Error

Target

Derivation

1. Split input into 3 regimes

2. if b < -3.1382247414568033e+125

   1. Initial program 51.4

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

   2. Simplified51.4

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

   3. Taylor expanded around -inf 8.8

      \[\leadsto \frac{\frac{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}{2}}{a}\]

   4. Simplified2.8

      \[\leadsto \frac{\frac{\color{blue}{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}}{2}}{a}\]

   if -3.1382247414568033e+125 < b < 8.111873011657015e-116

   1. Initial program 11.1

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

   2. Simplified11.1

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

   3. Taylor expanded around 0 11.1

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

   4. Simplified11.1

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

   if 8.111873011657015e-116 < b

   1. Initial program 51.7

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

   2. Simplified51.7

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

   4. Applied flip--51.8

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

   5. Applied associate-/l/51.8

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

   6. Simplified24.8

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

   8. Applied *-un-lft-identity24.8

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

   9. Applied associate-/l*24.9

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

   10. Taylor expanded around 0 11.1

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

   11. Simplified11.1

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

4. Final simplification10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.1382247414568033 \cdot 10^{+125}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le 8.111873011657015 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{\sqrt{(\left(c \cdot a\right) \cdot -4 + \left(b \cdot b\right))_*} - b}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019089 +o rules:numerics
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :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)))