Average Error: 52.5 → 6.2
Time: 25.6s
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 r4150629 = b;
        double r4150630 = -r4150629;
        double r4150631 = r4150629 * r4150629;
        double r4150632 = 3.0;
        double r4150633 = a;
        double r4150634 = r4150632 * r4150633;
        double r4150635 = c;
        double r4150636 = r4150634 * r4150635;
        double r4150637 = r4150631 - r4150636;
        double r4150638 = sqrt(r4150637);
        double r4150639 = r4150630 + r4150638;
        double r4150640 = r4150639 / r4150634;
        return r4150640;
}
↓
double f(double __attribute__((unused)) a, double b, double c) {
        double r4150641 = c;
        double r4150642 = b;
        double r4150643 = r4150641 / r4150642;
        double r4150644 = -0.5;
        double r4150645 = r4150643 * r4150644;
        return r4150645;
}

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

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

  2. Simplified52.5

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

  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 2019200 +o rules:numerics
(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)))