Average Error: 52.5 → 6.1
Time: 13.5s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[-1 \cdot \frac{c}{b}\]
\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 r15176 = b;
        double r15177 = -r15176;
        double r15178 = r15176 * r15176;
        double r15179 = 4.0;
        double r15180 = a;
        double r15181 = r15179 * r15180;
        double r15182 = c;
        double r15183 = r15181 * r15182;
        double r15184 = r15178 - r15183;
        double r15185 = sqrt(r15184);
        double r15186 = r15177 + r15185;
        double r15187 = 2.0;
        double r15188 = r15187 * r15180;
        double r15189 = r15186 / r15188;
        return r15189;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r15190 = -1.0;
        double r15191 = c;
        double r15192 = b;
        double r15193 = r15191 / r15192;
        double r15194 = r15190 * r15193;
        return r15194;
}

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

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

  3. Taylor expanded around inf 6.1

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

  4. Final simplification6.1

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

Reproduce

herbie shell --seed 2019208 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))