Average Error: 52.6 → 6.2
Time: 16.3s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c}{b} \cdot -0.5\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
\frac{c}{b} \cdot -0.5
double f(double a, double b, double c) {
        double r4460823 = b;
        double r4460824 = -r4460823;
        double r4460825 = r4460823 * r4460823;
        double r4460826 = 3.0;
        double r4460827 = a;
        double r4460828 = r4460826 * r4460827;
        double r4460829 = c;
        double r4460830 = r4460828 * r4460829;
        double r4460831 = r4460825 - r4460830;
        double r4460832 = sqrt(r4460831);
        double r4460833 = r4460824 + r4460832;
        double r4460834 = r4460833 / r4460828;
        return r4460834;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r4460835 = c;
        double r4460836 = b;
        double r4460837 = r4460835 / r4460836;
        double r4460838 = -0.5;
        double r4460839 = r4460837 * r4460838;
        return r4460839;
}

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 52.6

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

  2. Simplified52.6

    \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}{3 \cdot a}}\]

  3. Taylor expanded around inf 6.2

    \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

  4. Final simplification6.2

    \[\leadsto \frac{c}{b} \cdot -0.5\]

Reproduce

herbie shell --seed 2019170 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))