Cubic critical, wide range

Average Error: 52.6 → 6.2

Time: 16.5s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r81257 = b;
        double r81258 = -r81257;
        double r81259 = r81257 * r81257;
        double r81260 = 3.0;
        double r81261 = a;
        double r81262 = r81260 * r81261;
        double r81263 = c;
        double r81264 = r81262 * r81263;
        double r81265 = r81259 - r81264;
        double r81266 = sqrt(r81265);
        double r81267 = r81258 + r81266;
        double r81268 = r81267 / r81262;
        return r81268;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r81269 = -0.5;
        double r81270 = c;
        double r81271 = b;
        double r81272 = r81270 / r81271;
        double r81273 = r81269 * r81272;
        return r81273;
}

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 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019195 +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)))