Average Error: 43.7 → 12.1
Time: 18.2s
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 r1139268 = b;
        double r1139269 = -r1139268;
        double r1139270 = r1139268 * r1139268;
        double r1139271 = 4.0;
        double r1139272 = a;
        double r1139273 = r1139271 * r1139272;
        double r1139274 = c;
        double r1139275 = r1139273 * r1139274;
        double r1139276 = r1139270 - r1139275;
        double r1139277 = sqrt(r1139276);
        double r1139278 = r1139269 + r1139277;
        double r1139279 = 2.0;
        double r1139280 = r1139279 * r1139272;
        double r1139281 = r1139278 / r1139280;
        return r1139281;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r1139282 = c;
        double r1139283 = b;
        double r1139284 = r1139282 / r1139283;
        double r1139285 = -r1139284;
        return r1139285;
}

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 43.7

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

  2. Simplified43.7

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

  3. Taylor expanded around inf 12.1

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

  4. Simplified12.1

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

  5. Final simplification12.1

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

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(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)))