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Target

Derivation

1. Split input into 4 regimes

2. if b < -2.1317624771383882e+58

   1. Initial program 37.4

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

   2. Simplified37.4

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

   3. Taylor expanded around -inf 5.4

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

   if -2.1317624771383882e+58 < b < 1.080192248888389e-171

   1. Initial program 11.7

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

   2. Simplified11.7

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

   4. Applied div-sub11.7

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

   if 1.080192248888389e-171 < b < 1.3660877782595927e+23

   1. Initial program 32.3

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

   2. Simplified32.3

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

   4. Applied flip--32.4

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

   5. Applied associate-/l/37.4

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

   6. Simplified20.4

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

   8. Applied associate-/r*14.6

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

   9. Taylor expanded around 0 5.8

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

   11. Applied div-inv6.0

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

   if 1.3660877782595927e+23 < b

   1. Initial program 55.5

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

   2. Simplified55.5

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

   3. Taylor expanded around inf 4.4

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

   4. Simplified4.4

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

4. Final simplification7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.1317624771383882 \cdot 10^{+58}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 1.080192248888389 \cdot 10^{-171}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c}}{2 \cdot a} - \frac{b}{2 \cdot a}\\ \mathbf{elif}\;b \le 1.3660877782595927 \cdot 10^{+23}:\\ \;\;\;\;\left(c \cdot -2\right) \cdot \frac{1}{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019022 
(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)))

Details

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