Average Error: 52.5 → 6.2
Time: 6.8s
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 r24138 = b;
        double r24139 = -r24138;
        double r24140 = r24138 * r24138;
        double r24141 = 4.0;
        double r24142 = a;
        double r24143 = r24141 * r24142;
        double r24144 = c;
        double r24145 = r24143 * r24144;
        double r24146 = r24140 - r24145;
        double r24147 = sqrt(r24146);
        double r24148 = r24139 + r24147;
        double r24149 = 2.0;
        double r24150 = r24149 * r24142;
        double r24151 = r24148 / r24150;
        return r24151;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r24152 = -1.0;
        double r24153 = c;
        double r24154 = b;
        double r24155 = r24153 / r24154;
        double r24156 = r24152 * r24155;
        return r24156;
}

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 2020047 
(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)))