Cubic critical, medium range

Average Error: 44.0 → 11.9

Time: 18.6s

Precision: 64

\[1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt a \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt b \lt 9007199254740992 \land 1.1102230246251565404236316680908203125 \cdot 10^{-16} \lt c \lt 9007199254740992\]

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 r6212405 = b;
        double r6212406 = -r6212405;
        double r6212407 = r6212405 * r6212405;
        double r6212408 = 3.0;
        double r6212409 = a;
        double r6212410 = r6212408 * r6212409;
        double r6212411 = c;
        double r6212412 = r6212410 * r6212411;
        double r6212413 = r6212407 - r6212412;
        double r6212414 = sqrt(r6212413);
        double r6212415 = r6212406 + r6212414;
        double r6212416 = r6212415 / r6212410;
        return r6212416;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r6212417 = -0.5;
        double r6212418 = c;
        double r6212419 = b;
        double r6212420 = r6212418 / r6212419;
        double r6212421 = r6212417 * r6212420;
        return r6212421;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 44.0

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

2. Simplified 44.0

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

3. Taylor expanded around inf 11.9

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

4. Final simplification 11.9

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

Reproduce

herbie shell --seed 2019173 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))