Quadratic roots, medium range

Average Error: 43.7 → 12.1

Time: 15.7s

Precision: 64

\[1.1102230246251565 \cdot 10^{-16} \lt a \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt b \lt 9007199254740992.0 \land 1.1102230246251565 \cdot 10^{-16} \lt c \lt 9007199254740992.0\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

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

C

double f(double a, double b, double c) {
        double r1054277 = b;
        double r1054278 = -r1054277;
        double r1054279 = r1054277 * r1054277;
        double r1054280 = 4.0;
        double r1054281 = a;
        double r1054282 = r1054280 * r1054281;
        double r1054283 = c;
        double r1054284 = r1054282 * r1054283;
        double r1054285 = r1054279 - r1054284;
        double r1054286 = sqrt(r1054285);
        double r1054287 = r1054278 + r1054286;
        double r1054288 = 2.0;
        double r1054289 = r1054288 * r1054281;
        double r1054290 = r1054287 / r1054289;
        return r1054290;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1054291 = c;
        double r1054292 = b;
        double r1054293 = r1054291 / r1054292;
        double r1054294 = -r1054293;
        return r1054294;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

2. Simplified 43.7

   \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}}\]

3. Taylor expanded around inf 12.1

   \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]

4. Simplified 12.1

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

5. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019129 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))