Average Error: 52.7 → 6.1
Time: 13.1s
Precision: 64
\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[-0.5 \cdot \frac{c}{b}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}
↓
-0.5 \cdot \frac{c}{b}
double f(double a, double b, double c) {
        double r88350 = b;
        double r88351 = -r88350;
        double r88352 = r88350 * r88350;
        double r88353 = 3.0;
        double r88354 = a;
        double r88355 = r88353 * r88354;
        double r88356 = c;
        double r88357 = r88355 * r88356;
        double r88358 = r88352 - r88357;
        double r88359 = sqrt(r88358);
        double r88360 = r88351 + r88359;
        double r88361 = r88360 / r88355;
        return r88361;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r88362 = -0.5;
        double r88363 = c;
        double r88364 = b;
        double r88365 = r88363 / r88364;
        double r88366 = r88362 * r88365;
        return r88366;
}

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 52.7

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

  2. Simplified52.7

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

  3. Taylor expanded around inf 6.1

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

  4. Final simplification6.1

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))