Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 16.2s

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 r73519 = b;
        double r73520 = -r73519;
        double r73521 = r73519 * r73519;
        double r73522 = 3.0;
        double r73523 = a;
        double r73524 = r73522 * r73523;
        double r73525 = c;
        double r73526 = r73524 * r73525;
        double r73527 = r73521 - r73526;
        double r73528 = sqrt(r73527);
        double r73529 = r73520 + r73528;
        double r73530 = r73529 / r73524;
        return r73530;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r73531 = -0.5;
        double r73532 = c;
        double r73533 = b;
        double r73534 = r73532 / r73533;
        double r73535 = r73531 * r73534;
        return r73535;
}

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. Simplified 52.4

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

3. Taylor expanded around inf 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019326 
(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)))