Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 5.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

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 r73533 = b;
        double r73534 = -r73533;
        double r73535 = r73533 * r73533;
        double r73536 = 3.0;
        double r73537 = a;
        double r73538 = r73536 * r73537;
        double r73539 = c;
        double r73540 = r73538 * r73539;
        double r73541 = r73535 - r73540;
        double r73542 = sqrt(r73541);
        double r73543 = r73534 + r73542;
        double r73544 = r73543 / r73538;
        return r73544;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r73545 = -0.5;
        double r73546 = c;
        double r73547 = b;
        double r73548 = r73546 / r73547;
        double r73549 = r73545 * r73548;
        return r73549;
}

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. Simplified 52.4

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

3. Taylor expanded around inf 6.3

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

4. Final simplification 6.3

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

Reproduce

herbie shell --seed 2020046 
(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)))