Average Error: 34.4 → 5.5
Time: 2.1m
Precision: 64
Internal precision: 2944
\[\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
⬇
\[\begin{array}{l} \mathbf{if}\;b \le -2.049536640230252 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{c}{2} \cdot 4}{\frac{c + c}{\frac{b}{a}} - \left(b - \left(-b\right)\right)}\\ \mathbf{if}\;b \le 3.902728914492509 \cdot 10^{-158}:\\ \;\;\;\;\frac{4}{2} \cdot \frac{c}{\left(-b\right) + \sqrt{b \cdot b - a \cdot \left(c \cdot 4\right)}}\\ \mathbf{if}\;b \le 5.845042913155354 \cdot 10^{+61}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Target

Original34.4
Comparison21.5
Herbie5.5
\[ \begin{array}{l} \mathbf{if}\;b \lt 0:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) + \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b\right) - \sqrt{{b}^2 - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \end{array} \]

Derivation

  1. Split input into 4 regimes.

  2. if b < -2.049536640230252e+150

    1. Initial program 62.5

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

    3. Applied flip-- 62.6

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

    4. Applied simplify 36.5

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

    5. Applied taylor 14.1

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

    6. Taylor expanded around -inf 14.1

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

    7. Applied simplify 0.6

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

    8. Applied simplify 0.6

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

    if -2.049536640230252e+150 < b < 3.902728914492509e-158

    1. Initial program 30.9

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

    3. Applied flip-- 31.2

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

    4. Applied simplify 16.2

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

    6. Applied *-un-lft-identity 16.2

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

    7. Applied times-frac 16.2

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

    8. Applied times-frac 16.2

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

    9. Applied simplify 16.2

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

    10. Applied simplify 9.9

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

    if 3.902728914492509e-158 < b < 5.845042913155354e+61

    1. Initial program 5.6

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

    3. Applied clear-num 5.7

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

    if 5.845042913155354e+61 < b

    1. Initial program 41.6

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

    2. Applied taylor 11.9

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

    3. Taylor expanded around inf 11.9

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

    4. Applied simplify 0.0

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

    5. Applied simplify 0.0

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

Runtime

Time bar (total: 2.1m) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(772101555 1905824529 294602591 2478279198 2123125427 4197813737)'
(FPCore (a b c)
  :name "quadm (p42, negative)"

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

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