Cubic critical, wide range

Average Error: 52.6 → 6.1

Time: 14.8s

Precision: 64

\[4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt a \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt b \lt 20282409603651670423947251286016 \land 4.930380657631323783823303533017413935458 \cdot 10^{-32} \lt c \lt 20282409603651670423947251286016\]

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 r63567 = b;
        double r63568 = -r63567;
        double r63569 = r63567 * r63567;
        double r63570 = 3.0;
        double r63571 = a;
        double r63572 = r63570 * r63571;
        double r63573 = c;
        double r63574 = r63572 * r63573;
        double r63575 = r63569 - r63574;
        double r63576 = sqrt(r63575);
        double r63577 = r63568 + r63576;
        double r63578 = r63577 / r63572;
        return r63578;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r63579 = -0.5;
        double r63580 = c;
        double r63581 = b;
        double r63582 = r63580 / r63581;
        double r63583 = r63579 * r63582;
        return r63583;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.6

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

2. Simplified 52.6

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

3. Taylor expanded around inf 6.1

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

4. Final simplification 6.1

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

Reproduce

herbie shell --seed 2019208 +o rules:numerics
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e31) (< 4.93038e-32 b 2.02824e31) (< 4.93038e-32 c 2.02824e31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))