Average Error: 33.3 → 8.1
Time: 21.7s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -4.065450303684823 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.016967469290422 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 4.5644027550850554 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

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 < -4.065450303684823e+144

    1. Initial program 61.7

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

    2. Taylor expanded around -inf 1.4

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

    if -4.065450303684823e+144 < b_2 < -3.016967469290422e-184

    1. Initial program 38.8

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

    3. Applied div-inv38.9

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

    5. Applied flip--39.0

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

    6. Applied associate-*l/39.0

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

    7. Simplified14.0

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

    8. Taylor expanded around inf 6.7

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

    10. Applied add-sqr-sqrt7.0

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

    11. Applied associate-/r*7.0

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

    12. Simplified7.0

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

    if -3.016967469290422e-184 < b_2 < 4.5644027550850554e-14

    1. Initial program 12.5

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

    3. Applied *-un-lft-identity12.5

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

    4. Applied *-un-lft-identity12.5

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

    5. Applied distribute-lft-out--12.5

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

    6. Applied associate-/l*12.6

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

    if 4.5644027550850554e-14 < b_2

    1. Initial program 29.9

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

    3. Applied div-inv30.0

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

    4. Taylor expanded around inf 9.0

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -4.065450303684823 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{elif}\;b_2 \le -3.016967469290422 \cdot 10^{-184}:\\ \;\;\;\;\frac{\frac{c}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} + \left(-b_2\right)}}}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}\\ \mathbf{elif}\;b_2 \le 4.5644027550850554 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \end{array}\]

Runtime

Time bar (total: 21.7s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes30.18.15.924.291%
herbie shell --seed 2018351 
(FPCore (a b_2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))