Average Error: 0.4 → 0.4
Time: 26.3s
Precision: 64
\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]
\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
↓
\[\frac{\frac{\frac{\alpha + \left(\left(\beta + \beta \cdot \alpha\right) + 1.0\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]
\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{\alpha + \left(\left(\beta + \beta \cdot \alpha\right) + 1.0\right)}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\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 r2921605 = alpha;
        double r2921606 = beta;
        double r2921607 = r2921605 + r2921606;
        double r2921608 = r2921606 * r2921605;
        double r2921609 = r2921607 + r2921608;
        double r2921610 = 1.0;
        double r2921611 = /* ERROR: no posit support in C */;
        double r2921612 = r2921609 + r2921611;
        double r2921613 = 2.0;
        double r2921614 = /* ERROR: no posit support in C */;
        double r2921615 = 1.0;
        double r2921616 = /* ERROR: no posit support in C */;
        double r2921617 = r2921614 * r2921616;
        double r2921618 = r2921607 + r2921617;
        double r2921619 = r2921612 / r2921618;
        double r2921620 = r2921619 / r2921618;
        double r2921621 = r2921618 + r2921611;
        double r2921622 = r2921620 / r2921621;
        return r2921622;
}
↓
double f(double alpha, double beta) {
        double r2921623 = alpha;
        double r2921624 = beta;
        double r2921625 = r2921624 * r2921623;
        double r2921626 = r2921624 + r2921625;
        double r2921627 = 1.0;
        double r2921628 = r2921626 + r2921627;
        double r2921629 = r2921623 + r2921628;
        double r2921630 = r2921623 + r2921624;
        double r2921631 = 2.0;
        double r2921632 = 1.0;
        double r2921633 = r2921631 * r2921632;
        double r2921634 = r2921630 + r2921633;
        double r2921635 = r2921629 / r2921634;
        double r2921636 = r2921635 / r2921634;
        double r2921637 = r2921634 + r2921627;
        double r2921638 = r2921636 / r2921637;
        return r2921638;
}

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 associate-+l+0.4

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

  5. Applied associate-+l+0.4

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

  6. Final simplification0.4

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

Reproduce

herbie shell --seed 2019133 +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))))