Error

Target

Derivation

1. Split input into 4 regimes

2. if b < -2.30868906470999e+96

   1. Initial program 44.2

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

   2. Simplified44.2

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

   3. Taylor expanded around -inf 9.7

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

   4. Simplified3.6

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

   if -2.30868906470999e+96 < b < -7.561492702273163e-267

   1. Initial program 8.2

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

   2. Simplified8.2

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

   4. Applied add-cube-cbrt8.5

      \[\leadsto \frac{\frac{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*} - \color{blue}{\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}}{2}}{a}\]

   5. Applied add-sqr-sqrt8.5

      \[\leadsto \frac{\frac{\sqrt{\color{blue}{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*} \cdot \sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}} - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}{2}}{a}\]

   6. Applied sqrt-prod8.7

      \[\leadsto \frac{\frac{\color{blue}{\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}} \cdot \sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}} - \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}}{2}}{a}\]

   7. Applied prod-diff8.7

      \[\leadsto \frac{\frac{\color{blue}{(\left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}\right) \cdot \left(\sqrt{\sqrt{(b \cdot b + \left(\left(a \cdot c\right) \cdot -4\right))_*}}\right) + \left(-\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right))_* + (\left(-\sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) + \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right))_*}}{2}}{a}\]

   8. Simplified8.2

      \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*} + \left(-b\right)\right)} + (\left(-\sqrt[3]{b}\right) \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) + \left(\sqrt[3]{b} \cdot \left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)\right))_*}{2}}{a}\]

   9. Simplified8.2

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

   11. Applied clear-num8.4

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

   if -7.561492702273163e-267 < b < 9.799775814202614e+87

   1. Initial program 31.2

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

   2. Simplified31.2

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

   4. Applied flip--31.3

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

   5. Applied associate-/l/31.3

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

   6. Simplified16.6

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

   if 9.799775814202614e+87 < b

   1. Initial program 57.8

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

   2. Simplified57.8

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

   3. Taylor expanded around inf 14.8

      \[\leadsto \frac{\frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{2}}{a}\]
3. Recombined 4 regimes into one program.

4. Final simplification11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.30868906470999 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\left(\frac{c}{b} \cdot a - b\right) \cdot 2}{2}}{a}\\ \mathbf{elif}\;b \le -7.561492702273163 \cdot 10^{-267}:\\ \;\;\;\;\frac{1}{\frac{a}{\frac{\left(\left(-b\right) + b\right) + \left(\left(-b\right) + \sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*}\right)}{2}}}\\ \mathbf{elif}\;b \le 9.799775814202614 \cdot 10^{+87}:\\ \;\;\;\;\frac{\frac{c \cdot \left(-4 \cdot a\right)}{\left(b + \sqrt{(b \cdot b + \left(-4 \cdot \left(c \cdot a\right)\right))_*}\right) \cdot 2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-2 \cdot \frac{c \cdot a}{b}}{2}}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019100 +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)))