Average Error: 33.5 → 9.3
Time: 2.8m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -4.681232182683818 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(4 \cdot c\right)}{(c \cdot \left(\frac{a}{b} + \frac{a}{b}\right) + \left(\left(-b\right) - b\right))_*}\\ \mathbf{if}\;-b \le -2.118907474743662 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a + a}\\ \mathbf{if}\;-b \le 7.665518801524091 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{a + a}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_*}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (- b) < -4.681232182683818e+117

    1. Initial program 60.0

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

    3. Applied flip-+60.1

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

    4. Applied simplify33.9

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

    5. Taylor expanded around inf 14.6

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

    6. Applied simplify2.0

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

    if -4.681232182683818e+117 < (- b) < -2.118907474743662e-210

    1. Initial program 36.6

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

    3. Applied flip-+36.7

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

    4. Applied simplify16.4

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

    if -2.118907474743662e-210 < (- b) < 7.665518801524091e+112

    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 clear-num10.8

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

    if 7.665518801524091e+112 < (- b)

    1. Initial program 47.9

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

    2. Taylor expanded around -inf 9.6

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

    3. Applied simplify3.0

      \[\leadsto \color{blue}{1 \cdot \frac{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_*}{a}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -4.681232182683818 \cdot 10^{+117}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot \left(4 \cdot c\right)}{(c \cdot \left(\frac{a}{b} + \frac{a}{b}\right) + \left(\left(-b\right) - b\right))_*}\\ \mathbf{if}\;-b \le -2.118907474743662 \cdot 10^{-210}:\\ \;\;\;\;\frac{\frac{c \cdot \left(4 \cdot a\right)}{\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}{a + a}\\ \mathbf{if}\;-b \le 7.665518801524091 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{a + a}{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\frac{a}{b}\right) \cdot c + \left(-b\right))_*}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))