Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 18.5s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

Math

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

↓

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

C

double f(double a, double b, double c) {
        double r9355426 = b;
        double r9355427 = -r9355426;
        double r9355428 = r9355426 * r9355426;
        double r9355429 = 3.0;
        double r9355430 = a;
        double r9355431 = r9355429 * r9355430;
        double r9355432 = c;
        double r9355433 = r9355431 * r9355432;
        double r9355434 = r9355428 - r9355433;
        double r9355435 = sqrt(r9355434);
        double r9355436 = r9355427 + r9355435;
        double r9355437 = r9355436 / r9355431;
        return r9355437;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r9355438 = -0.5;
        double r9355439 = c;
        double r9355440 = b;
        double r9355441 = r9355439 / r9355440;
        double r9355442 = r9355438 * r9355441;
        return r9355442;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

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. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019173 +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)))