Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 4.0s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

↓

\[-0.5 \cdot \frac{c}{b}\]

C

double f(double a, double b, double c) {
        double r76650 = b;
        double r76651 = -r76650;
        double r76652 = r76650 * r76650;
        double r76653 = 3.0;
        double r76654 = a;
        double r76655 = r76653 * r76654;
        double r76656 = c;
        double r76657 = r76655 * r76656;
        double r76658 = r76652 - r76657;
        double r76659 = sqrt(r76658);
        double r76660 = r76651 + r76659;
        double r76661 = r76660 / r76655;
        return r76661;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r76662 = -0.5;
        double r76663 = c;
        double r76664 = b;
        double r76665 = r76663 / r76664;
        double r76666 = r76662 * r76665;
        return r76666;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]

2. Taylor expanded around inf 6.3

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}}\]

3. Final simplification 6.3

   \[\leadsto -0.5 \cdot \frac{c}{b}\]

Reproduce

herbie shell --seed 2020020 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))