Average Error: 28.4 → 0.3
Time: 44.4s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\frac{-2 \cdot c}{(\left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*}}\right) + b)_*}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 28.4

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

  2. Simplified28.4

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

  4. Applied *-un-lft-identity28.4

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

  5. Applied div-inv28.4

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

  6. Applied times-frac28.4

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

  7. Simplified28.3

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

  8. Simplified28.3

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

  10. Applied flip--28.4

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

  11. Applied associate-*l/28.4

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

  12. Simplified0.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{2}}{a} \cdot (a \cdot \left(c \cdot -4\right) + 0)_*}}{\sqrt{(b \cdot b + \left(\left(a \cdot -4\right) \cdot c\right))_*} + b}\]

  13. Taylor expanded around -inf 0.3

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

  15. Applied add-sqr-sqrt0.3

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

  16. Applied sqrt-prod0.4

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

  17. Applied fma-def0.3

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

  18. Final simplification0.3

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

Reproduce

herbie shell --seed 2019090 +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)))