Average Error: 43.8 → 0.4
Time: 56.7s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-1}{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} + b} \cdot \left(a \cdot \frac{c}{a}\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Results

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Derivation

  1. Initial program 43.8

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

  2. Initial simplification43.8

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

  4. Applied flip--43.8

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

  5. Applied associate-/l/43.8

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

  6. Simplified0.6

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

  8. Applied times-frac0.5

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

  9. Simplified0.4

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

  11. Applied div-inv0.5

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Runtime

Time bar (total: 56.7s)Debug logProfile

herbie shell --seed 2018234 
(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)))