Average Error: 52.4 → 6.3
Time: 17.9s
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 r3591447 = b;
        double r3591448 = -r3591447;
        double r3591449 = r3591447 * r3591447;
        double r3591450 = 3.0;
        double r3591451 = a;
        double r3591452 = r3591450 * r3591451;
        double r3591453 = c;
        double r3591454 = r3591452 * r3591453;
        double r3591455 = r3591449 - r3591454;
        double r3591456 = sqrt(r3591455);
        double r3591457 = r3591448 + r3591456;
        double r3591458 = r3591457 / r3591452;
        return r3591458;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r3591459 = c;
        double r3591460 = b;
        double r3591461 = r3591459 / r3591460;
        double r3591462 = -0.5;
        double r3591463 = r3591461 * r3591462;
        return r3591463;
}

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 52.4

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

  2. Simplified52.4

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

  3. Taylor expanded around inf 6.3

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

  4. Final simplification6.3

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

Reproduce

herbie shell --seed 2019172 
(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)))