Average Error: 52.5 → 6.2
Time: 16.7s
Precision: 64
\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\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 r344595 = b;
        double r344596 = -r344595;
        double r344597 = r344595 * r344595;
        double r344598 = 4.0;
        double r344599 = a;
        double r344600 = r344598 * r344599;
        double r344601 = c;
        double r344602 = r344600 * r344601;
        double r344603 = r344597 - r344602;
        double r344604 = sqrt(r344603);
        double r344605 = r344596 + r344604;
        double r344606 = 2.0;
        double r344607 = r344606 * r344599;
        double r344608 = r344605 / r344607;
        return r344608;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r344609 = c;
        double r344610 = b;
        double r344611 = r344609 / r344610;
        double r344612 = -2.0;
        double r344613 = r344611 * r344612;
        double r344614 = 2.0;
        double r344615 = r344613 / r344614;
        return r344615;
}

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

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

  3. Taylor expanded around inf 6.5

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

  4. Taylor expanded around 0 6.2

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

  5. Final simplification6.2

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

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))