Cubic critical, wide range

Average Error: 52.3 → 6.4

Time: 37.3s

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 r3873859 = b;
        double r3873860 = -r3873859;
        double r3873861 = r3873859 * r3873859;
        double r3873862 = 3.0;
        double r3873863 = a;
        double r3873864 = r3873862 * r3873863;
        double r3873865 = c;
        double r3873866 = r3873864 * r3873865;
        double r3873867 = r3873861 - r3873866;
        double r3873868 = sqrt(r3873867);
        double r3873869 = r3873860 + r3873868;
        double r3873870 = r3873869 / r3873864;
        return r3873870;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r3873871 = c;
        double r3873872 = b;
        double r3873873 = r3873871 / r3873872;
        double r3873874 = -0.5;
        double r3873875 = r3873873 * r3873874;
        return r3873875;
}

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. Taylor expanded around inf 6.4

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

3. Final simplification 6.4

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

Reproduce

herbie shell --seed 2019142 +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)))