Average Error: 1.6 → 1.5
Time: 21.8s
Precision: 64
\[\frac{\left(\left(-b\right) - \left(\sqrt{\left(\left(b \cdot b\right) - \left(\left(4\right) \cdot \left(a \cdot c\right)\right)\right)}\right)\right)}{\left(\left(2\right) \cdot a\right)}\]
↓
\[\frac{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}\]
\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{\frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2}}{a}
double f(double a, double b, double c) {
        double r859236 = b;
        double r859237 = -r859236;
        double r859238 = r859236 * r859236;
        double r859239 = 4.0;
        double r859240 = /* ERROR: no posit support in C */;
        double r859241 = a;
        double r859242 = c;
        double r859243 = r859241 * r859242;
        double r859244 = r859240 * r859243;
        double r859245 = r859238 - r859244;
        double r859246 = sqrt(r859245);
        double r859247 = r859237 - r859246;
        double r859248 = 2.0;
        double r859249 = /* ERROR: no posit support in C */;
        double r859250 = r859249 * r859241;
        double r859251 = r859247 / r859250;
        return r859251;
}
↓
double f(double a, double b, double c) {
        double r859252 = b;
        double r859253 = -r859252;
        double r859254 = r859252 * r859252;
        double r859255 = 4.0;
        double r859256 = a;
        double r859257 = c;
        double r859258 = r859256 * r859257;
        double r859259 = r859255 * r859258;
        double r859260 = r859254 - r859259;
        double r859261 = sqrt(r859260);
        double r859262 = r859253 - r859261;
        double r859263 = 2.0;
        double r859264 = r859262 / r859263;
        double r859265 = r859264 / r859256;
        return r859265;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 1.6

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

  3. Applied associate-/r*1.5

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

  4. Final simplification1.5

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

Reproduce

herbie shell --seed 2019121 
(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)))