Average Error: 28.7 → 28.7
Time: 21.4s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \sqrt[3]{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a}\]

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. Using strategy rm

  3. Applied add-cbrt-cube28.7

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

  4. Taylor expanded around inf 28.7

    \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \sqrt[3]{\color{blue}{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)} \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a}\]

  5. Final simplification28.7

    \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \sqrt[3]{\left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a}\]

Runtime

Time bar (total: 21.4s)Debug logProfile

herbie shell --seed 2018216 
(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)))