Average Error: 33.5 → 8.7
Time: 1.9m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le -1.3659996970185691 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.51588034752967 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.7030881267972487 \cdot 10^{-21} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 3.767402032370213 \cdot 10^{+50} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 1.323811352621703 \cdot 10^{+91}\right)\right):\\ \;\;\;\;\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b_2}{c}} - \left(b_2 + b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (/ -1/2 b_2) < -1.3659996970185691e-154

    1. Initial program 8.2

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

    3. Applied div-inv8.3

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

    if -1.3659996970185691e-154 < (/ -1/2 b_2) < 2.51588034752967e-309

    1. Initial program 60.6

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

    2. Taylor expanded around inf 12.1

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{c \cdot a}{b_2} - 2 \cdot b_2}}{a}\]

    3. Applied simplify1.9

      \[\leadsto \color{blue}{\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2}\]

    if 2.51588034752967e-309 < (/ -1/2 b_2) < 2.7030881267972487e-21 or 3.767402032370213e+50 < (/ -1/2 b_2) < 1.323811352621703e+91

    1. Initial program 53.8

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

    3. Applied flip--53.8

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

    4. Applied simplify26.3

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

    5. Applied simplify26.3

      \[\leadsto \frac{\frac{c \cdot a}{\color{blue}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}{a}\]
    6. Using strategy rm

    7. Applied clear-num26.5

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

    8. Taylor expanded around -inf 17.9

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

    9. Applied simplify6.5

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

    if 2.7030881267972487e-21 < (/ -1/2 b_2) < 3.767402032370213e+50 or 1.323811352621703e+91 < (/ -1/2 b_2)

    1. Initial program 27.1

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

    3. Applied flip--27.2

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

    4. Applied simplify17.0

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

    5. Applied simplify17.0

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

  4. Applied simplify8.7

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le -1.3659996970185691 \cdot 10^{-154}:\\ \;\;\;\;\left(\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - c \cdot a}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.51588034752967 \cdot 10^{-309}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le 2.7030881267972487 \cdot 10^{-21} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 3.767402032370213 \cdot 10^{+50} \lor \neg \left(\frac{\frac{-1}{2}}{b_2} \le 1.323811352621703 \cdot 10^{+91}\right)\right):\\ \;\;\;\;\frac{c}{\frac{a \cdot \frac{1}{2}}{\frac{b_2}{c}} - \left(b_2 + b_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{a}\\ \end{array}}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed '#(1071948828 1180510430 2986424009 997076509 406109801 420189285)' +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))