Cubic critical, medium range

Average Error: 44.0 → 11.9

Time: 19.4s

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 r7132999 = b;
        double r7133000 = -r7132999;
        double r7133001 = r7132999 * r7132999;
        double r7133002 = 3.0;
        double r7133003 = a;
        double r7133004 = r7133002 * r7133003;
        double r7133005 = c;
        double r7133006 = r7133004 * r7133005;
        double r7133007 = r7133001 - r7133006;
        double r7133008 = sqrt(r7133007);
        double r7133009 = r7133000 + r7133008;
        double r7133010 = r7133009 / r7133004;
        return r7133010;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r7133011 = -0.5;
        double r7133012 = c;
        double r7133013 = b;
        double r7133014 = r7133012 / r7133013;
        double r7133015 = r7133011 * r7133014;
        return r7133015;
}

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 +o rules:numerics
(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)))