Average Error: 44.0 → 0.2
Time: 2.0m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4}{2} \cdot \left(-c\right)}{b + \sqrt{\sqrt[3]{\left(\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right) \cdot \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)\right) \cdot \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Results

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Derivation

  1. Initial program 44.0

    \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]

  2. Initial simplification44.0

    \[\leadsto \frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b}{2 \cdot a}\]
  3. Using strategy rm

  4. Applied flip--44.0

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}}}{2 \cdot a}\]

  5. Applied associate-/l/44.0

    \[\leadsto \color{blue}{\frac{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} \cdot \sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} - b \cdot b}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}\]

  6. Simplified0.4

    \[\leadsto \frac{\color{blue}{\left(-4\right) \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]
  7. Using strategy rm

  8. Applied distribute-lft-neg-out0.4

    \[\leadsto \frac{\color{blue}{-4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}\]

  9. Applied distribute-frac-neg0.4

    \[\leadsto \color{blue}{-\frac{4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b\right)}}\]

  10. Simplified0.2

    \[\leadsto -\color{blue}{\frac{c \cdot \frac{4}{\frac{2}{1}}}{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + b}}\]
  11. Using strategy rm

  12. Applied add-cbrt-cube0.2

    \[\leadsto -\frac{c \cdot \frac{4}{\frac{2}{1}}}{\sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b - \left(4 \cdot a\right) \cdot c\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)\right) \cdot \left(b \cdot b - \left(4 \cdot a\right) \cdot c\right)}}} + b}\]

  13. Final simplification0.2

    \[\leadsto \frac{\frac{4}{2} \cdot \left(-c\right)}{b + \sqrt{\sqrt[3]{\left(\left(b \cdot b - c \cdot \left(4 \cdot a\right)\right) \cdot \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)\right) \cdot \left(b \cdot b - c \cdot \left(4 \cdot a\right)\right)}}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018227 
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))