Cubic critical, wide range

Average Error: 52.3 → 6.3

Time: 14.6s

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 r62400 = b;
        double r62401 = -r62400;
        double r62402 = r62400 * r62400;
        double r62403 = 3.0;
        double r62404 = a;
        double r62405 = r62403 * r62404;
        double r62406 = c;
        double r62407 = r62405 * r62406;
        double r62408 = r62402 - r62407;
        double r62409 = sqrt(r62408);
        double r62410 = r62401 + r62409;
        double r62411 = r62410 / r62405;
        return r62411;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r62412 = -0.5;
        double r62413 = c;
        double r62414 = b;
        double r62415 = r62413 / r62414;
        double r62416 = r62412 * r62415;
        return r62416;
}

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

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

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 2019209 +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)))