Average Error: 52.5 → 6.2
Time: 18.1s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{b} \cdot -2}{2}
double f(double a, double b, double c) {
        double r709850 = b;
        double r709851 = -r709850;
        double r709852 = r709850 * r709850;
        double r709853 = 4.0;
        double r709854 = a;
        double r709855 = r709853 * r709854;
        double r709856 = c;
        double r709857 = r709855 * r709856;
        double r709858 = r709852 - r709857;
        double r709859 = sqrt(r709858);
        double r709860 = r709851 + r709859;
        double r709861 = 2.0;
        double r709862 = r709861 * r709854;
        double r709863 = r709860 / r709862;
        return r709863;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r709864 = c;
        double r709865 = b;
        double r709866 = r709864 / r709865;
        double r709867 = -2.0;
        double r709868 = r709866 * r709867;
        double r709869 = 2.0;
        double r709870 = r709868 / r709869;
        return r709870;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 52.5

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

  2. Simplified52.5

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

  3. Taylor expanded around inf 6.2

    \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

  4. Final simplification6.2

    \[\leadsto \frac{\frac{c}{b} \cdot -2}{2}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))