Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 33.9s

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 r3660309 = b;
        double r3660310 = -r3660309;
        double r3660311 = r3660309 * r3660309;
        double r3660312 = 3.0;
        double r3660313 = a;
        double r3660314 = r3660312 * r3660313;
        double r3660315 = c;
        double r3660316 = r3660314 * r3660315;
        double r3660317 = r3660311 - r3660316;
        double r3660318 = sqrt(r3660317);
        double r3660319 = r3660310 + r3660318;
        double r3660320 = r3660319 / r3660314;
        return r3660320;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r3660321 = c;
        double r3660322 = b;
        double r3660323 = r3660321 / r3660322;
        double r3660324 = -0.5;
        double r3660325 = r3660323 * r3660324;
        return r3660325;
}

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 2019151 +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)))