Average Error: 33.5 → 10.3
Time: 2.0m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.5638780224706865 \cdot 10^{+87}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -6.931717002458052 \cdot 10^{-119}:\\ \;\;\;\;\log_* (1 + (e^{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} - 1)^*)\\ \mathbf{if}\;b_2 \le 3.9117660587132156 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -4.5638780224706865e+87

    1. Initial program 58.6

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

    2. Taylor expanded around -inf 40.8

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

    3. Applied simplify2.6

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

    if -4.5638780224706865e+87 < b_2 < -6.931717002458052e-119

    1. Initial program 40.7

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

    3. Applied flip--40.8

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

    4. Applied simplify15.2

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

    5. Applied simplify15.2

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

    7. Applied log1p-expm1-u32.7

      \[\leadsto \color{blue}{\log_* (1 + (e^{\frac{\frac{c \cdot a}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{a}} - 1)^*)}\]

    8. Applied simplify23.2

      \[\leadsto \log_* (1 + \color{blue}{(e^{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} - 1)^*})\]

    if -6.931717002458052e-119 < b_2 < 3.9117660587132156e+67

    1. Initial program 11.6

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

    3. Applied clear-num11.8

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

    if 3.9117660587132156e+67 < b_2

    1. Initial program 38.3

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

    2. Taylor expanded around inf 10.2

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

    3. Applied simplify5.1

      \[\leadsto \color{blue}{\frac{c}{b_2} \cdot \frac{1}{2} - \frac{b_2}{a} \cdot 2}\]
  3. Recombined 4 regimes into one program.

  4. Applied simplify10.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;b_2 \le -4.5638780224706865 \cdot 10^{+87}:\\ \;\;\;\;\left(-\frac{1}{2}\right) \cdot \frac{c}{b_2}\\ \mathbf{if}\;b_2 \le -6.931717002458052 \cdot 10^{-119}:\\ \;\;\;\;\log_* (1 + (e^{\frac{c}{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}} - 1)^*)\\ \mathbf{if}\;b_2 \le 3.9117660587132156 \cdot 10^{+67}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018178 +o rules:numerics
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))