Cubic critical, wide range

Average Error: 52.5 → 6.2

Time: 14.1s

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 r1241016 = b;
        double r1241017 = -r1241016;
        double r1241018 = r1241016 * r1241016;
        double r1241019 = 3.0;
        double r1241020 = a;
        double r1241021 = r1241019 * r1241020;
        double r1241022 = c;
        double r1241023 = r1241021 * r1241022;
        double r1241024 = r1241018 - r1241023;
        double r1241025 = sqrt(r1241024);
        double r1241026 = r1241017 + r1241025;
        double r1241027 = r1241026 / r1241021;
        return r1241027;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1241028 = c;
        double r1241029 = b;
        double r1241030 = r1241028 / r1241029;
        double r1241031 = -0.5;
        double r1241032 = r1241030 * r1241031;
        return r1241032;
}

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. Simplified 52.5

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

3. Taylor expanded around inf 6.2

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

4. Final simplification 6.2

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

Reproduce

herbie shell --seed 2019154 
(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)))