Cubic critical, medium range

Average Error: 44.0 → 11.9

Time: 19.0s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

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 r5642209 = b;
        double r5642210 = -r5642209;
        double r5642211 = r5642209 * r5642209;
        double r5642212 = 3.0;
        double r5642213 = a;
        double r5642214 = r5642212 * r5642213;
        double r5642215 = c;
        double r5642216 = r5642214 * r5642215;
        double r5642217 = r5642211 - r5642216;
        double r5642218 = sqrt(r5642217);
        double r5642219 = r5642210 + r5642218;
        double r5642220 = r5642219 / r5642214;
        return r5642220;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r5642221 = -0.5;
        double r5642222 = c;
        double r5642223 = b;
        double r5642224 = r5642222 / r5642223;
        double r5642225 = r5642221 * r5642224;
        return r5642225;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.0

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

2. Simplified 44.0

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

3. Taylor expanded around inf 11.9

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

4. Final simplification 11.9

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))