Average Error: 28.3 → 0.4
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{1}{\frac{\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(c \cdot a\right)\right) \cdot \left(4 \cdot \left(c \cdot a\right)\right)}{4 \cdot \left(c \cdot a\right) + b \cdot b}} + b}{\frac{\left(c \cdot -4\right) \cdot a}{a \cdot 2}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.3

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

  2. Initial simplification28.3

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

  4. Applied flip--28.4

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

  5. Applied associate-/l/28.4

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

  6. Simplified0.4

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

  8. Applied associate-/r*0.3

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

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

    \[\leadsto \frac{\color{blue}{1 \cdot \frac{\left(c \cdot -4\right) \cdot a}{2 \cdot a}}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

  11. Applied associate-/l*0.4

    \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}{\frac{\left(c \cdot -4\right) \cdot a}{2 \cdot a}}}}\]
  12. Using strategy rm

  13. Applied flip--0.4

    \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)}{b \cdot b + \left(c \cdot a\right) \cdot 4}}} + b}{\frac{\left(c \cdot -4\right) \cdot a}{2 \cdot a}}}\]

  14. Final simplification0.4

    \[\leadsto \frac{1}{\frac{\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(4 \cdot \left(c \cdot a\right)\right) \cdot \left(4 \cdot \left(c \cdot a\right)\right)}{4 \cdot \left(c \cdot a\right) + b \cdot b}} + b}{\frac{\left(c \cdot -4\right) \cdot a}{a \cdot 2}}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018252 
(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)))