Cubic critical, medium range

Average Error: 44.0 → 11.9

Time: 16.6s

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 r6271614 = b;
        double r6271615 = -r6271614;
        double r6271616 = r6271614 * r6271614;
        double r6271617 = 3.0;
        double r6271618 = a;
        double r6271619 = r6271617 * r6271618;
        double r6271620 = c;
        double r6271621 = r6271619 * r6271620;
        double r6271622 = r6271616 - r6271621;
        double r6271623 = sqrt(r6271622);
        double r6271624 = r6271615 + r6271623;
        double r6271625 = r6271624 / r6271619;
        return r6271625;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r6271626 = -0.5;
        double r6271627 = c;
        double r6271628 = b;
        double r6271629 = r6271627 / r6271628;
        double r6271630 = r6271626 * r6271629;
        return r6271630;
}

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 2019174 
(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)))