Average Error: 1.6 → 1.5
Time: 1.1m
Precision: 64
\[\frac{\left(\frac{\left(-b\right)}{\left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)}\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}\]
\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{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{2 \cdot a}
double f(double a, double b, double c) {
        double r1361283 = b;
        double r1361284 = -r1361283;
        double r1361285 = r1361283 * r1361283;
        double r1361286 = 4.0;
        double r1361287 = /* ERROR: no posit support in C */;
        double r1361288 = a;
        double r1361289 = c;
        double r1361290 = r1361288 * r1361289;
        double r1361291 = r1361287 * r1361290;
        double r1361292 = r1361285 - r1361291;
        double r1361293 = sqrt(r1361292);
        double r1361294 = r1361284 + r1361293;
        double r1361295 = 2.0;
        double r1361296 = /* ERROR: no posit support in C */;
        double r1361297 = r1361296 * r1361288;
        double r1361298 = r1361294 / r1361297;
        return r1361298;
}
↓
double f(double a, double b, double c) {
        double r1361299 = b;
        double r1361300 = r1361299 * r1361299;
        double r1361301 = c;
        double r1361302 = a;
        double r1361303 = 4.0;
        double r1361304 = r1361302 * r1361303;
        double r1361305 = r1361301 * r1361304;
        double r1361306 = r1361300 - r1361305;
        double r1361307 = sqrt(r1361306);
        double r1361308 = r1361307 - r1361299;
        double r1361309 = 2.0;
        double r1361310 = r1361309 * r1361302;
        double r1361311 = r1361308 / r1361310;
        return r1361311;
}

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 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. Final simplification1.5

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(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)))