Cubic critical, wide range

Average Error: 52.8 → 6.0

Time: 22.6s

Precision: 64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

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 r76343 = b;
        double r76344 = -r76343;
        double r76345 = r76343 * r76343;
        double r76346 = 3.0;
        double r76347 = a;
        double r76348 = r76346 * r76347;
        double r76349 = c;
        double r76350 = r76348 * r76349;
        double r76351 = r76345 - r76350;
        double r76352 = sqrt(r76351);
        double r76353 = r76344 + r76352;
        double r76354 = r76353 / r76348;
        return r76354;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r76355 = -0.5;
        double r76356 = c;
        double r76357 = b;
        double r76358 = r76356 / r76357;
        double r76359 = r76355 * r76358;
        return r76359;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.8

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

2. Simplified 52.8

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

3. Taylor expanded around inf 6.0

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

4. Final simplification 6.0

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

Reproduce

herbie shell --seed 2019199 +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) (* (* 3.0 a) c)))) (* 3.0 a)))