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Target

Derivation

1. Split input into 4 regimes

2. if b < -1.3338600865169524e+85

   1. Initial program 42.2

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

   2. Initial simplification42.2

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

   3. Taylor expanded around -inf 3.9

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

   4. Simplified3.9

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

   if -1.3338600865169524e+85 < b < 1.5616348041811878e-106

   1. Initial program 12.6

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

   2. Initial simplification12.6

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

   4. Applied div-inv12.7

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

   if 1.5616348041811878e-106 < b < 2.9849414744995752e+82

   1. Initial program 42.7

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

   2. Initial simplification42.7

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

   4. Applied div-inv42.7

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

   6. Applied flip--42.8

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

   7. Applied associate-*l/42.8

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

   8. Simplified15.9

      \[\leadsto \frac{\color{blue}{\frac{\left(c \cdot a\right) \cdot \left(-4\right) + 0}{2 \cdot a}}}{\sqrt{b \cdot b - \left(c \cdot a\right) \cdot 4} + b}\]

   if 2.9849414744995752e+82 < b

   1. Initial program 57.8

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

   2. Initial simplification57.7

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

   4. Applied div-inv57.7

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

   5. Taylor expanded around inf 3.2

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

   6. Simplified3.2

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

4. Final simplification9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.3338600865169524 \cdot 10^{+85}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 1.5616348041811878 \cdot 10^{-106}:\\ \;\;\;\;\left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} - b\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{elif}\;b \le 2.9849414744995752 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\left(a \cdot c\right) \cdot \left(-4\right)}{a \cdot 2}}{\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + b}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{b}\\ \end{array}\]

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018225 
(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)))