Average Error: 19.7 → 13.1
Time: 31.5s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 3.3290229124776183 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\]

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 2 regimes

  2. if b < 3.3290229124776183e+142

    1. Initial program 14.5

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

    2. Initial simplification14.5

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

    4. Applied add-sqr-sqrt14.5

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

    5. Applied sqrt-prod14.6

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

    if 3.3290229124776183e+142 < b

    1. Initial program 57.0

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

    2. Initial simplification57.0

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

    3. Taylor expanded around 0 2.3

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

  4. Final simplification13.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 3.3290229124776183 \cdot 10^{+142}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}} \cdot \sqrt{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)}}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{\left(-b\right) - b}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot c}{\sqrt{b \cdot b + c \cdot \left(a \cdot -4\right)} - b}\\ \end{array}\]

Runtime

Time bar (total: 31.5s)Debug logProfile

herbie shell --seed 2018251 
(FPCore (a b c)
  :name "jeff quadratic root 1"
  (if (>= b 0) (/ (- (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)) (/ (* 2 c) (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))))))