Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 32.5s

Precision: 64

\[4.930380657631324 \cdot 10^{-32} \lt a \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt b \lt 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} \lt c \lt 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

\[\frac{c}{b} \cdot \frac{-1}{2}\]

C

double f(double a, double b, double c) {
        double r3674715 = b;
        double r3674716 = -r3674715;
        double r3674717 = r3674715 * r3674715;
        double r3674718 = 3.0;
        double r3674719 = a;
        double r3674720 = r3674718 * r3674719;
        double r3674721 = c;
        double r3674722 = r3674720 * r3674721;
        double r3674723 = r3674717 - r3674722;
        double r3674724 = sqrt(r3674723);
        double r3674725 = r3674716 + r3674724;
        double r3674726 = r3674725 / r3674720;
        return r3674726;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r3674727 = c;
        double r3674728 = b;
        double r3674729 = r3674727 / r3674728;
        double r3674730 = -0.5;
        double r3674731 = r3674729 * r3674730;
        return r3674731;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Taylor expanded around inf 6.2

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}}\]

3. Final simplification 6.2

   \[\leadsto \frac{c}{b} \cdot \frac{-1}{2}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))