Average Error: 44.0 → 0.5
Time: 5.6s
Precision: binary64
\[1.11022 \cdot 10^{-16} \lt a \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt b \lt 9.0072 \cdot 10^{15} \land 1.11022 \cdot 10^{-16} \lt c \lt 9.0072 \cdot 10^{15}\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\left(3 \cdot a\right) \cdot \frac{\frac{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}}{c}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 44.0

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

  3. Applied flip-+44.0

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

  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 clear-num0.5

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

  9. Simplified0.5

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

  11. Applied clear-num0.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2020171 
(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)))