Average Error: 43.9 → 0.3
Time: 1.6m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c}{\frac{-1}{2}} \cdot \frac{1}{b + \sqrt{b \cdot b + \left(-4 \cdot c\right) \cdot a}}\]

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 43.9

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

  2. Initial simplification43.9

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

  4. Applied flip--43.9

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

  5. Applied associate-/l/43.9

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

  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*0.2

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

  10. Applied *-un-lft-identity0.2

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

  11. Applied times-frac0.2

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

  12. Applied times-frac0.3

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

  13. Simplified0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018251 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :pre (and (< 1.1102230246251565e-16 a 9007199254740992.0) (< 1.1102230246251565e-16 b 9007199254740992.0) (< 1.1102230246251565e-16 c 9007199254740992.0))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))