Average Error: 52.3 → 6.4
Time: 17.9s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c}{b} \cdot -2}{2}\]
\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 r795708 = b;
        double r795709 = -r795708;
        double r795710 = r795708 * r795708;
        double r795711 = 4.0;
        double r795712 = a;
        double r795713 = r795711 * r795712;
        double r795714 = c;
        double r795715 = r795713 * r795714;
        double r795716 = r795710 - r795715;
        double r795717 = sqrt(r795716);
        double r795718 = r795709 + r795717;
        double r795719 = 2.0;
        double r795720 = r795719 * r795712;
        double r795721 = r795718 / r795720;
        return r795721;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r795722 = c;
        double r795723 = b;
        double r795724 = r795722 / r795723;
        double r795725 = -2.0;
        double r795726 = r795724 * r795725;
        double r795727 = 2.0;
        double r795728 = r795726 / r795727;
        return r795728;
}

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 52.3

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

  2. Simplified52.3

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

  3. Taylor expanded around inf 6.4

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

  4. Final simplification6.4

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

Reproduce

herbie shell --seed 2019192 +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) (* (* 4.0 a) c)))) (* 2.0 a)))