Cubic critical, medium range

Average Error: 43.7 → 0.2

Time: 8.5min

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

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

↓

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

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

1. Initial program 43.7

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

2. Simplified 43.7

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

4. Applied flip-- 43.7

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

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 distribute-frac-neg 0.4

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

8. Applied distribute-frac-neg 0.4

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

9. Simplified 0.2

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

10. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))