Error

Target

Derivation

1. Split input into 4 regimes

2. if b < -1.2475815988719495e+108

   1. Initial program 46.0

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

   2. Simplified46.0

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

   3. Taylor expanded around -inf 3.5

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

   if -1.2475815988719495e+108 < b < 1.0773379242733233e-296

   1. Initial program 9.7

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

   2. Simplified9.7

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

   4. Applied *-un-lft-identity9.7

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

   5. Applied div-inv9.7

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

   6. Applied times-frac9.8

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

   7. Simplified9.8

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

   8. Simplified9.8

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

   10. Applied *-commutative9.8

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

   if 1.0773379242733233e-296 < b < 5.1457556548524475e+64

   1. Initial program 31.5

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

   2. Simplified31.5

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

   4. Applied *-un-lft-identity31.5

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

   5. Applied div-inv31.5

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

   6. Applied times-frac31.5

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

   7. Simplified31.5

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

   8. Simplified31.5

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

   10. Applied *-commutative31.5

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

   12. Applied flip--31.6

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

   13. Applied *-un-lft-identity31.6

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

   14. Applied associate-/r*31.6

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

   15. Applied frac-times36.2

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

   16. Simplified22.5

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

   18. Applied *-un-lft-identity22.5

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

   19. Applied associate-*l*22.5

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

   20. Applied *-commutative22.5

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

   21. Applied times-frac22.5

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

   22. Simplified22.5

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

   23. Simplified10.0

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

   if 5.1457556548524475e+64 < b

   1. Initial program 57.4

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

   2. Simplified57.4

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

   3. Taylor expanded around inf 3.3

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

   4. Simplified3.3

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

4. Final simplification7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.2475815988719495 \cdot 10^{+108}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.0773379242733233 \cdot 10^{-296}:\\ \;\;\;\;\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*} - b\right)\\ \mathbf{elif}\;b \le 5.1457556548524475 \cdot 10^{+64}:\\ \;\;\;\;-2 \cdot \frac{c}{\sqrt{(-4 \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019088 +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)))