Average Error: 52.7 → 0.2
Time: 5.5s
Precision: binary64
\[4.93038 \cdot 10^{-32} \lt a \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt b \lt 2.02824 \cdot 10^{31} \land 4.93038 \cdot 10^{-32} \lt c \lt 2.02824 \cdot 10^{31}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{3}{\frac{3}{a}} \cdot \frac{\frac{c}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  3. Applied flip-+52.7

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

  4. Simplified0.5

    \[\leadsto \frac{\frac{\color{blue}{0 + 3 \cdot \left(a \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  5. Using strategy rm

  6. Applied associate-*r*0.4

    \[\leadsto \frac{\frac{0 + \color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a}\]
  7. Using strategy rm

  8. Applied *-un-lft-identity0.4

    \[\leadsto \frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{\color{blue}{1 \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}}{3 \cdot a}\]

  9. Applied associate-/r*0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{0 + \left(3 \cdot a\right) \cdot c}{1}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

  10. Simplified0.4

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

  12. Applied *-un-lft-identity0.4

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

  13. Applied times-frac0.2

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

  14. Applied times-frac0.3

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

  15. Simplified0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020162 
(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)))