Error

Target

Derivation

1. Split input into 3 regimes

2. if b < -2.6981693858302217e+153

   1. Initial program 60.5

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

   2. Simplified60.5

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

   3. Taylor expanded around -inf 1.6

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

   if -2.6981693858302217e+153 < b < 9.972464149054613e-68

   1. Initial program 11.8

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

   2. Simplified11.8

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

   4. Applied add-cube-cbrt12.0

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

   5. Applied add-sqr-sqrt12.0

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

   6. Applied sqrt-prod12.2

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

   7. Applied prod-diff12.2

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

   8. Simplified11.8

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

   9. Simplified11.8

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

   if 9.972464149054613e-68 < b

   1. Initial program 52.8

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

   2. Simplified52.8

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

   3. Taylor expanded around inf 8.7

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

   4. Simplified8.7

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

4. Final simplification9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -2.6981693858302217 \cdot 10^{+153}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \le 9.972464149054613 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\left(\left(-b\right) + b\right) + \left(\sqrt{(\left(-4 \cdot c\right) \cdot a + \left(b \cdot b\right))_*} + \left(-b\right)\right)}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019094 +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)))