Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 33.1s

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 r101623 = b;
        double r101624 = -r101623;
        double r101625 = r101623 * r101623;
        double r101626 = 3.0;
        double r101627 = a;
        double r101628 = r101626 * r101627;
        double r101629 = c;
        double r101630 = r101628 * r101629;
        double r101631 = r101625 - r101630;
        double r101632 = sqrt(r101631);
        double r101633 = r101624 + r101632;
        double r101634 = r101633 / r101628;
        return r101634;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r101635 = -0.5;
        double r101636 = c;
        double r101637 = b;
        double r101638 = r101636 / r101637;
        double r101639 = r101635 * r101638;
        return r101639;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

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

3. Taylor expanded around inf 6.4

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

4. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))