Quadratic roots, medium range

Average Error: 43.9 → 12.0

Time: 39.2s

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 r6008708 = b;
        double r6008709 = -r6008708;
        double r6008710 = r6008708 * r6008708;
        double r6008711 = 4.0;
        double r6008712 = a;
        double r6008713 = r6008711 * r6008712;
        double r6008714 = c;
        double r6008715 = r6008713 * r6008714;
        double r6008716 = r6008710 - r6008715;
        double r6008717 = sqrt(r6008716);
        double r6008718 = r6008709 + r6008717;
        double r6008719 = 2.0;
        double r6008720 = r6008719 * r6008712;
        double r6008721 = r6008718 / r6008720;
        return r6008721;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r6008722 = c;
        double r6008723 = b;
        double r6008724 = r6008722 / r6008723;
        double r6008725 = -r6008724;
        return r6008725;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

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

3. Taylor expanded around inf 12.0

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

4. Simplified 12.0

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

5. Final simplification 12.0

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

Reproduce

herbie shell --seed 2019112 
(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)))