Average Error: 52.3 → 6.3
Time: 18.5s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c}{b} \cdot -1\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}
↓
\frac{c}{b} \cdot -1
double f(double a, double b, double c) {
        double r910666 = b;
        double r910667 = -r910666;
        double r910668 = r910666 * r910666;
        double r910669 = 4.0;
        double r910670 = a;
        double r910671 = r910669 * r910670;
        double r910672 = c;
        double r910673 = r910671 * r910672;
        double r910674 = r910668 - r910673;
        double r910675 = sqrt(r910674);
        double r910676 = r910667 + r910675;
        double r910677 = 2.0;
        double r910678 = r910677 * r910670;
        double r910679 = r910676 / r910678;
        return r910679;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r910680 = c;
        double r910681 = b;
        double r910682 = r910680 / r910681;
        double r910683 = -1.0;
        double r910684 = r910682 * r910683;
        return r910684;
}

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

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

  3. Taylor expanded around inf 6.3

    \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

  4. Final simplification6.3

    \[\leadsto \frac{c}{b} \cdot -1\]

Reproduce

herbie shell --seed 2019172 +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)))