Average Error: 1.5 → 1.5
Time: 30.5s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4 \cdot a\right), c\right)\right)}}{2} \cdot \frac{1.0}{a}\]
\frac{\left(\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(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(4 \cdot a\right), c\right)\right)}}{2} \cdot \frac{1.0}{a}
double f(double a, double b, double c) {
        double r450451 = b;
        double r450452 = -r450451;
        double r450453 = r450451 * r450451;
        double r450454 = 4.0;
        double r450455 = /* ERROR: no posit support in C */;
        double r450456 = a;
        double r450457 = c;
        double r450458 = r450456 * r450457;
        double r450459 = r450455 * r450458;
        double r450460 = r450453 - r450459;
        double r450461 = sqrt(r450460);
        double r450462 = r450452 - r450461;
        double r450463 = 2.0;
        double r450464 = /* ERROR: no posit support in C */;
        double r450465 = r450464 * r450456;
        double r450466 = r450462 / r450465;
        return r450466;
}
↓
double f(double a, double b, double c) {
        double r450467 = b;
        double r450468 = -r450467;
        double r450469 = r450467 * r450467;
        double r450470 = /*Error: no posit support in C */;
        double r450471 = 4.0;
        double r450472 = a;
        double r450473 = r450471 * r450472;
        double r450474 = c;
        double r450475 = /*Error: no posit support in C */;
        double r450476 = /*Error: no posit support in C */;
        double r450477 = sqrt(r450476);
        double r450478 = r450468 - r450477;
        double r450479 = 2.0;
        double r450480 = r450478 / r450479;
        double r450481 = 1.0;
        double r450482 = r450481 / r450472;
        double r450483 = r450480 * r450482;
        return r450483;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.5

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

  3. Applied associate-*r*1.5

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

  5. Applied introduce-quire1.5

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

  6. Applied insert-quire-fdp-sub1.5

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

  8. Applied *p16-rgt-identity-expand1.5

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

  9. Applied p16-times-frac1.5

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

  10. Final simplification1.5

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

Reproduce

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