Quadratic roots, wide range

Average Error: 52.2 → 6.5

Time: 17.9s

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 r590680 = b;
        double r590681 = -r590680;
        double r590682 = r590680 * r590680;
        double r590683 = 4.0;
        double r590684 = a;
        double r590685 = r590683 * r590684;
        double r590686 = c;
        double r590687 = r590685 * r590686;
        double r590688 = r590682 - r590687;
        double r590689 = sqrt(r590688);
        double r590690 = r590681 + r590689;
        double r590691 = 2.0;
        double r590692 = r590691 * r590684;
        double r590693 = r590690 / r590692;
        return r590693;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r590694 = c;
        double r590695 = b;
        double r590696 = r590694 / r590695;
        double r590697 = -r590696;
        return r590697;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.2

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

2. Simplified 52.2

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

3. Taylor expanded around inf 6.5

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

4. Simplified 6.5

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

5. Final simplification 6.5

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

Reproduce

herbie shell --seed 2019134 
(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)))