Quadratic roots, wide range

Average Error: 52.4 → 0.2

Time: 1.4min

Precision: binary64

Cost: 2880

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

Math

\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

↓

\[\frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)\right)}}}}{a \cdot 2}\]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

↓

(FPCore (a b c)
 :precision binary64
 (/
  (*
   (* 4.0 a)
   (/
    c
    (-
     (- b)
     (sqrt
      (/
       (- (pow b 6.0) (pow (* (* 4.0 a) c) 3.0))
       (+ (pow b 4.0) (* 4.0 (* (* a c) (+ (* (* 4.0 a) c) (* b b))))))))))
  (* a 2.0)))

C

double code(double a, double b, double c) {
	return (-b + sqrt((b * b) - ((4.0 * a) * c))) / (2.0 * a);
}

↓

double code(double a, double b, double c) {
	return ((4.0 * a) * (c / (-b - sqrt((pow(b, 6.0) - pow(((4.0 * a) * c), 3.0)) / (pow(b, 4.0) + (4.0 * ((a * c) * (((4.0 * a) * c) + (b * b))))))))) / (a * 2.0);
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Alternative 1
Accuracy	0.2
Cost	4928

\[\frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{{b}^{8} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{2} \cdot \left(\left(\left(4 \cdot a\right) \cdot c + b \cdot b\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)\right)} \cdot \left({b}^{4} - 4 \cdot \left(\left(\left(4 \cdot a\right) \cdot c + b \cdot b\right) \cdot \left(a \cdot c\right)\right)\right)}}}{a \cdot 2}\]

Alternative 2
Accuracy	0.4
Cost	1472

\[\frac{\frac{\left(4 \cdot a\right) \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2}\]

Derivation

1. Initial program 52.4

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
2. Using strategy rm

3. Applied flip-+_binary64_52 52.4

   \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

4. Simplified 0.4

   \[\leadsto \frac{\frac{\color{blue}{\left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]
5. Using strategy rm

6. Applied *-un-lft-identity_binary64_78 0.4

   \[\leadsto \frac{\frac{\left(4 \cdot a\right) \cdot c}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}\right)}}}{2 \cdot a}\]

7. Applied times-frac_binary64_84 0.2

   \[\leadsto \frac{\color{blue}{\frac{4 \cdot a}{1} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}}{2 \cdot a}\]

8. Simplified 0.2

   \[\leadsto \frac{\color{blue}{\left(4 \cdot a\right)} \cdot \frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}}{2 \cdot a}\]

9. Simplified 0.2

   \[\leadsto \frac{\left(4 \cdot a\right) \cdot \color{blue}{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}}{2 \cdot a}\]
10. Using strategy rm

11. Applied flip3--_binary64_82 0.2

    \[\leadsto \frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\color{blue}{\frac{{\left(b \cdot b\right)}^{3} - {\left(c \cdot \left(4 \cdot a\right)\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right)\right)}}}}}{2 \cdot a}\]

12. Simplified 0.2

    \[\leadsto \frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{\color{blue}{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(c \cdot \left(4 \cdot a\right)\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right) + \left(b \cdot b\right) \cdot \left(c \cdot \left(4 \cdot a\right)\right)\right)}}}}{2 \cdot a}\]

13. Simplified 0.2

    \[\leadsto \frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{\color{blue}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}}}}}{2 \cdot a}\]
14. Using strategy rm

15. Applied *-un-lft-identity_binary64_78 0.2

    \[\leadsto \frac{\color{blue}{1 \cdot \left(\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(b \cdot b + \left(4 \cdot a\right) \cdot c\right)\right)}}}\right)}}{2 \cdot a}\]

16. Final simplification 0.2

    \[\leadsto \frac{\left(4 \cdot a\right) \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{b}^{6} - {\left(\left(4 \cdot a\right) \cdot c\right)}^{3}}{{b}^{4} + 4 \cdot \left(\left(a \cdot c\right) \cdot \left(\left(4 \cdot a\right) \cdot c + b \cdot b\right)\right)}}}}{a \cdot 2}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4.0 a) c)))) (* 2.0 a)))