Error

Target

Derivation

1. Split input into 4 regimes

2. if b < -1.6964169257649298e+81

   1. Initial program 57.5

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

   2. Taylor expanded around -inf 2.6

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

   3. Simplified2.6

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

   if -1.6964169257649298e+81 < b < -2.650832923705524e-105

   1. Initial program 39.2

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

   3. Applied flip--39.3

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

   4. Applied associate-/l/42.6

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

   5. Simplified17.7

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

   if -2.650832923705524e-105 < b < 2.696518439055105e+143

   1. Initial program 11.6

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

   3. Applied associate-/r*11.5

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

   5. Applied div-inv11.7

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

   if 2.696518439055105e+143 < b

   1. Initial program 56.7

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

   3. Applied associate-/r*56.7

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

   5. Applied *-un-lft-identity56.7

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

   6. Applied associate-/l*56.7

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

   7. Taylor expanded around 0 1.9

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

   8. Simplified1.9

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

4. Final simplification9.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6964169257649298 \cdot 10^{+81}:\\ \;\;\;\;-\frac{c}{b}\\ \mathbf{elif}\;b \le -2.650832923705524 \cdot 10^{-105}:\\ \;\;\;\;\frac{\left(a \cdot c\right) \cdot 4}{\left(2 \cdot a\right) \cdot \left(\sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4} + \left(-b\right)\right)}\\ \mathbf{elif}\;b \le 2.696518439055105 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{a} \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - \left(a \cdot c\right) \cdot 4}}{2}\\ \mathbf{else}:\\ \;\;\;\;-\frac{b}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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)))