quadp (p42, positive)

Average Error: 1.6 → 1.6

Time: 29.7s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r1237038 = b;
        double r1237039 = -r1237038;
        double r1237040 = r1237038 * r1237038;
        double r1237041 = 4.0;
        double r1237042 = /* ERROR: no posit support in C */;
        double r1237043 = a;
        double r1237044 = c;
        double r1237045 = r1237043 * r1237044;
        double r1237046 = r1237042 * r1237045;
        double r1237047 = r1237040 - r1237046;
        double r1237048 = sqrt(r1237047);
        double r1237049 = r1237039 + r1237048;
        double r1237050 = 2.0;
        double r1237051 = /* ERROR: no posit support in C */;
        double r1237052 = r1237051 * r1237043;
        double r1237053 = r1237049 / r1237052;
        return r1237053;
}

↓

double f(double a, double b, double c) {
        double r1237054 = b;
        double r1237055 = r1237054 * r1237054;
        double r1237056 = /*Error: no posit support in C */;
        double r1237057 = c;
        double r1237058 = a;
        double r1237059 = r1237057 * r1237058;
        double r1237060 = 4.0;
        double r1237061 = /*Error: no posit support in C */;
        double r1237062 = /*Error: no posit support in C */;
        double r1237063 = sqrt(r1237062);
        double r1237064 = r1237063 - r1237054;
        double r1237065 = 1.0;
        double r1237066 = r1237064 / r1237065;
        double r1237067 = r1237065 / r1237058;
        double r1237068 = 2.0;
        double r1237069 = r1237067 / r1237068;
        double r1237070 = r1237066 * r1237069;
        return r1237070;
}

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 introduce-quire 1.6

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

5. Applied insert-quire-fdp-sub 1.6

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

7. Applied *p16-rgt-identity-expand 1.6

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

8. Applied p16-times-frac 1.6

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

10. Applied p16-*-un-lft-identity 1.6

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

11. Applied *p16-rgt-identity-expand 1.6

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

12. Applied p16-times-frac 1.6

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

13. Applied associate-*l* 1.6

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

14. Simplified 1.6

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

15. Final simplification 1.6

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

Reproduce

herbie shell --seed 2019165 
(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)))