r3463431 = r3463430 / r3463419;
        return r3463431;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Initial program 0.7

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

3. Applied *p16-rgt-identity-expand 0.7

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

4. Applied p16-*-un-lft-identity 0.7

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

5. Applied p16-times-frac 0.8

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

6. Simplified 0.8

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

8. Applied sub-neg 0.8

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

9. Applied distribute-rgt-in 0.7

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

11. Applied introduce-quire 0.7

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

12. Applied insert-quire-add 0.7

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

13. Applied insert-quire-add 0.7

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

14. Simplified 0.6

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

15. Final simplification 0.6

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

Reproduce

herbie shell --seed 2019162 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (+.p16 (/.p16 (-.p16 beta alpha) (+.p16 (+.p16 alpha beta) (real->posit16 2.0))) (real->posit16 1.0)) (real->posit16 2.0)))