Quadratic roots, medium range

Average Error: 44.1 → 11.8

Time: 20.1s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r2055777 = b;
        double r2055778 = -r2055777;
        double r2055779 = r2055777 * r2055777;
        double r2055780 = 4.0;
        double r2055781 = a;
        double r2055782 = r2055780 * r2055781;
        double r2055783 = c;
        double r2055784 = r2055782 * r2055783;
        double r2055785 = r2055779 - r2055784;
        double r2055786 = sqrt(r2055785);
        double r2055787 = r2055778 + r2055786;
        double r2055788 = 2.0;
        double r2055789 = r2055788 * r2055781;
        double r2055790 = r2055787 / r2055789;
        return r2055790;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r2055791 = c;
        double r2055792 = b;
        double r2055793 = r2055791 / r2055792;
        double r2055794 = -2.0;
        double r2055795 = r2055793 * r2055794;
        double r2055796 = 2.0;
        double r2055797 = r2055795 / r2055796;
        return r2055797;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.1

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

2. Simplified 44.1

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

3. Taylor expanded around inf 11.8

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

4. Final simplification 11.8

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

Reproduce

herbie shell --seed 2019169 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))