Cubic critical, wide range

Average Error: 52.6 → 6.2

Time: 5.3s

Precision: binary64

\[4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31} \land 4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\]

Math

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

↓

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

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 (((double) (((double) -(b)) + ((double) sqrt(((double) (((double) (b * b)) - ((double) (((double) (3.0 * a)) * c)))))))) / ((double) (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 52.6

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

2. Simplified 52.6

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

3. Taylor expanded around inf 6.5

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

5. Applied associate-/l*_binary64 6.5

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

6. Simplified 6.2

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

7. Final simplification 6.2

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

Reproduce

herbie shell --seed 2020210 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :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) (* (* 3.0 a) c)))) (* 3.0 a)))