Quadratic roots, narrow range

Average Error: 28.5 → 0.3

Time: 36.8min

Precision: binary64

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

\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]

Math

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 28.5

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

2. Simplified 28.5

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

4. Applied flip-- 28.6

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

5. Simplified 0.5

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

7. Applied *-un-lft-identity 0.5

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

8. Applied distribute-lft-neg-in 0.5

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

9. Applied times-frac 0.3

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

10. Simplified 0.3

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

12. Applied distribute-lft-neg-out 0.3

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

13. Applied distribute-frac-neg 0.3

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

14. Applied distribute-frac-neg 0.3

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

15. Simplified 0.3

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

16. Final simplification 0.3

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

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Quadratic roots, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))