Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 15.6s

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

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

↓

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

C

double f(double a, double b, double c) {
        double r85690 = b;
        double r85691 = -r85690;
        double r85692 = r85690 * r85690;
        double r85693 = 3.0;
        double r85694 = a;
        double r85695 = r85693 * r85694;
        double r85696 = c;
        double r85697 = r85695 * r85696;
        double r85698 = r85692 - r85697;
        double r85699 = sqrt(r85698);
        double r85700 = r85691 + r85699;
        double r85701 = r85700 / r85695;
        return r85701;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r85702 = c;
        double r85703 = b;
        double r85704 = r85702 / r85703;
        double r85705 = -0.5;
        double r85706 = r85704 * r85705;
        return r85706;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Taylor expanded around inf 6.3

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

3. Simplified 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019194 +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)))