Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 21.5s

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 r6221365 = b;
        double r6221366 = -r6221365;
        double r6221367 = r6221365 * r6221365;
        double r6221368 = 3.0;
        double r6221369 = a;
        double r6221370 = r6221368 * r6221369;
        double r6221371 = c;
        double r6221372 = r6221370 * r6221371;
        double r6221373 = r6221367 - r6221372;
        double r6221374 = sqrt(r6221373);
        double r6221375 = r6221366 + r6221374;
        double r6221376 = r6221375 / r6221370;
        return r6221376;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r6221377 = -0.5;
        double r6221378 = c;
        double r6221379 = b;
        double r6221380 = r6221378 / r6221379;
        double r6221381 = r6221377 * r6221380;
        return r6221381;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

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. Simplified 52.6

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

3. Taylor expanded around inf 6.1

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

4. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019174 +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)))