Average Error: 28.3 → 0.3
Time: 1.3m
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{4}{2} \cdot \left(-c\right)}{b + \sqrt{\sqrt[3]{\left((\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_* \cdot (\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*\right) \cdot (\left(a \cdot 4\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*}}}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.3

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

  2. Initial simplification28.3

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

  4. Applied flip--28.3

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

  5. Applied associate-/l/28.3

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

  6. Simplified0.4

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

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

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

  9. Applied distribute-frac-neg0.4

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

  10. Simplified0.3

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

  12. Applied add-cbrt-cube0.3

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

  13. Final simplification0.3

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed 2018225 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, 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) (* (* 4 a) c)))) (* 2 a)))