Cubic critical, medium range

Average Error: 43.9 → 12.0

Time: 13.1s

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 r67729 = b;
        double r67730 = -r67729;
        double r67731 = r67729 * r67729;
        double r67732 = 3.0;
        double r67733 = a;
        double r67734 = r67732 * r67733;
        double r67735 = c;
        double r67736 = r67734 * r67735;
        double r67737 = r67731 - r67736;
        double r67738 = sqrt(r67737);
        double r67739 = r67730 + r67738;
        double r67740 = r67739 / r67734;
        return r67740;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r67741 = -0.5;
        double r67742 = c;
        double r67743 = b;
        double r67744 = r67742 / r67743;
        double r67745 = r67741 * r67744;
        return r67745;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.9

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

2. Simplified 43.9

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

3. Taylor expanded around inf 12.0

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

4. Final simplification 12.0

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

Reproduce

herbie shell --seed 2019235 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e15) (< 1.11022e-16 b 9.0072e15) (< 1.11022e-16 c 9.0072e15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))