quadp (p42, positive)

Average Error: 1.5 → 1.5

Time: 39.0s

Precision: 64

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r1124454 = b;
        double r1124455 = -r1124454;
        double r1124456 = r1124454 * r1124454;
        double r1124457 = 4.0;
        double r1124458 = /* ERROR: no posit support in C */;
        double r1124459 = a;
        double r1124460 = c;
        double r1124461 = r1124459 * r1124460;
        double r1124462 = r1124458 * r1124461;
        double r1124463 = r1124456 - r1124462;
        double r1124464 = sqrt(r1124463);
        double r1124465 = r1124455 + r1124464;
        double r1124466 = 2.0;
        double r1124467 = /* ERROR: no posit support in C */;
        double r1124468 = r1124467 * r1124459;
        double r1124469 = r1124465 / r1124468;
        return r1124469;
}

↓

double f(double a, double b, double c) {
        double r1124470 = b;
        double r1124471 = r1124470 * r1124470;
        double r1124472 = /*Error: no posit support in C */;
        double r1124473 = c;
        double r1124474 = a;
        double r1124475 = 4.0;
        double r1124476 = /* ERROR: no posit support in C */;
        double r1124477 = r1124474 * r1124476;
        double r1124478 = /*Error: no posit support in C */;
        double r1124479 = /*Error: no posit support in C */;
        double r1124480 = sqrt(r1124479);
        double r1124481 = r1124480 - r1124470;
        double r1124482 = 2.0;
        double r1124483 = /* ERROR: no posit support in C */;
        double r1124484 = r1124483 * r1124474;
        double r1124485 = r1124481 / r1124484;
        return r1124485;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 1.5

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

2. Simplified 1.5

   \[\leadsto \color{blue}{\frac{\left(\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot \left(4\right)\right)\right)}\right) - b\right)}{\left(\left(2\right) \cdot a\right)}}\]
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 introduce-quire 1.5

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

7. Applied insert-quire-fdp-sub 1.5

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

8. Final simplification 1.5

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

Reproduce

herbie shell --seed 2019164 
(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)))