Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 24.7s

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 r1673495 = b;
        double r1673496 = -r1673495;
        double r1673497 = r1673495 * r1673495;
        double r1673498 = 3.0;
        double r1673499 = a;
        double r1673500 = r1673498 * r1673499;
        double r1673501 = c;
        double r1673502 = r1673500 * r1673501;
        double r1673503 = r1673497 - r1673502;
        double r1673504 = sqrt(r1673503);
        double r1673505 = r1673496 + r1673504;
        double r1673506 = r1673505 / r1673500;
        return r1673506;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1673507 = c;
        double r1673508 = b;
        double r1673509 = r1673507 / r1673508;
        double r1673510 = -0.5;
        double r1673511 = r1673509 * r1673510;
        return r1673511;
}

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)))