Average Error: 43.4 → 0.2
Time: 38.8s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{\sqrt{b \cdot b + \left(-3 \cdot c\right) \cdot a} + b}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 43.4

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

  2. Initial simplification43.4

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

  4. Applied flip--43.4

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

  5. Applied associate-/l/43.4

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

  6. Simplified0.5

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

  8. Applied distribute-lft-in0.5

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

  10. Applied *-un-lft-identity0.5

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

  11. Applied times-frac0.5

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

  12. Simplified0.5

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

  13. Simplified0.5

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

  15. Applied pow10.5

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

  16. Applied pow10.5

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

  17. Applied pow-prod-down0.5

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

  18. Simplified0.2

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

  19. Final simplification0.2

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

Runtime

Time bar (total: 38.8s)Debug logProfile

herbie shell --seed 2018258 
(FPCore (a b c d)
  :name "Cubic critical, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))