Average Error: 52.5 → 6.2
Time: 11.0s
Precision: 64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-1 \cdot \frac{c}{b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-1 \cdot \frac{c}{b}
double f(double a, double b, double c) {
        double r49808 = b;
        double r49809 = -r49808;
        double r49810 = r49808 * r49808;
        double r49811 = 4.0;
        double r49812 = a;
        double r49813 = r49811 * r49812;
        double r49814 = c;
        double r49815 = r49813 * r49814;
        double r49816 = r49810 - r49815;
        double r49817 = sqrt(r49816);
        double r49818 = r49809 + r49817;
        double r49819 = 2.0;
        double r49820 = r49819 * r49812;
        double r49821 = r49818 / r49820;
        return r49821;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r49822 = -1.0;
        double r49823 = c;
        double r49824 = b;
        double r49825 = r49823 / r49824;
        double r49826 = r49822 * r49825;
        return r49826;
}

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

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

  3. Taylor expanded around inf 6.2

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

  4. Final simplification6.2

    \[\leadsto -1 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))