Error

Target

Derivation

1. Split input into 4 regimes

2. if b < -6.3284880384768394e+119

   1. Initial program 50.1

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

   2. Taylor expanded around -inf 9.9

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

   3. Applied simplify2.9

      \[\leadsto \color{blue}{\frac{(\left(-c\right) \cdot \left(\frac{a}{b}\right) + \left(-b\right))_*}{a}}\]

   if -6.3284880384768394e+119 < b < 8.31926581131786e-309

   1. Initial program 8.9

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

   3. Applied div-inv9.1

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

   if 8.31926581131786e-309 < b < 1.4399774592443239e+37

   1. Initial program 29.0

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

   3. Applied flip-+29.1

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

   4. Applied simplify17.9

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

   6. Applied *-un-lft-identity17.9

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

   7. Applied times-frac17.9

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

   8. Applied simplify10.6

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

   if 1.4399774592443239e+37 < b

   1. Initial program 56.3

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

   3. Applied flip-+56.4

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

   4. Applied simplify27.5

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

   6. Applied *-un-lft-identity27.5

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

   7. Applied times-frac27.5

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

   8. Applied simplify24.1

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

   9. Taylor expanded around inf 7.2

      \[\leadsto \frac{1}{2} \cdot \frac{\frac{c}{\frac{1}{4}}}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\]

   10. Applied simplify3.8

       \[\leadsto \color{blue}{\frac{\frac{4 \cdot c}{2 \cdot 2}}{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_*}}\]
3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2020178 +o rules:numerics
(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)))