Error

Derivation

1. Initial program 44.0

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

2. Simplified44.0

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

4. Applied add-sqr-sqrt44.0

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

5. Applied sqrt-prod44.0

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

6. Applied fma-neg43.4

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

8. Applied add-cbrt-cube43.4

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

10. Applied *-commutative43.4

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

11. Final simplification43.4

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

Reproduce

herbie shell --seed 2019100 +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)))