Cubic critical, medium range

Average Error: 43.7 → 12.1

Time: 6.7s

Precision: 64

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

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

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 r81757 = b;
        double r81758 = -r81757;
        double r81759 = r81757 * r81757;
        double r81760 = 3.0;
        double r81761 = a;
        double r81762 = r81760 * r81761;
        double r81763 = c;
        double r81764 = r81762 * r81763;
        double r81765 = r81759 - r81764;
        double r81766 = sqrt(r81765);
        double r81767 = r81758 + r81766;
        double r81768 = r81767 / r81762;
        return r81768;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r81769 = -0.5;
        double r81770 = c;
        double r81771 = b;
        double r81772 = r81770 / r81771;
        double r81773 = r81769 * r81772;
        return r81773;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

2. Taylor expanded around inf 12.1

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

3. Final simplification 12.1

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))