Average Error: 28.8 → 0.5
Time: 4.5s
Precision: binary64
\[1.05367121277235087 \cdot 10^{-8} \lt a \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt b \lt 94906265.6242515594 \land 1.05367121277235087 \cdot 10^{-8} \lt c \lt 94906265.6242515594\]
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{c \cdot \frac{a}{\left(-b\right) - \sqrt{\sqrt[3]{{\left(b \cdot b - 3 \cdot \left(c \cdot a\right)\right)}^{3}}}}}{a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.8

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

  3. Applied flip-+28.8

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

  4. Simplified0.6

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

  5. Simplified0.6

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

  7. Applied associate-/r*0.6

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

  8. Simplified0.4

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

  10. Applied add-cbrt-cube0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2020181 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (neg b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))