Average Error: 33.3 → 6.9
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -8.318102466069036 \cdot 10^{+150}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.677906325314508 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{elif}\;b \le 6.594734415149172 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b < -8.318102466069036e+150

    1. Initial program 59.6

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -8.318102466069036e+150 < b < -2.677906325314508e-265

    1. Initial program 7.9

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

    3. Applied *-un-lft-identity7.9

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

    4. Applied associate-/l*8.1

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

    if -2.677906325314508e-265 < b < 6.594734415149172e+81

    1. Initial program 30.1

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

    3. Applied flip-+30.3

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

    4. Applied associate-/l/34.7

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

    5. Simplified20.8

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

    7. Applied associate-/r*15.9

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

    8. Simplified15.9

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

    9. Taylor expanded around inf 9.8

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

    11. Applied associate-/l*10.1

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

    if 6.594734415149172e+81 < b

    1. Initial program 58.0

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

    3. Applied flip-+58.1

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

    4. Applied associate-/l/58.4

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

    5. Simplified31.9

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

    7. Applied associate-/r*30.0

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

    8. Simplified30.0

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

    9. Taylor expanded around inf 29.1

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

    10. Taylor expanded around 0 3.6

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -8.318102466069036 \cdot 10^{+150}:\\ \;\;\;\;-\frac{b}{a}\\ \mathbf{elif}\;b \le -2.677906325314508 \cdot 10^{-265}:\\ \;\;\;\;\frac{1}{\frac{a \cdot 2}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)}}}\\ \mathbf{elif}\;b \le 6.594734415149172 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{\left(-b\right) - \sqrt{(\left(-4 \cdot a\right) \cdot c + \left(b \cdot b\right))_*}}{c}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot 2}{\left(-b\right) - b}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018249 +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))