Average Error: 43.8 → 12.0
Time: 17.6s
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 r1458649 = b;
        double r1458650 = -r1458649;
        double r1458651 = r1458649 * r1458649;
        double r1458652 = 4.0;
        double r1458653 = a;
        double r1458654 = r1458652 * r1458653;
        double r1458655 = c;
        double r1458656 = r1458654 * r1458655;
        double r1458657 = r1458651 - r1458656;
        double r1458658 = sqrt(r1458657);
        double r1458659 = r1458650 + r1458658;
        double r1458660 = 2.0;
        double r1458661 = r1458660 * r1458653;
        double r1458662 = r1458659 / r1458661;
        return r1458662;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r1458663 = c;
        double r1458664 = b;
        double r1458665 = r1458663 / r1458664;
        double r1458666 = -2.0;
        double r1458667 = r1458665 * r1458666;
        double r1458668 = 2.0;
        double r1458669 = r1458667 / r1458668;
        return r1458669;
}

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 43.8

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

  2. Simplified43.8

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

  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 2019163 
(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)))