Cubic critical, medium range

Average Error: 43.6 → 12.2

Time: 4.8s

Precision: binary64

\[1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992 \land 1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\]

Math

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

↓

\[\frac{-1.5}{\frac{3}{\frac{c}{b}}}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

↓

(FPCore (a b c) :precision binary64 (/ -1.5 (/ 3.0 (/ c b))))

C

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((3.0 * a) * c))) / (3.0 * a);
}

↓

double code(double a, double b, double c) {
	return -1.5 / (3.0 / (c / b));
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.6

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

2. Simplified 43.6

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

3. Taylor expanded around inf 12.4

   \[\leadsto \frac{\color{blue}{-1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a}\]
4. Using strategy rm

5. Applied associate-/l*_binary64 12.4

   \[\leadsto \color{blue}{\frac{-1.5}{\frac{3 \cdot a}{\frac{a \cdot c}{b}}}}\]

6. Simplified 12.2

   \[\leadsto \frac{-1.5}{\color{blue}{\frac{3}{\frac{c}{b}}}}\]

7. Final simplification 12.2

   \[\leadsto \frac{-1.5}{\frac{3}{\frac{c}{b}}}\]

Reproduce

herbie shell --seed 2020224 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))