Cubic critical, wide range

Average Error: 52.6 → 0.2

Time: 16.4min

Precision: binary64

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

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

Math

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

↓

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

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. Using strategy rm

4. Applied flip-- 52.6

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

5. Simplified 0.4

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

7. Applied *-un-lft-identity 0.4

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

8. Applied distribute-lft-neg-in 0.4

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

9. Applied times-frac 0.2

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

10. Applied times-frac 0.3

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

11. Simplified 0.2

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

12. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020181 
(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))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))