Average Error: 1.5 → 1.5
Time: 28.3s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
\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{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
double f(double a, double b, double c) {
        double r2013224 = b;
        double r2013225 = -r2013224;
        double r2013226 = r2013224 * r2013224;
        double r2013227 = 4.0;
        double r2013228 = /* ERROR: no posit support in C */;
        double r2013229 = a;
        double r2013230 = c;
        double r2013231 = r2013229 * r2013230;
        double r2013232 = r2013228 * r2013231;
        double r2013233 = r2013226 - r2013232;
        double r2013234 = sqrt(r2013233);
        double r2013235 = r2013225 - r2013234;
        double r2013236 = 2.0;
        double r2013237 = /* ERROR: no posit support in C */;
        double r2013238 = r2013237 * r2013229;
        double r2013239 = r2013235 / r2013238;
        return r2013239;
}
↓
double f(double a, double b, double c) {
        double r2013240 = b;
        double r2013241 = -r2013240;
        double r2013242 = r2013240 * r2013240;
        double r2013243 = 4.0;
        double r2013244 = a;
        double r2013245 = r2013243 * r2013244;
        double r2013246 = c;
        double r2013247 = r2013245 * r2013246;
        double r2013248 = r2013242 - r2013247;
        double r2013249 = sqrt(r2013248);
        double r2013250 = r2013241 - r2013249;
        double r2013251 = 2.0;
        double r2013252 = r2013251 * r2013244;
        double r2013253 = r2013250 / r2013252;
        return r2013253;
}

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. Final simplification1.5

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))