Quadratic roots, medium range

Average Error: 44.2 → 11.8

Time: 59.1s

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 r8447771 = b;
        double r8447772 = -r8447771;
        double r8447773 = r8447771 * r8447771;
        double r8447774 = 4.0;
        double r8447775 = a;
        double r8447776 = r8447774 * r8447775;
        double r8447777 = c;
        double r8447778 = r8447776 * r8447777;
        double r8447779 = r8447773 - r8447778;
        double r8447780 = sqrt(r8447779);
        double r8447781 = r8447772 + r8447780;
        double r8447782 = 2.0;
        double r8447783 = r8447782 * r8447775;
        double r8447784 = r8447781 / r8447783;
        return r8447784;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r8447785 = c;
        double r8447786 = b;
        double r8447787 = r8447785 / r8447786;
        double r8447788 = -r8447787;
        return r8447788;
}

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

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

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 2019104 
(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)))