Average Error: 52.3 → 6.4
Time: 15.3s
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 r16867 = b;
        double r16868 = -r16867;
        double r16869 = r16867 * r16867;
        double r16870 = 4.0;
        double r16871 = a;
        double r16872 = r16870 * r16871;
        double r16873 = c;
        double r16874 = r16872 * r16873;
        double r16875 = r16869 - r16874;
        double r16876 = sqrt(r16875);
        double r16877 = r16868 + r16876;
        double r16878 = 2.0;
        double r16879 = r16878 * r16871;
        double r16880 = r16877 / r16879;
        return r16880;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r16881 = -1.0;
        double r16882 = c;
        double r16883 = b;
        double r16884 = r16882 / r16883;
        double r16885 = r16881 * r16884;
        return r16885;
}

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

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

  3. Taylor expanded around inf 6.4

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

  4. Final simplification6.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))