Error

Target

Derivation

1. Split input into 2 regimes

2. if b < 1.4625177337647182e-163

   1. Initial program 20.5

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

   2. Initial simplification20.4

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

   4. Applied div-sub20.5

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

   6. Applied div-inv20.5

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

   7. Applied add-sqr-sqrt20.7

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

   8. Applied times-frac20.7

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

   9. Applied prod-diff20.6

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

   10. Simplified20.5

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

   11. Simplified20.5

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

   if 1.4625177337647182e-163 < b

   1. Initial program 48.6

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

   2. Initial simplification48.6

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

   4. Applied flip--48.6

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

   5. Applied associate-/l/50.5

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

   6. Simplified25.7

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

4. Final simplification22.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.4625177337647182 \cdot 10^{-163}:\\ \;\;\;\;(b \cdot \left(\frac{\frac{-1}{2}}{a}\right) + \left(\frac{\sqrt{(a \cdot \left(-4 \cdot c\right) + \left(b \cdot b\right))_*}}{\frac{a}{\frac{1}{2}}}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-4 \cdot a\right) \cdot c}{\left(a \cdot 2\right) \cdot \left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} + b\right)}\\ \end{array}\]

Runtime

Time bar (total: 31.9s)Debug logProfile

herbie shell --seed 2018295 +o rules:numerics
(FPCore (a b c)
  :name "The quadratic formula (r1)"

  :herbie-target
  (if (< b 0) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))