Average Error: 33.2 → 10.8
Time: 4.4m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.2004129928316368 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2 \cdot a}{\frac{b}{c}} - \left(b + b\right)}{2 \cdot a}\\ \mathbf{if}\;b \le 756484004754347.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b < -1.2004129928316368e+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. Using strategy rm

    3. Applied div-inv60.9

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

    4. Taylor expanded around -inf 11.1

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

    5. Applied simplify2.2

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

    if -1.2004129928316368e+154 < b < 756484004754347.6

    1. Initial program 15.5

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

    if 756484004754347.6 < b

    1. Initial program 55.5

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

    2. Taylor expanded around inf 43.8

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

    3. Applied simplify5.3

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

  4. Applied simplify10.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b \le -1.2004129928316368 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2 \cdot a}{\frac{b}{c}} - \left(b + b\right)}{2 \cdot a}\\ \mathbf{if}\;b \le 756484004754347.6:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-\frac{c}{b}\\ \end{array}}\]

Runtime

Time bar (total: 4.4m)Debug logProfile

herbie shell --seed '#(1072840222 1305617769 1692503039 1353360431 4178980589 1488672652)' 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))