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Target

Derivation

1. Split input into 4 regimes

2. if b < -4.2485149476691936e+130

   1. Initial program 60.8

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

   2. Initial simplification60.8

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

   3. Taylor expanded around -inf 2.0

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

   4. Simplified2.0

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

   if -4.2485149476691936e+130 < b < -9.532534227544595e-103

   1. Initial program 42.3

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

   2. Initial simplification42.3

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

   4. Applied flip--42.3

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

   5. Applied associate-/l/44.3

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

   6. Simplified16.8

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

   8. Applied associate-/r*13.1

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

   9. Simplified13.1

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

   if -9.532534227544595e-103 < b < 4.62148638684652e+78

   1. Initial program 11.5

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

   2. Initial simplification11.5

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

   3. Taylor expanded around 0 11.5

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

   5. Applied div-sub11.5

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

   if 4.62148638684652e+78 < b

   1. Initial program 40.4

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

   2. Initial simplification40.4

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

   3. Taylor expanded around inf 5.1

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

4. Final simplification8.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -4.2485149476691936 \cdot 10^{+130}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{elif}\;b \le -9.532534227544595 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{4 \cdot \left(c \cdot a\right)}{a \cdot 2}}{\sqrt{a \cdot \left(-4 \cdot c\right) + b \cdot b} - b}\\ \mathbf{elif}\;b \le 4.62148638684652 \cdot 10^{+78}:\\ \;\;\;\;\frac{-b}{a \cdot 2} - \frac{\sqrt{{b}^{2} - 4 \cdot \left(c \cdot a\right)}}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 
(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

Time bar (total: 18.4s)Debug log