Error

Derivation

1. Split input into 2 regimes

2. if b < 4.107445960089145e-23

   1. Initial program 22.4

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

   2. Simplified22.4

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

   4. Applied add-sqr-sqrt22.4

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

   5. Applied sqrt-prod22.6

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

   6. Applied fma-neg22.5

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

   if 4.107445960089145e-23 < b

   1. Initial program 54.3

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

   2. Simplified54.3

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

   4. Applied *-un-lft-identity54.3

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

   5. Applied associate-/l*54.3

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

   6. Taylor expanded around 0 7.6

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c}}}\]

   7. Simplified7.6

      \[\leadsto \frac{1}{\color{blue}{\frac{-b}{c}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification17.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 4.107445960089145 \cdot 10^{-23}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(\left(a \cdot -4\right) \cdot c + \left(b \cdot b\right))_*}}\right) + \left(-b\right))_*}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))

Details

Time bar (total: 14.6s)Debug log