Average Error: 28.6 → 0.4
Time: 28.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)}{\left(a \cdot c\right) \cdot 4 + b \cdot b}} + b\right)}{a}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.6

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

  2. Simplified28.6

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

  4. Applied flip--28.6

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

  5. Applied associate-/l/28.6

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

  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-/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 flip--0.4

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

  11. Taylor expanded around 0 0.4

    \[\leadsto \frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - \color{blue}{16 \cdot \left({a}^{2} \cdot {c}^{2}\right)}}{b \cdot b + \left(c \cdot a\right) \cdot 4}} + b\right)}{a}}\]

  12. Final simplification0.4

    \[\leadsto \frac{c \cdot -4}{\frac{\left(2 \cdot a\right) \cdot \left(\sqrt{\frac{\left(b \cdot b\right) \cdot \left(b \cdot b\right) - 16 \cdot \left({a}^{2} \cdot {c}^{2}\right)}{\left(a \cdot c\right) \cdot 4 + b \cdot b}} + b\right)}{a}}\]

Reproduce

herbie shell --seed 2019026 
(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)))