Average Error: 1.6 → 1.5
Time: 36.6s
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{1.0}{2} \cdot \frac{\sqrt{\left(\mathsf{qms}\left(\left(\left(b \cdot b\right)\right), \left(c \cdot a\right), 4\right)\right)} - b}{a}\]
\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), \left(c \cdot a\right), 4\right)\right)} - b}{a}
double f(double a, double b, double c) {
        double r560249 = b;
        double r560250 = -r560249;
        double r560251 = r560249 * r560249;
        double r560252 = 4.0;
        double r560253 = /* ERROR: no posit support in C */;
        double r560254 = a;
        double r560255 = c;
        double r560256 = r560254 * r560255;
        double r560257 = r560253 * r560256;
        double r560258 = r560251 - r560257;
        double r560259 = sqrt(r560258);
        double r560260 = r560250 + r560259;
        double r560261 = 2.0;
        double r560262 = /* ERROR: no posit support in C */;
        double r560263 = r560262 * r560254;
        double r560264 = r560260 / r560263;
        return r560264;
}
↓
double f(double a, double b, double c) {
        double r560265 = 1.0;
        double r560266 = 2.0;
        double r560267 = r560265 / r560266;
        double r560268 = b;
        double r560269 = r560268 * r560268;
        double r560270 = /*Error: no posit support in C */;
        double r560271 = c;
        double r560272 = a;
        double r560273 = r560271 * r560272;
        double r560274 = 4.0;
        double r560275 = /*Error: no posit support in C */;
        double r560276 = /*Error: no posit support in C */;
        double r560277 = sqrt(r560276);
        double r560278 = r560277 - r560268;
        double r560279 = r560278 / r560272;
        double r560280 = r560267 * r560279;
        return r560280;
}

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 introduce-quire1.6

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

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

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

  7. Applied p16-*-un-lft-identity1.5

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

  8. Applied p16-times-frac1.5

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

  9. Final simplification1.5

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

Reproduce

herbie shell --seed 2019153 
(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)))