Error

Target

Derivation

1. Split input into 4 regimes

2. if b < -1.5373800263892595e+87

   1. Initial program 42.6

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

   2. Taylor expanded around -inf 3.5

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

   if -1.5373800263892595e+87 < b < -4.586451634118627e-185

   1. Initial program 6.7

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

   3. Applied *-un-lft-identity6.7

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

   4. Applied *-un-lft-identity6.7

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

   5. Applied distribute-lft-out6.7

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

   6. Applied associate-/l*6.8

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

   if -4.586451634118627e-185 < b < 2.7893378365974675e+143

   1. Initial program 31.4

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

   3. Applied flip-+31.6

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

   4. Applied associate-/l/36.0

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

   5. Simplified20.4

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

   7. Applied times-frac14.8

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

   8. Simplified9.7

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

   if 2.7893378365974675e+143 < b

   1. Initial program 61.7

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

   2. Taylor expanded around inf 1.2

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

   3. Simplified1.2

      \[\leadsto \color{blue}{-\frac{c}{b}}\]
3. Recombined 4 regimes into one program.

4. Final simplification6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.5373800263892595 \cdot 10^{+87}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le -4.586451634118627 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)} + \left(-b\right)}}\\ \mathbf{elif}\;b \le 2.7893378365974675 \cdot 10^{+143}:\\ \;\;\;\;\frac{4}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(c \cdot a\right)}} \cdot \left(c \cdot \frac{1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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)))