Average Error: 52.7 → 0.1
Time: 38.7s
Precision: 64
Internal Precision: 832
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{\frac{c \cdot -4}{(\left(\sqrt{\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(a \cdot \left(c \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + b)_*}}{2}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 52.7

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

  2. Initial simplification52.7

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

  4. Applied flip--52.8

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

  5. Applied associate-/l/52.8

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

  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 *-un-lft-identity0.2

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

  11. Applied associate-/r*0.2

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

  12. Simplified0.1

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

  14. Applied add-sqr-sqrt0.1

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

  15. Applied sqrt-prod0.3

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

  16. Applied fma-def0.1

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

  17. Final simplification0.1

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

Runtime

Time bar (total: 38.7s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, wide range"
  :pre (and (< 4.930380657631324e-32 a 2.028240960365167e+31) (< 4.930380657631324e-32 b 2.028240960365167e+31) (< 4.930380657631324e-32 c 2.028240960365167e+31))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))