Average Error: 44.2 → 11.7
Time: 53.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-\frac{c}{b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
-\frac{c}{b}
double f(double a, double b, double c) {
        double r5633918 = b;
        double r5633919 = -r5633918;
        double r5633920 = r5633918 * r5633918;
        double r5633921 = 4.0;
        double r5633922 = a;
        double r5633923 = r5633921 * r5633922;
        double r5633924 = c;
        double r5633925 = r5633923 * r5633924;
        double r5633926 = r5633920 - r5633925;
        double r5633927 = sqrt(r5633926);
        double r5633928 = r5633919 + r5633927;
        double r5633929 = 2.0;
        double r5633930 = r5633929 * r5633922;
        double r5633931 = r5633928 / r5633930;
        return r5633931;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r5633932 = c;
        double r5633933 = b;
        double r5633934 = r5633932 / r5633933;
        double r5633935 = -r5633934;
        return r5633935;
}

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 44.2

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

  2. Simplified44.2

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

  3. Taylor expanded around inf 11.7

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

  4. Simplified11.7

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

  5. Final simplification11.7

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

Reproduce

herbie shell --seed 2019128 
(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 a) c)))) (* 2 a)))