Average Error: 52.5 → 6.2
Time: 7.4s
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 r22454 = b;
        double r22455 = -r22454;
        double r22456 = r22454 * r22454;
        double r22457 = 4.0;
        double r22458 = a;
        double r22459 = r22457 * r22458;
        double r22460 = c;
        double r22461 = r22459 * r22460;
        double r22462 = r22456 - r22461;
        double r22463 = sqrt(r22462);
        double r22464 = r22455 + r22463;
        double r22465 = 2.0;
        double r22466 = r22465 * r22458;
        double r22467 = r22464 / r22466;
        return r22467;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r22468 = -1.0;
        double r22469 = c;
        double r22470 = b;
        double r22471 = r22469 / r22470;
        double r22472 = r22468 * r22471;
        return r22472;
}

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 2020042 
(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)))