Average Error: 34.9 → 5.3
Time: 20.3s
Precision: 64
Internal precision: 3200
\[\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -9.188864339554766 \cdot 10^{+63}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{if}\;b \le -7.412474314304801 \cdot 10^{-284}:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{if}\;b \le 2.8184599889584243 \cdot 10^{+148}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c \cdot 4}{\left(-b\right) - \sqrt{b \cdot b - \left(c \cdot 4\right) \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot \frac{-2}{2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.9
Comparison21.9
Herbie5.3
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes.

  2. if b < -9.188864339554766e+63

    1. Initial program 41.2

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

    2. Applied taylor 10.5

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

    3. Taylor expanded around -inf 10.5

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

    4. Applied simplify 0.0

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

    5. Applied simplify 0.0

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

    if -9.188864339554766e+63 < b < -7.412474314304801e-284

    1. Initial program 9.1

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

    if -7.412474314304801e-284 < b < 2.8184599889584243e+148

    1. Initial program 33.7

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

    3. Applied flip-+ 33.8

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

    4. Applied simplify 16.4

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

    6. Applied *-un-lft-identity 16.4

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

    7. Applied times-frac 16.3

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

    8. Applied simplify 8.6

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

    if 2.8184599889584243e+148 < b

    1. Initial program 62.5

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

    2. Applied taylor 14.1

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

    3. Taylor expanded around inf 14.1

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

    4. Applied simplify 0

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

Runtime

Time bar (total: 20.3s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1066500295 745726447 3908002351 725592315 4114972361 2368915013)'
(FPCore (a b c)
  :name "quadp (p42, positive)"

  :target
  (if (< b 0) (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (sqr b) (* 4 (* a c))))) (* 2 a)))