Average Error: 33.0 → 8.9
Time: 3.1m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -5.606083790106066 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a}{2} \cdot \frac{4}{a}\right)}{(\left(\frac{a}{\frac{b}{c}}\right) \cdot 2 + \left(\left(-b\right) - b\right))_*}\\ \mathbf{if}\;-b \le -2.671473252609305 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\left(b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}\right) \cdot \frac{\frac{a \cdot 2}{-4}}{c \cdot a}}\\ \mathbf{if}\;-b \le 1.0915827421615809 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -5.606083790106066e+25

    1. Initial program 55.5

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

    3. Applied flip-+55.5

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

    4. Applied simplify27.4

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

    5. Taylor expanded around inf 15.4

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

    6. Applied simplify5.1

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

    if -5.606083790106066e+25 < (- b) < -2.671473252609305e-90

    1. Initial program 39.5

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

    3. Applied clear-num39.5

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

    4. Applied simplify39.5

      \[\leadsto \frac{1}{\color{blue}{\frac{a \cdot 2}{\sqrt{(\left(-4\right) \cdot \left(a \cdot c\right) + \left(b \cdot b\right))_*} - b}}}\]
    5. Using strategy rm

    6. Applied flip--39.5

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

    7. Applied associate-/r/39.6

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

    8. Applied simplify15.3

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

    if -2.671473252609305e-90 < (- b) < 1.0915827421615809e+111

    1. Initial program 12.0

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

    3. Applied clear-num12.1

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

    4. Applied simplify12.1

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

    if 1.0915827421615809e+111 < (- b)

    1. Initial program 46.6

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

    2. Taylor expanded around -inf 9.7

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

    3. Applied simplify3.1

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - \frac{b + b}{2 \cdot a}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify8.9

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -5.606083790106066 \cdot 10^{+25}:\\ \;\;\;\;\frac{c \cdot \left(\frac{a}{2} \cdot \frac{4}{a}\right)}{(\left(\frac{a}{\frac{b}{c}}\right) \cdot 2 + \left(\left(-b\right) - b\right))_*}\\ \mathbf{if}\;-b \le -2.671473252609305 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{\left(b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}\right) \cdot \frac{\frac{a \cdot 2}{-4}}{c \cdot a}}\\ \mathbf{if}\;-b \le 1.0915827421615809 \cdot 10^{+111}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b + b}{a \cdot 2}\\ \end{array}}\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018166 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))