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Target

Derivation

1. Split input into 4 regimes

2. if b < -4.9529633933266e-08 or -1.432147794020809e-104 < b < -5.51185453096989e-127

   1. Initial program 53.5

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

   2. Taylor expanded around -inf 7.8

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

   3. Simplified7.8

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

   if -4.9529633933266e-08 < b < -1.432147794020809e-104

   1. Initial program 34.9

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

   3. Applied flip--34.9

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

   4. Applied associate-/l/38.7

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

   5. Simplified21.5

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

   if -5.51185453096989e-127 < b < 8.046684077965657e+99

   1. Initial program 11.0

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

   3. Applied *-un-lft-identity11.0

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

   4. Applied associate-/l*11.1

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

   if 8.046684077965657e+99 < b

   1. Initial program 44.6

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

   3. Applied *-un-lft-identity44.6

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

   4. Applied associate-/l*44.7

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

   6. Applied div-inv44.7

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

   7. Applied associate-/r*44.7

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

   8. Taylor expanded around 0 3.6

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

   9. Simplified3.6

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

4. Final simplification9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.9529633933266 \cdot 10^{-08}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -1.432147794020809 \cdot 10^{-104}:\\ \;\;\;\;\frac{\left(4 \cdot c\right) \cdot a}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}\right)}\\ \mathbf{elif}\;b \le -5.51185453096989 \cdot 10^{-127}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 8.046684077965657 \cdot 10^{+99}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(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

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