Error

Try it out

Target

Derivation

1. Split input into 4 regimes

2. if b < -4.854031372780893e+121 or -9.565134490438525e-14 < b < -1.4327714371069e-133

   1. Initial program 51.7

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

   2. Taylor expanded around inf 51.7

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

   3. Taylor expanded around -inf 11.5

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

   4. Simplified11.5

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

   if -4.854031372780893e+121 < b < -9.565134490438525e-14

   1. Initial program 46.3

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

   2. Taylor expanded around inf 46.3

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

   4. Applied *-un-lft-identity46.3

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

   5. Applied associate-/l*46.3

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

   7. Applied associate-/r/46.3

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

   9. Applied flip--46.4

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

   10. Applied frac-times48.0

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

   11. Simplified14.1

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

   12. Simplified14.1

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

   if -1.4327714371069e-133 < b < 1.6367105644834092e+113

   1. Initial program 11.5

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

   2. Taylor expanded around inf 11.5

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

   4. Applied *-un-lft-identity11.5

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

   5. Applied associate-/l*11.6

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

   7. Applied associate-/r/11.6

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

   if 1.6367105644834092e+113 < b

   1. Initial program 46.8

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

   2. Taylor expanded around inf 46.8

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

   4. Applied *-un-lft-identity46.8

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

   5. Applied associate-/l*46.8

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

   7. Applied associate-/r/46.8

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

   8. Taylor expanded around 0 2.8

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

   9. Simplified2.8

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

4. Final simplification10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.854031372780893 \cdot 10^{+121}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -9.565134490438525 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(4 \cdot c\right) \cdot a}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b + a \cdot \left(-4 \cdot c\right)} - b\right)}\\ \mathbf{elif}\;b \le -1.4327714371069 \cdot 10^{-133}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 1.6367105644834092 \cdot 10^{+113}:\\ \;\;\;\;\left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (a b c)
  :name "The quadratic formula (r2)"

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

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

Details

Time bar (total: 21.6s)Debug log