Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 13.6s

Precision: 64

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

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 r68223 = b;
        double r68224 = -r68223;
        double r68225 = r68223 * r68223;
        double r68226 = 3.0;
        double r68227 = a;
        double r68228 = r68226 * r68227;
        double r68229 = c;
        double r68230 = r68228 * r68229;
        double r68231 = r68225 - r68230;
        double r68232 = sqrt(r68231);
        double r68233 = r68224 + r68232;
        double r68234 = r68233 / r68228;
        return r68234;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r68235 = -0.5;
        double r68236 = c;
        double r68237 = b;
        double r68238 = r68236 / r68237;
        double r68239 = r68235 * r68238;
        return r68239;
}

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

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

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 2019303 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))