Average Error: 33.8 → 14.7
Time: 3.8m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.374743781623492 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{b}{a}\\ \mathbf{if}\;b \le 1.7966867293409628 \cdot 10^{-177}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} - b}}\\ \mathbf{if}\;b \le 1.7514490224172964 \cdot 10^{+145}:\\ \;\;\;\;\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}} \cdot \sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}\right)\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{2 \cdot b}}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -1.374743781623492e+154

    1. Initial program 60.9

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

    2. Applied simplify60.8

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

    3. Taylor expanded around -inf 52.2

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{b}{a}}\]

    if -1.374743781623492e+154 < b < 1.7966867293409628e-177

    1. Initial program 10.2

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

    2. Applied simplify10.2

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

    4. Applied clear-num10.4

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

    if 1.7966867293409628e-177 < b < 1.7514490224172964e+145

    1. Initial program 39.0

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

    2. Applied simplify39.0

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

    4. Applied flip--39.1

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

    5. Applied simplify15.0

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

    7. Applied add-cube-cbrt15.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\right) \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}}\]

    8. Applied simplify15.7

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right)} \cdot \sqrt[3]{\frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\sqrt{(\left(4 \cdot a\right) \cdot \left(-c\right) + \left(b \cdot b\right))_*} + b}}{2 \cdot a}}\]

    9. Applied simplify6.9

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt7.0

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}}\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\]

    12. Applied cbrt-prod7.1

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\right)}\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\]

    13. Applied simplify7.1

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \left(\color{blue}{\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}} \cdot \sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}}\right)\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\]

    14. Applied simplify7.1

      \[\leadsto \left(\sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}} \cdot \sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*}}}} \cdot \color{blue}{\sqrt[3]{\sqrt[3]{\frac{\left(-c\right) \cdot \frac{4}{2}}{b + \sqrt{(\left(-4\right) \cdot \left(c \cdot a\right) + \left(b \cdot b\right))_*}}}}}\right)\right) \cdot \sqrt[3]{\frac{\left(4 \cdot \left(-c\right)\right) \cdot \frac{1}{2}}{\sqrt{(\left(-c\right) \cdot \left(4 \cdot a\right) + \left(b \cdot b\right))_*} + b}}\]

    if 1.7514490224172964e+145 < b

    1. Initial program 61.9

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

    2. Applied simplify62.0

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

    4. Applied flip--62.0

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

    5. Applied simplify36.9

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

    6. Taylor expanded around 0 13.9

      \[\leadsto \frac{\frac{\left(c \cdot a\right) \cdot \left(-4\right)}{\color{blue}{2 \cdot b}}}{2 \cdot a}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed '#(1071215679 2002590028 935158157 1944352234 2656991306 2955288481)' +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))