Average Error: 28.7 → 0.3
Time: 1.9m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{4}{2} \cdot \frac{c}{\left(-b\right) - \sqrt{\frac{{\left(b \cdot b\right)}^{3} - {\left(\left(c \cdot 4\right) \cdot a\right)}^{3}}{\left(\left(c \cdot a\right) \cdot 4 + b \cdot b\right) \cdot \left(\left(c \cdot a\right) \cdot 4\right) + \left(b \cdot b\right) \cdot \left(b \cdot b\right)}}}\]

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 28.7

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

  3. Applied flip-+28.7

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

  4. Applied simplify0.5

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

  6. Applied *-un-lft-identity0.5

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

  7. Applied times-frac0.5

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

  8. Applied times-frac0.5

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

  9. Applied simplify0.5

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

  10. Applied simplify0.3

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

  12. Applied flip3--0.3

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

  13. Applied simplify0.3

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2019053 
(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)))