Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 45.3s

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 r161084 = b;
        double r161085 = -r161084;
        double r161086 = r161084 * r161084;
        double r161087 = 3.0;
        double r161088 = a;
        double r161089 = r161087 * r161088;
        double r161090 = c;
        double r161091 = r161089 * r161090;
        double r161092 = r161086 - r161091;
        double r161093 = sqrt(r161092);
        double r161094 = r161085 + r161093;
        double r161095 = r161094 / r161089;
        return r161095;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r161096 = -0.5;
        double r161097 = c;
        double r161098 = b;
        double r161099 = r161097 / r161098;
        double r161100 = r161096 * r161099;
        return r161100;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Simplified 52.3

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

3. Taylor expanded around inf 6.4

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

4. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.93038e-32 a 2.02824e+31) (< 4.93038e-32 b 2.02824e+31) (< 4.93038e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))