Average Error: 1.6 → 1.6
Time: 57.9s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot 4\right)\right)\right)} - b}{a}\]
\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{1.0}{2} \cdot \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), c, \left(a \cdot 4\right)\right)\right)} - b}{a}
double f(double a, double b, double c) {
        double r955312 = b;
        double r955313 = -r955312;
        double r955314 = r955312 * r955312;
        double r955315 = 4.0;
        double r955316 = /* ERROR: no posit support in C */;
        double r955317 = a;
        double r955318 = c;
        double r955319 = r955317 * r955318;
        double r955320 = r955316 * r955319;
        double r955321 = r955314 - r955320;
        double r955322 = sqrt(r955321);
        double r955323 = r955313 + r955322;
        double r955324 = 2.0;
        double r955325 = /* ERROR: no posit support in C */;
        double r955326 = r955325 * r955317;
        double r955327 = r955323 / r955326;
        return r955327;
}
↓
double f(double a, double b, double c) {
        double r955328 = 1.0;
        double r955329 = 2.0;
        double r955330 = r955328 / r955329;
        double r955331 = b;
        double r955332 = r955331 * r955331;
        double r955333 = /*Error: no posit support in C */;
        double r955334 = c;
        double r955335 = a;
        double r955336 = 4.0;
        double r955337 = r955335 * r955336;
        double r955338 = /*Error: no posit support in C */;
        double r955339 = /*Error: no posit support in C */;
        double r955340 = sqrt(r955339);
        double r955341 = r955340 - r955331;
        double r955342 = r955341 / r955335;
        double r955343 = r955330 * r955342;
        return r955343;
}

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. Simplified1.6

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

  4. Applied p16-*-un-lft-identity1.6

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

  5. Applied p16-times-frac1.6

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

  7. Applied associate-*l*1.6

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

  9. Applied introduce-quire1.6

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

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

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

  11. Final simplification1.6

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

Reproduce

herbie shell --seed 2019162 
(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)))