Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 15.2s

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 r107198 = b;
        double r107199 = -r107198;
        double r107200 = r107198 * r107198;
        double r107201 = 3.0;
        double r107202 = a;
        double r107203 = r107201 * r107202;
        double r107204 = c;
        double r107205 = r107203 * r107204;
        double r107206 = r107200 - r107205;
        double r107207 = sqrt(r107206);
        double r107208 = r107199 + r107207;
        double r107209 = r107208 / r107203;
        return r107209;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r107210 = -0.5;
        double r107211 = c;
        double r107212 = b;
        double r107213 = r107211 / r107212;
        double r107214 = r107210 * r107213;
        return r107214;
}

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 2019235 +o rules:numerics
(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)))