Average Error: 43.7 → 0.5
Time: 45.7s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(c \cdot a\right) \cdot -3}{\left(\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b\right) \cdot \left(3 \cdot a\right)}\]

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 43.7

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

  2. Initial simplification43.7

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

  4. Applied flip--43.7

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

  5. Applied associate-/l/43.7

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

  6. Simplified0.4

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

  8. Applied associate-*r*0.6

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

  9. Taylor expanded around inf 0.5

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

  10. Final simplification0.5

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

Runtime

Time bar (total: 45.7s)Debug logProfile

herbie shell --seed 2018242 
(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)))