Average Error: 28.8 → 0.4
Time: 2.2m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-1}{\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} + b}{c}}\]

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

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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. Initial simplification28.8

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

  4. Applied flip--28.8

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

  5. Applied associate-/l/28.8

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

  6. Simplified0.4

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

  8. Applied distribute-lft-neg-out0.4

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

  9. Applied distribute-frac-neg0.4

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

  10. Simplified0.3

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

  12. Applied *-un-lft-identity0.3

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

  13. Applied associate-/l*0.4

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

  14. Final simplification0.4

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018234 
(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)))