Average Error: 33.2 → 10.5
Time: 28.3s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.5692996466264766 \cdot 10^{+148}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 1.01261065157807 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

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 < -3.5692996466264766e+148

    1. Initial program 58.3

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*58.3

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2}}{a}}\]

    4. Taylor expanded around -inf 2.4

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}}\]

    5. Simplified2.4

      \[\leadsto \color{blue}{\frac{-b}{a}}\]

    if -3.5692996466264766e+148 < b < 1.01261065157807e-130

    1. Initial program 10.7

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*10.7

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2}}{a}}\]

    if 1.01261065157807e-130 < b

    1. Initial program 50.4

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*50.4

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2}}{a}}\]
    4. Using strategy rm

    5. Applied clear-num50.4

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2}}}}\]

    6. Taylor expanded around 0 12.5

      \[\leadsto \frac{1}{\color{blue}{-1 \cdot \frac{b}{c}}}\]

    7. Simplified12.5

      \[\leadsto \frac{1}{\color{blue}{\frac{-b}{c}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.5692996466264766 \cdot 10^{+148}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le 1.01261065157807 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-\frac{b}{c}}\\ \end{array}\]

Runtime

Time bar (total: 28.3s)Debug logProfile

herbie shell --seed 2018235 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))