Average Error: 44.0 → 12.0
Time: 14.2s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{\frac{c}{b} \cdot -2}{2}
double f(double a, double b, double c) {
        double r809409 = b;
        double r809410 = -r809409;
        double r809411 = r809409 * r809409;
        double r809412 = 4.0;
        double r809413 = a;
        double r809414 = r809412 * r809413;
        double r809415 = c;
        double r809416 = r809414 * r809415;
        double r809417 = r809411 - r809416;
        double r809418 = sqrt(r809417);
        double r809419 = r809410 + r809418;
        double r809420 = 2.0;
        double r809421 = r809420 * r809413;
        double r809422 = r809419 / r809421;
        return r809422;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r809423 = c;
        double r809424 = b;
        double r809425 = r809423 / r809424;
        double r809426 = -2.0;
        double r809427 = r809425 * r809426;
        double r809428 = 2.0;
        double r809429 = r809427 / r809428;
        return r809429;
}

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 44.0

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

  2. Simplified44.0

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

  3. Taylor expanded around inf 12.0

    \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c}{b}}}{2}\]

  4. Final simplification12.0

    \[\leadsto \frac{\frac{c}{b} \cdot -2}{2}\]

Reproduce

herbie shell --seed 2019153 +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)))