Cubic critical, wide range

Average Error: 52.2 → 6.4

Time: 35.6s

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 r4094987 = b;
        double r4094988 = -r4094987;
        double r4094989 = r4094987 * r4094987;
        double r4094990 = 3.0;
        double r4094991 = a;
        double r4094992 = r4094990 * r4094991;
        double r4094993 = c;
        double r4094994 = r4094992 * r4094993;
        double r4094995 = r4094989 - r4094994;
        double r4094996 = sqrt(r4094995);
        double r4094997 = r4094988 + r4094996;
        double r4094998 = r4094997 / r4094992;
        return r4094998;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r4094999 = c;
        double r4095000 = b;
        double r4095001 = r4094999 / r4095000;
        double r4095002 = -0.5;
        double r4095003 = r4095001 * r4095002;
        return r4095003;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.2

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

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 2019143 +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)))