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Target

Derivation

1. Split input into 4 regimes

2. if b < -3.9662470494954583e+148

   1. Initial program 58.7

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

   2. Simplified58.7

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

   3. Taylor expanded around -inf 2.0

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

   if -3.9662470494954583e+148 < b < 2.2500462861079674e-93

   1. Initial program 11.4

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

   2. Simplified11.4

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

   if 2.2500462861079674e-93 < b < 1.6889965443539738e+108

   1. Initial program 42.7

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

   2. Simplified42.7

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

   4. Applied flip--42.7

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

   5. Applied associate-/l/45.2

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

   6. Simplified16.7

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

   8. Applied associate-/l*9.2

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

   if 1.6889965443539738e+108 < b

   1. Initial program 59.4

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

   2. Simplified59.4

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

   3. Taylor expanded around inf 2.3

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

   4. Simplified2.3

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

4. Final simplification7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.9662470494954583 \cdot 10^{+148}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 2.2500462861079674 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} - b}{a \cdot 2}\\ \mathbf{elif}\;b \le 1.6889965443539738 \cdot 10^{+108}:\\ \;\;\;\;\frac{c \cdot -4}{\frac{\left(\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + b\right) \cdot \left(a \cdot 2\right)}{a}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 323378853 
(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

Time bar (total: 30.5s)Debug log