Average Error: 19.1 → 7.7
Time: 22.2s
Precision: 64
Internal Precision: 576
\[\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\\ \end{array}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.381288527642394 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 2.5815353648624203 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{a}{\frac{b}{c}} \cdot 2 - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if b < -1.381288527642394e+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. Using strategy rm

    4. Applied add-sqr-sqrt60.9

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

    5. Applied sqrt-prod60.9

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

    6. Taylor expanded around inf 60.9

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

    7. Taylor expanded around -inf 12.3

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

    if -1.381288527642394e+154 < b < 2.5815353648624203e+98

    1. Initial program 8.7

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

    2. Initial simplification8.7

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

    4. Applied add-sqr-sqrt8.7

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

    5. Applied sqrt-prod8.8

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

    if 2.5815353648624203e+98 < b

    1. Initial program 28.7

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

    2. Initial simplification28.7

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

    4. Applied add-sqr-sqrt28.7

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

    5. Applied sqrt-prod28.7

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

    6. Taylor expanded around inf 6.1

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

    8. Applied associate-/l*2.6

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

  4. Final simplification7.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.381288527642394 \cdot 10^{+154}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \le 2.5815353648624203 \cdot 10^{+98}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\left(-b\right) - \sqrt{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c}} \cdot \sqrt{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}{a \cdot 2}\\ \end{array}\\ \mathbf{elif}\;b \ge 0:\\ \;\;\;\;\frac{2 \cdot c}{\frac{a}{\frac{b}{c}} \cdot 2 - 2 \cdot b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{b \cdot b + \left(a \cdot -4\right) \cdot c} - b}{a \cdot 2}\\ \end{array}\]

Runtime

Time bar (total: 22.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes19.17.76.812.492.2%
herbie shell --seed 2018295 
(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))))