Quadratic roots, medium range

Average Error: 43.7 → 12.1

Time: 12.7s

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 r33102 = b;
        double r33103 = -r33102;
        double r33104 = r33102 * r33102;
        double r33105 = 4.0;
        double r33106 = a;
        double r33107 = r33105 * r33106;
        double r33108 = c;
        double r33109 = r33107 * r33108;
        double r33110 = r33104 - r33109;
        double r33111 = sqrt(r33110);
        double r33112 = r33103 + r33111;
        double r33113 = 2.0;
        double r33114 = r33113 * r33106;
        double r33115 = r33112 / r33114;
        return r33115;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r33116 = -1.0;
        double r33117 = c;
        double r33118 = b;
        double r33119 = r33117 / r33118;
        double r33120 = r33116 * r33119;
        return r33120;
}

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

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

3. Taylor expanded around inf 12.1

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

4. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))