Average Error: 43.6 → 0.2
Time: 31.6s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{-c}{\sqrt{\frac{\left(\left(a \cdot c\right) \cdot -3\right) \cdot \left(\left(a \cdot c\right) \cdot -3\right) - \left(b \cdot b\right) \cdot \left(b \cdot b\right)}{\left(a \cdot c\right) \cdot -3 - b \cdot b}} + b}\]

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 43.6

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

  2. Initial simplification43.6

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

  4. Applied flip--43.6

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

  5. Applied associate-/l/43.6

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

  6. Simplified0.6

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

  8. Applied associate-/r*0.5

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

  9. Taylor expanded around inf 0.2

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

  10. Simplified0.2

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

  12. Applied flip-+0.2

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

  13. Final simplification0.2

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

Runtime

Time bar (total: 31.6s)Debug logProfile

herbie shell --seed 2018304 
(FPCore (a b c d)
  :name "Cubic critical, 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) (* (* 3 a) c)))) (* 3 a)))