Average Error: 33.5 → 6.5
Time: 39.9s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -2.7277782301101006 \cdot 10^{+149}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le -2.9479498849466343 \cdot 10^{-288}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right) \cdot \left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right) + \left(-b_2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 2.3995554738573616 \cdot 10^{+127}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{(\frac{-1}{2} \cdot \left(\frac{c}{\frac{b_2}{a}}\right) + \left(b_2 \cdot 2\right))_*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if b_2 < -2.7277782301101006e+149

    1. Initial program 59.1

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

    2. Initial simplification59.1

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

    3. Taylor expanded around -inf 11.0

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

    4. Simplified2.2

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

    if -2.7277782301101006e+149 < b_2 < -2.9479498849466343e-288

    1. Initial program 8.4

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

    2. Initial simplification8.4

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

    4. Applied add-sqr-sqrt8.6

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

    5. Applied fma-neg8.5

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

    if -2.9479498849466343e-288 < b_2 < 2.3995554738573616e+127

    1. Initial program 32.9

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

    2. Initial simplification32.9

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

    4. Applied flip--33.0

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

    5. Applied associate-/l/37.4

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

    6. Simplified19.5

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

    8. Applied distribute-frac-neg19.5

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

    9. Simplified8.4

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

    if 2.3995554738573616e+127 < b_2

    1. Initial program 60.3

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

    2. Initial simplification60.3

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

    4. Applied flip--60.3

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

    5. Applied associate-/l/60.3

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

    6. Simplified34.6

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

    8. Applied distribute-frac-neg34.6

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

    9. Simplified33.6

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

    10. Taylor expanded around inf 6.9

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

    11. Simplified2.3

      \[\leadsto -\frac{c}{\color{blue}{(\frac{-1}{2} \cdot \left(\frac{c}{\frac{b_2}{a}}\right) + \left(2 \cdot b_2\right))_*}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification6.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -2.7277782301101006 \cdot 10^{+149}:\\ \;\;\;\;\frac{(\frac{1}{2} \cdot \left(\frac{a}{\frac{b_2}{c}}\right) + \left(b_2 \cdot -2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le -2.9479498849466343 \cdot 10^{-288}:\\ \;\;\;\;\frac{(\left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right) \cdot \left(\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c}}\right) + \left(-b_2\right))_*}{a}\\ \mathbf{elif}\;b_2 \le 2.3995554738573616 \cdot 10^{+127}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{(\frac{-1}{2} \cdot \left(\frac{c}{\frac{b_2}{a}}\right) + \left(b_2 \cdot 2\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 39.9s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))