Quadratic roots, wide range

Average Error: 52.7 → 6.1

Time: 1.8m

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

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 r9154867 = b;
        double r9154868 = -r9154867;
        double r9154869 = r9154867 * r9154867;
        double r9154870 = 4.0;
        double r9154871 = a;
        double r9154872 = r9154870 * r9154871;
        double r9154873 = c;
        double r9154874 = r9154872 * r9154873;
        double r9154875 = r9154869 - r9154874;
        double r9154876 = sqrt(r9154875);
        double r9154877 = r9154868 + r9154876;
        double r9154878 = 2.0;
        double r9154879 = r9154878 * r9154871;
        double r9154880 = r9154877 / r9154879;
        return r9154880;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r9154881 = c;
        double r9154882 = b;
        double r9154883 = r9154881 / r9154882;
        double r9154884 = -r9154883;
        return r9154884;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.7

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

2. Simplified 52.7

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

3. Taylor expanded around inf 6.1

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

4. Simplified 6.1

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

5. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019104 
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))