Average Error: 33.6 → 6.7
Time: 51.7s
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -3.0830793037231415 \cdot 10^{+141}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le 2.267827358099368 \cdot 10^{-293}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}}{\frac{1}{c}}\\ \mathbf{if}\;b/2 \le 2.7727792816348866 \cdot 10^{+84}:\\ \;\;\;\;\left(\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b/2} - 2 \cdot \frac{b/2}{a}\\ \end{array}\]

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 < -3.0830793037231415e+141

    1. Initial program 61.4

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

    2. Taylor expanded around -inf 13.6

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

    3. Applied simplify1.9

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

    if -3.0830793037231415e+141 < b/2 < 2.267827358099368e-293

    1. Initial program 33.4

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

    3. Applied flip--33.5

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

    4. Applied simplify15.7

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

    6. Applied clear-num15.9

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

    7. Applied simplify8.8

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{b/2 \cdot b/2 - c \cdot a} + \left(-b/2\right)}{c}}}\]
    8. Using strategy rm

    9. Applied div-inv8.9

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

    10. Applied associate-/r*8.5

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

    if 2.267827358099368e-293 < b/2 < 2.7727792816348866e+84

    1. Initial program 9.1

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

    3. Applied div-inv9.3

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

    if 2.7727792816348866e+84 < b/2

    1. Initial program 42.8

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

    3. Applied flip--61.2

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

    4. Applied simplify61.4

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

    6. Applied clear-num61.4

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

    7. Applied simplify61.3

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

    8. Taylor expanded around inf 4.1

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

Runtime

Time bar (total: 51.7s)Debug logProfile

herbie shell --seed '#(1063313015 2771194459 1594909340 1344785158 2223560818 546365448)' 
(FPCore (a b/2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))