Average Error: 52.6 → 0.3
Time: 48.7s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[c \cdot \frac{-1}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + b}\]

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

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Derivation

  1. Initial program 52.6

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

  2. Initial simplification52.6

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

  4. Applied flip--52.6

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

  5. Applied associate-/l/52.6

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

  6. Simplified0.5

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

  8. Applied associate-/r*0.4

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

  9. Taylor expanded around inf 0.1

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

  10. Simplified0.1

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

  12. Applied div-inv0.3

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

  13. Final simplification0.3

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

Runtime

Time bar (total: 48.7s)Debug logProfile

herbie shell --seed 2018234 
(FPCore (a b c d)
  :name "Cubic critical, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))