Cubic critical, wide range

Average Error: 52.4 → 6.3

Time: 6.1s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

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

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 r70117 = b;
        double r70118 = -r70117;
        double r70119 = r70117 * r70117;
        double r70120 = 3.0;
        double r70121 = a;
        double r70122 = r70120 * r70121;
        double r70123 = c;
        double r70124 = r70122 * r70123;
        double r70125 = r70119 - r70124;
        double r70126 = sqrt(r70125);
        double r70127 = r70118 + r70126;
        double r70128 = r70127 / r70122;
        return r70128;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r70129 = -0.5;
        double r70130 = c;
        double r70131 = b;
        double r70132 = r70130 / r70131;
        double r70133 = r70129 * r70132;
        return r70133;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.4

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

2. Taylor expanded around inf 6.3

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

3. Final simplification 6.3

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (< 4.9303800000000003e-32 a 2.02824e+31) (< 4.9303800000000003e-32 b 2.02824e+31) (< 4.9303800000000003e-32 c 2.02824e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))