Average Error: 33.6 → 6.9
Time: 26.6s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -5.087865568067213 \cdot 10^{+131}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.4741986809709392 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{a}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 3.7439912098571924 \cdot 10^{+47}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{(2 \cdot b_2 + \left(\frac{a \cdot \frac{-1}{2}}{\frac{b_2}{c}}\right))_*}\\ \end{array}\]

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 < -5.087865568067213e+131

    1. Initial program 52.9

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

    2. Initial simplification52.9

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

    3. Taylor expanded around -inf 2.6

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

    if -5.087865568067213e+131 < b_2 < 1.4741986809709392e-275

    1. Initial program 9.4

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

    2. Initial simplification9.4

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

    4. Applied add-sqr-sqrt9.8

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

    5. Applied associate-/l*9.8

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

    if 1.4741986809709392e-275 < b_2 < 3.7439912098571924e+47

    1. Initial program 30.2

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

    2. Initial simplification30.2

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

    4. Applied flip--30.3

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

    5. Applied associate-/l/35.3

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

    6. Simplified22.8

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

    8. Applied distribute-frac-neg22.8

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

    9. Simplified9.2

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

    if 3.7439912098571924e+47 < b_2

    1. Initial program 56.7

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

    2. Initial simplification56.7

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

    4. Applied flip--56.7

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

    5. Applied associate-/l/57.2

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

    6. Simplified28.2

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

    8. Applied distribute-frac-neg28.2

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

    9. Simplified25.0

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

    10. Taylor expanded around inf 6.6

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

    11. Simplified3.7

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

  4. Final simplification6.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -5.087865568067213 \cdot 10^{+131}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 1.4741986809709392 \cdot 10^{-275}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}{\frac{a}{\sqrt{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}}}\\ \mathbf{elif}\;b_2 \le 3.7439912098571924 \cdot 10^{+47}:\\ \;\;\;\;\frac{-c}{\sqrt{b_2 \cdot b_2 - a \cdot c} + b_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{-c}{(2 \cdot b_2 + \left(\frac{a \cdot \frac{-1}{2}}{\frac{b_2}{c}}\right))_*}\\ \end{array}\]

Runtime

Time bar (total: 26.6s)Debug logProfile

herbie shell --seed 2018250 +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))