quadp (p42, positive)

Average Error: 1.6 → 1.6

Time: 33.9s

Precision: 64

Math

\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

↓

\[\frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot 4\right)\right)\right)} - b}{1.0} \cdot \frac{1.0}{2 \cdot a}\]

C

double f(double a, double b, double c) {
        double r581858 = b;
        double r581859 = -r581858;
        double r581860 = r581858 * r581858;
        double r581861 = 4.0;
        double r581862 = /* ERROR: no posit support in C */;
        double r581863 = a;
        double r581864 = c;
        double r581865 = r581863 * r581864;
        double r581866 = r581862 * r581865;
        double r581867 = r581860 - r581866;
        double r581868 = sqrt(r581867);
        double r581869 = r581859 + r581868;
        double r581870 = 2.0;
        double r581871 = /* ERROR: no posit support in C */;
        double r581872 = r581871 * r581863;
        double r581873 = r581869 / r581872;
        return r581873;
}

↓

double f(double a, double b, double c) {
        double r581874 = b;
        double r581875 = r581874 * r581874;
        double r581876 = /*Error: no posit support in C */;
        double r581877 = c;
        double r581878 = a;
        double r581879 = 4.0;
        double r581880 = r581878 * r581879;
        double r581881 = /*Error: no posit support in C */;
        double r581882 = /*Error: no posit support in C */;
        double r581883 = sqrt(r581882);
        double r581884 = r581883 - r581874;
        double r581885 = 1.0;
        double r581886 = r581884 / r581885;
        double r581887 = 2.0;
        double r581888 = r581887 * r581878;
        double r581889 = r581885 / r581888;
        double r581890 = r581886 * r581889;
        return r581890;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 1.6

   \[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]

2. Simplified 1.6

   \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\left(2\right) \cdot a\right)}}\]
3. Using strategy rm

4. Applied associate-*l* 1.5

   \[\leadsto \frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \color{blue}{\left(c \cdot \left(a \cdot \left(4\right)\right)\right)}\right)}\right) - b\right)}{\left(\left(2\right) \cdot a\right)}\]
5. Using strategy rm

6. Applied *p16-rgt-identity-expand 1.5

   \[\leadsto \frac{\color{blue}{\left(\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(c \cdot \left(a \cdot \left(4\right)\right)\right)\right)}\right) - b\right) \cdot \left(1.0\right)\right)}}{\left(\left(2\right) \cdot a\right)}\]

7. Applied p16-times-frac 1.6

   \[\leadsto \color{blue}{\left(\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(c \cdot \left(a \cdot \left(4\right)\right)\right)\right)}\right) - b\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)}\]
8. Using strategy rm

9. Applied introduce-quire 1.6

   \[\leadsto \left(\frac{\left(\left(\sqrt{\left(\color{blue}{\left(\left(\left(b \cdot b\right)\right)\right)} - \left(c \cdot \left(a \cdot \left(4\right)\right)\right)\right)}\right) - b\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

10. Applied insert-quire-fdp-sub 1.5

    \[\leadsto \left(\frac{\left(\left(\sqrt{\color{blue}{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}}\right) - b\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]
11. Using strategy rm

12. Applied p16-*-un-lft-identity 1.5

    \[\leadsto \left(\frac{\left(\left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}\right) - b\right)}{\color{blue}{\left(\left(1.0\right) \cdot \left(2\right)\right)}}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

13. Applied *p16-rgt-identity-expand 1.5

    \[\leadsto \left(\frac{\color{blue}{\left(\left(\left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}\right) - b\right) \cdot \left(1.0\right)\right)}}{\left(\left(1.0\right) \cdot \left(2\right)\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

14. Applied p16-times-frac 1.5

    \[\leadsto \color{blue}{\left(\left(\frac{\left(\left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}\right) - b\right)}{\left(1.0\right)}\right) \cdot \left(\frac{\left(1.0\right)}{\left(2\right)}\right)\right)} \cdot \left(\frac{\left(1.0\right)}{a}\right)\]

15. Applied associate-*l* 1.6

    \[\leadsto \color{blue}{\left(\frac{\left(\left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}\right) - b\right)}{\left(1.0\right)}\right) \cdot \left(\left(\frac{\left(1.0\right)}{\left(2\right)}\right) \cdot \left(\frac{\left(1.0\right)}{a}\right)\right)}\]

16. Simplified 1.6

    \[\leadsto \left(\frac{\left(\left(\sqrt{\left(\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot \left(4\right)\right)\right)\right)\right)}\right) - b\right)}{\left(1.0\right)}\right) \cdot \color{blue}{\left(\frac{\left(1.0\right)}{\left(\left(2\right) \cdot a\right)}\right)}\]

17. Final simplification 1.6

    \[\leadsto \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot 4\right)\right)\right)} - b}{1.0} \cdot \frac{1.0}{2 \cdot a}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  (/.p16 (+.p16 (neg.p16 b) (sqrt.p16 (-.p16 (*.p16 b b) (*.p16 (real->posit16 4) (*.p16 a c))))) (*.p16 (real->posit16 2) a)))