Average Error: 20.0 → 8.7
Time: 28.5s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.336244498910505 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.2771361016259362 \cdot 10^{+45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{a}{\frac{b}{c}} \cdot 2 - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b} \cdot \sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}}{2 \cdot a}\\ \end{array}\]

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 < -1.336244498910505e+154

    1. Initial program 60.9

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

    2. Initial simplification60.9

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

    3. Taylor expanded around -inf 11.3

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

    if -1.336244498910505e+154 < b < 2.2771361016259362e+45

    1. Initial program 9.4

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

    2. Initial simplification9.4

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

    4. Applied add-exp-log10.7

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

    if 2.2771361016259362e+45 < b

    1. Initial program 25.9

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

    2. Initial simplification25.9

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

    3. Taylor expanded around inf 7.0

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

    5. Applied associate-/l*3.8

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

    7. Applied add-sqr-sqrt3.8

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.336244498910505 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{a \cdot c}{b} - b\right) - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \le 2.2771361016259362 \cdot 10^{+45}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - e^{\log \left(\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}{2 \cdot a}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{a}{\frac{b}{c}} \cdot 2 - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b} \cdot \sqrt{\sqrt{\left(-4 \cdot a\right) \cdot c + b \cdot b} - b}}{2 \cdot a}\\ \end{array}\]

Runtime

Time bar (total: 28.5s)Debug logProfile

herbie shell --seed 2018248 
(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))))