Average Error: 28.2 → 0.4
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{1}{{\left(\frac{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 3\right) \cdot a}}{c}\right)}^{1}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 28.2

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

  3. Applied flip-+28.2

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

  4. Applied simplify0.6

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

  6. Applied clear-num0.6

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

  7. Applied simplify0.5

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

  9. Applied pow10.5

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

  10. Applied pow10.5

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

  11. Applied pow-prod-down0.5

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

  12. Applied simplify0.4

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018296 
(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)))