Average Error: 33.4 → 9.4
Time: 2.7m
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b \le -1.4732232065949135 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 1.80809249792908 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if (- b) < -1.4732232065949135e-71

    1. Initial program 53.5

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

    2. Taylor expanded around inf 46.5

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

    3. Applied simplify7.8

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

    if -1.4732232065949135e-71 < (- b) < 1.80809249792908e+139

    1. Initial program 12.1

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

    if 1.80809249792908e+139 < (- b)

    1. Initial program 55.1

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

    2. Taylor expanded around -inf 10.9

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

    3. Applied simplify3.3

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

  4. Applied simplify9.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;-b \le -1.4732232065949135 \cdot 10^{-71}:\\ \;\;\;\;\frac{-c}{b}\\ \mathbf{if}\;-b \le 1.80809249792908 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 4\right) \cdot c} + \left(-b\right)}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c}{b}}{1} - \frac{b + b}{2 \cdot a}\\ \end{array}}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1070991898 1055468627 4280279443 640792587 928206309 3646738750)' 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4 a) c)))) (* 2 a)))