Average Error: 28.5 → 0.3
Time: 46.9s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{-4 \cdot c}{\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{b \cdot b + \left(a \cdot c\right) \cdot 4}} + b}}{2}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Results

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Derivation

  1. Initial program 28.5

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Initial simplification28.5

    \[\leadsto \frac{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{a}}{2}\]
  3. Using strategy rm

  4. Applied flip--28.5

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{a}}{2}\]

  5. Applied associate-/l/28.5

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}}{2}\]

  6. Simplified0.4

    \[\leadsto \frac{\frac{\color{blue}{\left(c \cdot -4\right) \cdot a}}{a \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}{2}\]
  7. Using strategy rm

  8. Applied associate-/r*0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\left(c \cdot -4\right) \cdot a}{a}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{2}\]

  9. Taylor expanded around inf 0.3

    \[\leadsto \frac{\frac{\color{blue}{-4 \cdot c}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}{2}\]
  10. Using strategy rm

  11. Applied flip--0.3

    \[\leadsto \frac{\frac{-4 \cdot c}{\sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)}{b \cdot b + \left(c \cdot a\right) \cdot 4}}} + b}}{2}\]

  12. Final simplification0.3

    \[\leadsto \frac{\frac{-4 \cdot c}{\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)}{b \cdot b + \left(a \cdot c\right) \cdot 4}} + b}}{2}\]

Runtime

Time bar (total: 46.9s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))