Average Error: 43.8 → 0.3
Time: 39.7s
Precision: 64
Internal Precision: 576
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot -4}{\frac{(\left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\right) + b)_* \cdot a}{a}}}{2}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.8

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

  2. Initial simplification43.8

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

  4. Applied flip--43.9

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

  5. Applied associate-/l/43.9

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

  6. Simplified0.4

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

  8. Applied associate-/l*0.2

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied fma-def0.3

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

  12. Final simplification0.3

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

Runtime

Time bar (total: 39.7s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(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)))