Cubic critical, medium range

Average Error: 43.7 → 12.1

Time: 10.9s

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 r64999 = b;
        double r65000 = -r64999;
        double r65001 = r64999 * r64999;
        double r65002 = 3.0;
        double r65003 = a;
        double r65004 = r65002 * r65003;
        double r65005 = c;
        double r65006 = r65004 * r65005;
        double r65007 = r65001 - r65006;
        double r65008 = sqrt(r65007);
        double r65009 = r65000 + r65008;
        double r65010 = r65009 / r65004;
        return r65010;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r65011 = -0.5;
        double r65012 = c;
        double r65013 = b;
        double r65014 = r65012 / r65013;
        double r65015 = r65011 * r65014;
        return r65015;
}

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. Simplified 43.7

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

3. Taylor expanded around inf 12.1

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

4. Final simplification 12.1

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(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)))