Average Error: 28.4 → 0.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\left(-c\right) \cdot \frac{4}{2}}{\frac{\sqrt{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \left(\left(a \cdot 4\right) \cdot c\right) \cdot \left(\left(a \cdot 4\right) \cdot c\right)}}{\sqrt{\left(a \cdot 4\right) \cdot c + b \cdot b}} + b}\]

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 28.4

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

  2. Initial simplification28.4

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

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

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

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

  9. Applied distribute-frac-neg0.4

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

  10. Simplified0.3

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

  12. Applied flip--0.3

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

  13. Applied sqrt-div0.3

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

  14. Final simplification0.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018230 
(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)))