Average Error: 43.8 → 43.8
Time: 26.9s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt{b \cdot b - \sqrt[3]{\left(\left(c \cdot a\right) \cdot \left(9 \cdot a\right)\right) \cdot \left(\left(c \cdot a\right) \cdot \left(c \cdot 3\right)\right)}} + \left(-b\right)}{a \cdot 3}\]

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 43.8

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

  3. Applied add-sqr-sqrt43.8

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

  5. Applied add-cbrt-cube43.8

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

  6. Applied add-cbrt-cube43.8

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

  7. Applied cbrt-unprod43.8

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

  8. Simplified43.8

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

  9. Final simplification43.8

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

Runtime

Time bar (total: 26.9s)Debug logProfile

herbie shell --seed 2018250 
(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)))