Average Error: 33.1 → 9.6
Time: 2.7m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -1.6366096550735478 \cdot 10^{-45}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \mathbf{if}\;-b \le 6.0685917336100174 \cdot 10^{+94}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

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) < -1.6366096550735478e-45

    1. Initial program 53.6

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

    2. Taylor expanded around inf 18.4

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{c \cdot a}{b}}}{2 \cdot a}\]

    3. Applied simplify7.3

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

    if -1.6366096550735478e-45 < (- b) < 6.0685917336100174e+94

    1. Initial program 13.3

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

    3. Applied div-inv13.4

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

    if 6.0685917336100174e+94 < (- b)

    1. Initial program 44.0

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

    2. Taylor expanded around -inf 10.0

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{c \cdot a}{b} - 2 \cdot b}}{2 \cdot a}\]

    3. Applied simplify3.7

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

  4. Applied simplify9.6

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -1.6366096550735478 \cdot 10^{-45}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \mathbf{if}\;-b \le 6.0685917336100174 \cdot 10^{+94}:\\ \;\;\;\;\left(\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} + \left(-b\right)\right) \cdot \frac{1}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))