Average Error: 19.3 → 6.6
Time: 30.4s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5692996466264766 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 4.4193134745531346 \cdot 10^{+135}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}} \cdot \sqrt{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\sqrt[3]{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)} \cdot \left|\sqrt[3]{b \cdot b + \left(-4 \cdot c\right) \cdot a}\right| - b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -3.5692996466264766e+148

    1. Initial program 58.3

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Initial simplification58.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

    3. Taylor expanded around inf 58.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
    4. Using strategy rm

    5. Applied associate-/l*58.3

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
    6. Using strategy rm

    7. Applied add-exp-log58.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\right)} - b}{2 \cdot a}\\ \end{array}\]

    8. Taylor expanded around -inf 7.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{-\log \left(\frac{-1}{b}\right)} - b}{2 \cdot a}\\ \end{array}\]

    9. Simplified2.6

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \end{array}\]

    if -3.5692996466264766e+148 < b < 4.4193134745531346e+135

    1. Initial program 8.5

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Initial simplification8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt8.5

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\color{blue}{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

    5. Applied sqrt-prod8.7

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \color{blue}{\sqrt{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

    if 4.4193134745531346e+135 < b

    1. Initial program 34.8

      \[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]

    2. Initial simplification34.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]

    3. Taylor expanded around inf 5.9

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\color{blue}{2 \cdot \frac{a \cdot c}{b} - 2 \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
    4. Using strategy rm

    5. Applied associate-/l*1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c} - b}{2 \cdot a}\\ \end{array}\]
    6. Using strategy rm

    7. Applied add-exp-log1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{b \cdot b + \left(-4 \cdot a\right) \cdot c}\right)} - b}{2 \cdot a}\\ \end{array}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\left(\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}\right) \cdot \sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} - b}{2 \cdot a}\\ \end{array}\]

    10. Applied sqrt-prod1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}} \cdot \sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} - b}{2 \cdot a}\\ \end{array}\]

    11. Applied log-prod1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right) + \log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} - b}{2 \cdot a}\\ \end{array}\]

    12. Applied exp-sum1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c} \cdot \sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} \cdot e^{\log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} - b}{2 \cdot a}\\ \end{array}\]

    13. Simplified1.8

      \[\leadsto \begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt[3]{b \cdot b + a \cdot \left(-4 \cdot c\right)}\right| \cdot e^{\log \left(\sqrt{\sqrt[3]{b \cdot b + \left(-4 \cdot a\right) \cdot c}}\right)} - b}{2 \cdot a}\\ \end{array}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification6.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5692996466264766 \cdot 10^{+148}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 4.4193134745531346 \cdot 10^{+135}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}} \cdot \sqrt{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -4\right) + b \cdot b} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{a}{\frac{b}{c}} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\sqrt{\sqrt[3]{c \cdot \left(a \cdot -4\right) + b \cdot b}}\right)} \cdot \left|\sqrt[3]{b \cdot b + \left(-4 \cdot c\right) \cdot a}\right| - b}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 30.4s)Debug logProfile

herbie shell --seed 2018235 
(FPCore (a b c)
  :name "jeff quadratic root 2"
  (if (>= b 0) (/ (* 2 c) (- (- b) (sqrt (- (* b b) (* (* 4 a) c))))) (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a))))