Quadratic roots, medium range

Average Error: 43.6 → 0.2

Time: 49.8min

Precision: binary64

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

\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]

Math

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.6

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

2. Simplified 43.6

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

4. Applied flip-- 43.6

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

5. Simplified 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied *-un-lft-identity 0.4

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

9. Applied *-un-lft-identity 0.4

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

10. Applied distribute-lft-neg-in 0.4

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

11. Applied times-frac 0.2

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

12. Applied times-frac 0.2

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

13. Applied times-frac 0.2

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

14. Simplified 0.2

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

15. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))