Average Error: 28.6 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{b + \sqrt{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}} \cdot \sqrt{\sqrt{(\left(a \cdot 3\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Derivation

  1. Initial program 28.6

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

  2. Initial simplification28.5

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

  4. Applied flip--28.6

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

  5. Applied associate-/l/28.6

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

  6. Simplified0.5

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

  8. Applied distribute-lft-neg-out0.5

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

  9. Applied distribute-frac-neg0.5

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

  10. Simplified0.3

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

  12. Applied add-sqr-sqrt0.3

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

  13. Applied sqrt-prod0.4

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

  14. Final simplification0.4

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018227 +o rules:numerics
(FPCore (a b c d)
  :name "Cubic critical, narrow range"
  :pre (and (< 1.0536712127723509e-08 a 94906265.62425156) (< 1.0536712127723509e-08 b 94906265.62425156) (< 1.0536712127723509e-08 c 94906265.62425156))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))