Average Error: 34.6 → 8.8
Time: 11.9s
Precision: 64
Internal precision: 2688
\[\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b/2 \le -1.3595260203786992 \cdot 10^{+104}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \mathbf{if}\;b/2 \le 4.157295140109413 \cdot 10^{-131}:\\ \;\;\;\;\frac{\left(-b/2\right) + \sqrt{{b/2}^2 - a \cdot c}}{a}\\ \mathbf{if}\;b/2 \le 3.5012060427894713 \cdot 10^{+160}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b/2\right) - \sqrt{{b/2}^2 - a \cdot c}}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 4 regimes.

  2. if b/2 < -1.3595260203786992e+104

    1. Initial program 47.8

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

    2. Applied taylor 0

      \[\leadsto -2 \cdot \frac{b/2}{a}\]

    3. Taylor expanded around -inf 0

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

    if -1.3595260203786992e+104 < b/2 < 4.157295140109413e-131

    1. Initial program 11.4

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

    if 4.157295140109413e-131 < b/2 < 3.5012060427894713e+160

    1. Initial program 42.2

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

    3. Applied flip-+ 42.2

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

    4. Applied simplify 16.3

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

    if 3.5012060427894713e+160 < b/2

    1. Initial program 62.9

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

    2. Applied taylor 16.2

      \[\leadsto \frac{\frac{-1}{2} \cdot \frac{c \cdot a}{b/2}}{a}\]

    3. Taylor expanded around inf 16.2

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

    4. Applied simplify 0.0

      \[\leadsto \color{blue}{\frac{c}{b/2} \cdot \frac{-1}{2}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 11.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1067488128 1186562765 67428877 2948030040 4169093828 3206893112)'
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (sqr b/2) (* a c)))) a))