Cubic critical, wide range

Average Error: 52.3 → 6.3

Time: 15.0s

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 r1158931 = b;
        double r1158932 = -r1158931;
        double r1158933 = r1158931 * r1158931;
        double r1158934 = 3.0;
        double r1158935 = a;
        double r1158936 = r1158934 * r1158935;
        double r1158937 = c;
        double r1158938 = r1158936 * r1158937;
        double r1158939 = r1158933 - r1158938;
        double r1158940 = sqrt(r1158939);
        double r1158941 = r1158932 + r1158940;
        double r1158942 = r1158941 / r1158936;
        return r1158942;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r1158943 = c;
        double r1158944 = b;
        double r1158945 = r1158943 / r1158944;
        double r1158946 = -0.5;
        double r1158947 = r1158945 * r1158946;
        return r1158947;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.3

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

2. Taylor expanded around inf 6.3

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

3. Final simplification 6.3

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

Reproduce

herbie shell --seed 2019153 +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)))