Average Error: 44.0 → 11.9
Time: 18.0s
Precision: 64
\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\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 r687872 = b;
        double r687873 = -r687872;
        double r687874 = r687872 * r687872;
        double r687875 = 4.0;
        double r687876 = a;
        double r687877 = r687875 * r687876;
        double r687878 = c;
        double r687879 = r687877 * r687878;
        double r687880 = r687874 - r687879;
        double r687881 = sqrt(r687880);
        double r687882 = r687873 + r687881;
        double r687883 = 2.0;
        double r687884 = r687883 * r687876;
        double r687885 = r687882 / r687884;
        return r687885;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r687886 = c;
        double r687887 = b;
        double r687888 = r687886 / r687887;
        double r687889 = -2.0;
        double r687890 = r687888 * r687889;
        double r687891 = 2.0;
        double r687892 = r687890 / r687891;
        return r687892;
}

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. Simplified43.9

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

  3. Taylor expanded around inf 11.9

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

  4. Final simplification11.9

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

Reproduce

herbie shell --seed 2019155 +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)))