Average Error: 33.9 → 9.8
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le -1.9823885828713301 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.647022606142172 \cdot 10^{-307}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 1.029880118028398 \cdot 10^{-66}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.951247700907802 \cdot 10^{-22} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 1.2111898138507246 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (/ -1/2 b_2) < -1.9823885828713301e-140

    1. Initial program 7.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv7.9

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]

    if -1.9823885828713301e-140 < (/ -1/2 b_2) < 2.647022606142172e-307

    1. Initial program 56.1

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 4.0

      \[\leadsto \color{blue}{-2 \cdot \frac{b_2}{a}}\]

    if 2.647022606142172e-307 < (/ -1/2 b_2) < 1.029880118028398e-66 or 2.951247700907802e-22 < (/ -1/2 b_2) < 1.2111898138507246e+118

    1. Initial program 51.9

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv51.9

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied flip--52.0

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]

    6. Applied associate-*l/52.0

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    7. Applied simplify25.2

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]

    8. Taylor expanded around -inf 18.0

      \[\leadsto \frac{\frac{c \cdot a}{a}}{\left(-b_2\right) + \color{blue}{\left(\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - b_2\right)}}\]

    9. Applied simplify10.5

      \[\leadsto \color{blue}{\frac{c}{\frac{c \cdot \frac{1}{2}}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}}\]

    if 1.029880118028398e-66 < (/ -1/2 b_2) < 2.951247700907802e-22 or 1.2111898138507246e+118 < (/ -1/2 b_2)

    1. Initial program 26.7

      \[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
    2. Using strategy rm

    3. Applied div-inv26.7

      \[\leadsto \color{blue}{\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}\]
    4. Using strategy rm

    5. Applied flip--26.8

      \[\leadsto \color{blue}{\frac{\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}} \cdot \frac{1}{a}\]

    6. Applied associate-*l/26.9

      \[\leadsto \color{blue}{\frac{\left(\left(-b_2\right) \cdot \left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c} \cdot \sqrt{b_2 \cdot b_2 - a \cdot c}\right) \cdot \frac{1}{a}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}}\]

    7. Applied simplify17.1

      \[\leadsto \frac{\color{blue}{\frac{c \cdot a}{a}}}{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify9.8

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le -1.9823885828713301 \cdot 10^{-140}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.647022606142172 \cdot 10^{-307}:\\ \;\;\;\;\frac{b_2}{a} \cdot -2\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 1.029880118028398 \cdot 10^{-66}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.951247700907802 \cdot 10^{-22} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 1.2111898138507246 \cdot 10^{+118}\right):\\ \;\;\;\;\frac{\frac{c \cdot a}{a}}{\sqrt{b_2 \cdot b_2 - c \cdot a} + \left(-b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{\frac{\frac{1}{2} \cdot c}{\frac{b_2}{a}} - \left(b_2 + b_2\right)}\\ \end{array}}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))