Average Error: 43.7 → 43.2
Time: 28.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\left(\sqrt[3]{\sqrt[3]{\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}} \cdot \sqrt[3]{\frac{\frac{(\left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(c \cdot \left(a \cdot -4\right) + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{2}}{a}}\right)\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Initial program 43.7

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

  2. Simplified43.7

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

  4. Applied add-sqr-sqrt43.8

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

  5. Applied fma-neg43.2

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

  7. Applied add-cube-cbrt43.2

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

  9. Applied add-cube-cbrt43.2

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

  10. Final simplification43.2

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

Reproduce

herbie shell --seed 2019068 +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)))