Cubic critical, medium range

Average Error: 43.7 → 12.1

Time: 6.0s

Precision: 64

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

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

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 r78379 = b;
        double r78380 = -r78379;
        double r78381 = r78379 * r78379;
        double r78382 = 3.0;
        double r78383 = a;
        double r78384 = r78382 * r78383;
        double r78385 = c;
        double r78386 = r78384 * r78385;
        double r78387 = r78381 - r78386;
        double r78388 = sqrt(r78387);
        double r78389 = r78380 + r78388;
        double r78390 = r78389 / r78384;
        return r78390;
}

↓

double f(double __attribute__((unused)) a, double b, double c) {
        double r78391 = -0.5;
        double r78392 = c;
        double r78393 = b;
        double r78394 = r78392 / r78393;
        double r78395 = r78391 * r78394;
        return r78395;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

2. Simplified 43.7

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

3. Taylor expanded around inf 12.1

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

4. Final simplification 12.1

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

Reproduce

herbie shell --seed 2020046 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.11022e-16 a 9.0072e+15) (< 1.11022e-16 b 9.0072e+15) (< 1.11022e-16 c 9.0072e+15))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))