Average Error: 19.3 → 7.5
Time: 1.2m
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -2.475582491061442 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \le 7.574837362105014 \cdot 10^{-10}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{c \cdot \frac{a}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a \cdot 2}\\ \end{array}\]

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 < -2.475582491061442e+98

    1. Initial program 44.9

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

    2. Applied simplify44.9

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

    3. Taylor expanded around -inf 11.2

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

    4. Applied simplify4.6

      \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - a \cdot \left(4 \cdot c\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot 1 - \frac{b + b}{a \cdot 2}\\ \end{array}}\]

    if -2.475582491061442e+98 < b < 7.574837362105014e-10

    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. Applied simplify9.4

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

    4. Applied add-sqr-sqrt9.4

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

    5. Applied sqrt-prod9.5

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

    if 7.574837362105014e-10 < b

    1. Initial program 22.0

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

    2. Applied simplify22.0

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

    3. Taylor expanded around inf 8.2

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

    4. Applied simplify5.8

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

  4. Applied simplify7.5

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -2.475582491061442 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \le 7.574837362105014 \cdot 10^{-10}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}} \cdot \sqrt{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{c}{c \cdot \frac{a}{b} - b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot c\right) \cdot a} - b}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018170 
(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))))