Average Error: 44.1 → 44.1
Time: 44.2s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\frac{\sqrt{b \cdot b - \sqrt[3]{{\left(a \cdot \left(3 \cdot c\right)\right)}^{3}}} - b}{3 \cdot a}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

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Results

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Derivation

  1. Initial program 44.1

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

  2. Applied simplify44.1

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

  4. Applied add-cbrt-cube44.1

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

  5. Applied simplify44.1

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

  7. Applied add-cbrt-cube44.1

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

  8. Applied simplify44.1

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

Runtime

Time bar (total: 44.2s)Debug logProfile

herbie shell --seed 2018198 
(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)))