Average Error: 43.9 → 12.0
Time: 12.4s
Precision: 64
\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-1 \cdot \frac{c}{b}\]
\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 r28164 = b;
        double r28165 = -r28164;
        double r28166 = r28164 * r28164;
        double r28167 = 4.0;
        double r28168 = a;
        double r28169 = r28167 * r28168;
        double r28170 = c;
        double r28171 = r28169 * r28170;
        double r28172 = r28166 - r28171;
        double r28173 = sqrt(r28172);
        double r28174 = r28165 + r28173;
        double r28175 = 2.0;
        double r28176 = r28175 * r28168;
        double r28177 = r28174 / r28176;
        return r28177;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r28178 = -1.0;
        double r28179 = c;
        double r28180 = b;
        double r28181 = r28179 / r28180;
        double r28182 = r28178 * r28181;
        return r28182;
}

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 43.9

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

  2. Simplified43.9

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

  3. Taylor expanded around inf 12.0

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

  4. Final simplification12.0

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

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))