Quadratic roots, medium range

Average Error: 44.1 → 11.8

Time: 16.9s

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 r1093647 = b;
        double r1093648 = -r1093647;
        double r1093649 = r1093647 * r1093647;
        double r1093650 = 4.0;
        double r1093651 = a;
        double r1093652 = r1093650 * r1093651;
        double r1093653 = c;
        double r1093654 = r1093652 * r1093653;
        double r1093655 = r1093649 - r1093654;
        double r1093656 = sqrt(r1093655);
        double r1093657 = r1093648 + r1093656;
        double r1093658 = 2.0;
        double r1093659 = r1093658 * r1093651;
        double r1093660 = r1093657 / r1093659;
        return r1093660;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1093661 = c;
        double r1093662 = b;
        double r1093663 = r1093661 / r1093662;
        double r1093664 = -r1093663;
        return r1093664;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.1

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

2. Simplified 44.1

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

3. Taylor expanded around inf 11.8

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

4. Simplified 11.8

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

5. Final simplification 11.8

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

Reproduce

herbie shell --seed 2019134 
(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)))