Quadratic roots, wide range

Average Error: 52.5 → 0.1

Time: 4.7s

Precision: binary64

\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]

Math

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 52.5

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

2. Simplified 52.5

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

4. Applied flip-- 52.5

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

5. Simplified 52.2

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

6. Simplified 52.2

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

8. Applied div-inv 52.2

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

9. Applied times-frac 52.2

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

10. Simplified 0.4

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

11. Simplified 0.4

    \[\leadsto \frac{4 \cdot \left(a \cdot \left(-c\right)\right)}{a} \cdot \color{blue}{\frac{1}{\left(b + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right) \cdot 2}}\]
12. Using strategy rm

13. Applied distribute-rgt-neg-out 0.4

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

14. Applied distribute-rgt-neg-out 0.4

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

15. Applied distribute-frac-neg 0.4

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

16. Applied distribute-lft-neg-out 0.4

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

17. Simplified 0.1

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

18. Final simplification 0.1

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

Reproduce

herbie shell --seed 2020184 
(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))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))