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Target

Derivation

1. Split input into 3 regimes

2. if b < -1.6339360070887115e-105

   1. Initial program 51.2

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

   2. Initial simplification51.2

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

   4. Applied div-inv51.2

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

   5. Taylor expanded around -inf 10.6

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

   6. Simplified10.6

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

   if -1.6339360070887115e-105 < b < 8.694455854141842e+139

   1. Initial program 12.0

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

   2. Initial simplification12.0

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

   4. Applied div-inv12.1

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

   6. Applied pow112.1

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

   7. Applied pow112.1

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

   8. Applied pow-prod-down12.1

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

   9. Simplified12.0

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

   if 8.694455854141842e+139 < b

   1. Initial program 55.9

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

   2. Initial simplification55.9

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

   3. Taylor expanded around inf 2.1

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

   4. Simplified2.1

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

4. Final simplification10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6339360070887115 \cdot 10^{-105}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le 8.694455854141842 \cdot 10^{+139}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\left(c \cdot -4\right) \cdot a + b \cdot b}}{\frac{a}{\frac{1}{2}}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018251 
(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)))