Average Error: 28.6 → 0.3
Time: 23.3s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{c}{b + \sqrt{\sqrt[3]{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot \left((\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_* \cdot (\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*\right)}}} \cdot -2\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.6

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

  2. Simplified28.6

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

  4. Applied flip--28.6

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

  5. Applied associate-/l/28.6

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

  6. Simplified0.4

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

  8. Applied times-frac0.3

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

  9. Simplified0.3

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

  11. Applied add-cbrt-cube0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019026 +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)))