Average Error: 28.4 → 0.4
Time: 45.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot -4}{\frac{\left(\sqrt{\frac{\left(b \cdot b\right) \cdot {b}^{4} - {\left(\left(a \cdot c\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)\right)}} + b\right) \cdot \left(2 \cdot a\right)}{a}}\]

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. Simplified0.5

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

  8. Applied associate-/l*0.3

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

  10. Applied flip3--0.4

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

  12. Applied add-cbrt-cube0.6

    \[\leadsto \frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{{\color{blue}{\left(\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}\right)}}^{3} - {\left(\left(c \cdot a\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)}} + b\right)}{a}}\]

  13. Applied rem-cube-cbrt0.4

    \[\leadsto \frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{\color{blue}{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)} - {\left(\left(c \cdot a\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)}} + b\right)}{a}}\]

  14. Simplified0.4

    \[\leadsto \frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{\color{blue}{{b}^{4}} \cdot \left(b \cdot b\right) - {\left(\left(c \cdot a\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(c \cdot a\right) \cdot 4\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right)\right)}} + b\right)}{a}}\]

  15. Final simplification0.4

    \[\leadsto \frac{c \cdot -4}{\frac{\left(\sqrt{\frac{\left(b \cdot b\right) \cdot {b}^{4} - {\left(\left(a \cdot c\right) \cdot 4\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(a \cdot c\right) \cdot 4\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(\left(a \cdot c\right) \cdot 4\right)\right)}} + b\right) \cdot \left(2 \cdot a\right)}{a}}\]

Runtime

Time bar (total: 45.3s)Debug logProfile

herbie shell --seed 2018349 
(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)))