Average Error: 1.5 → 1.5
Time: 34.3s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), 4, \left(c \cdot a\right)\right)\right)}}{2 \cdot a}\]
\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), 4, \left(c \cdot a\right)\right)\right)}}{2 \cdot a}
double f(double a, double b, double c) {
        double r1177329 = b;
        double r1177330 = -r1177329;
        double r1177331 = r1177329 * r1177329;
        double r1177332 = 4.0;
        double r1177333 = /* ERROR: no posit support in C */;
        double r1177334 = a;
        double r1177335 = c;
        double r1177336 = r1177334 * r1177335;
        double r1177337 = r1177333 * r1177336;
        double r1177338 = r1177331 - r1177337;
        double r1177339 = sqrt(r1177338);
        double r1177340 = r1177330 - r1177339;
        double r1177341 = 2.0;
        double r1177342 = /* ERROR: no posit support in C */;
        double r1177343 = r1177342 * r1177334;
        double r1177344 = r1177340 / r1177343;
        return r1177344;
}
↓
double f(double a, double b, double c) {
        double r1177345 = b;
        double r1177346 = -r1177345;
        double r1177347 = r1177345 * r1177345;
        double r1177348 = /*Error: no posit support in C */;
        double r1177349 = 4.0;
        double r1177350 = c;
        double r1177351 = a;
        double r1177352 = r1177350 * r1177351;
        double r1177353 = /*Error: no posit support in C */;
        double r1177354 = /*Error: no posit support in C */;
        double r1177355 = sqrt(r1177354);
        double r1177356 = r1177346 - r1177355;
        double r1177357 = 2.0;
        double r1177358 = r1177357 * r1177351;
        double r1177359 = r1177356 / r1177358;
        return r1177359;
}

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 *-commutative1.5

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

  4. Applied associate-*r*1.5

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

  6. Applied associate-*l*1.5

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

  7. Applied introduce-quire1.5

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

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

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019158 
(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)))