Quadratic roots, medium range

Average Error: 44.2 → 11.7

Time: 1.1m

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

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

↓

\[-1 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r21729 = b;
        double r21730 = -r21729;
        double r21731 = r21729 * r21729;
        double r21732 = 4.0;
        double r21733 = a;
        double r21734 = r21732 * r21733;
        double r21735 = c;
        double r21736 = r21734 * r21735;
        double r21737 = r21731 - r21736;
        double r21738 = sqrt(r21737);
        double r21739 = r21730 + r21738;
        double r21740 = 2.0;
        double r21741 = r21740 * r21733;
        double r21742 = r21739 / r21741;
        return r21742;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r21743 = -1.0;
        double r21744 = c;
        double r21745 = b;
        double r21746 = r21744 / r21745;
        double r21747 = r21743 * r21746;
        return r21747;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.2

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

2. Simplified 44.2

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

3. Taylor expanded around inf 11.7

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

4. Final simplification 11.7

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

Reproduce

herbie shell --seed 2019202 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))