Average Error: 1.6 → 1.5
Time: 26.2s
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 r1840853 = b;
        double r1840854 = -r1840853;
        double r1840855 = r1840853 * r1840853;
        double r1840856 = 4.0;
        double r1840857 = /* ERROR: no posit support in C */;
        double r1840858 = a;
        double r1840859 = c;
        double r1840860 = r1840858 * r1840859;
        double r1840861 = r1840857 * r1840860;
        double r1840862 = r1840855 - r1840861;
        double r1840863 = sqrt(r1840862);
        double r1840864 = r1840854 + r1840863;
        double r1840865 = 2.0;
        double r1840866 = /* ERROR: no posit support in C */;
        double r1840867 = r1840866 * r1840858;
        double r1840868 = r1840864 / r1840867;
        return r1840868;
}
↓
double f(double a, double b, double c) {
        double r1840869 = b;
        double r1840870 = r1840869 * r1840869;
        double r1840871 = c;
        double r1840872 = a;
        double r1840873 = 4.0;
        double r1840874 = r1840872 * r1840873;
        double r1840875 = r1840871 * r1840874;
        double r1840876 = r1840870 - r1840875;
        double r1840877 = sqrt(r1840876);
        double r1840878 = r1840877 - r1840869;
        double r1840879 = 2.0;
        double r1840880 = r1840879 * r1840872;
        double r1840881 = r1840878 / r1840880;
        return r1840881;
}

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 2019125 
(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)))