Error

Target

Derivation

1. Split input into 3 regimes

2. if b < 2.2808806815259953e-214

   1. Initial program 20.7

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

   2. Initial simplification20.6

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

   4. Applied clear-num20.8

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

   6. Applied div-inv20.8

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

   7. Applied associate-/r*20.8

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

   9. Applied associate-/r/20.8

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

   if 2.2808806815259953e-214 < b < 6.544733698813264e+98

   1. Initial program 34.4

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

   2. Initial simplification34.4

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

   4. Applied clear-num34.4

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

   6. Applied div-inv34.4

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

   7. Applied associate-/r*34.4

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

   9. Applied associate-/r/34.4

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

   11. Applied flip--34.5

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

   12. Applied frac-times34.5

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

   13. Simplified15.6

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

   14. Simplified15.5

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

   if 6.544733698813264e+98 < b

   1. Initial program 59.1

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

   2. Initial simplification59.1

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

   4. Applied clear-num59.2

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

   5. Taylor expanded around 0 3.6

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

   6. Simplified3.6

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

4. Final simplification15.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 2.2808806815259953 \cdot 10^{-214}:\\ \;\;\;\;\left(\sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*} - b\right) \cdot \frac{1}{2 \cdot a}\\ \mathbf{elif}\;b \le 6.544733698813264 \cdot 10^{+98}:\\ \;\;\;\;\frac{\left(\left(a \cdot c\right) \cdot -4\right) \cdot \frac{\frac{1}{2}}{a}}{b + \sqrt{(-4 \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-b}{c}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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)))

Details

Time bar (total: 26.0s)Debug log