Average Error: 28.7 → 0.4
Time: 1.7m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{-c} + \frac{b}{-c}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.7

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

  2. Applied simplify28.7

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

  4. Applied flip--28.8

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

  5. Applied simplify0.6

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

  7. Applied clear-num0.6

    \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\frac{\left(-a\right) \cdot \left(c \cdot 3\right)}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} + b}}}}\]

  8. Applied simplify0.5

    \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{c} \cdot \frac{a}{-a}\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} + b\right)}}\]
  9. Using strategy rm

  10. Applied distribute-lft-in0.4

    \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{c} \cdot \frac{a}{-a}\right) \cdot \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 3} + \left(\frac{1}{c} \cdot \frac{a}{-a}\right) \cdot b}}\]

  11. Applied simplify0.4

    \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{-c}} + \left(\frac{1}{c} \cdot \frac{a}{-a}\right) \cdot b}\]

  12. Applied simplify0.4

    \[\leadsto \frac{1}{\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)}}{-c} + \color{blue}{\frac{b}{-c}}}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018208 
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))