Average Error: 33.4 → 10.1
Time: 26.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -3.8082344975212443 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3348675093554875 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\frac{a}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

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Your Program's Arguments

Results

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Derivation

  1. Split input into 3 regimes

  2. if b_2 < -3.8082344975212443e+84

    1. Initial program 41.3

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

    2. Initial simplification41.3

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

    3. Taylor expanded around -inf 4.5

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

    if -3.8082344975212443e+84 < b_2 < 1.3348675093554875e-65

    1. Initial program 13.3

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

    2. Initial simplification13.3

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

    4. Applied add-sqr-sqrt13.7

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

    5. Applied associate-/l*13.7

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

    if 1.3348675093554875e-65 < b_2

    1. Initial program 52.6

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

    2. Initial simplification52.6

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

    3. Taylor expanded around inf 8.4

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

  4. Final simplification10.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -3.8082344975212443 \cdot 10^{+84}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b_2} - \frac{b_2}{a} \cdot 2\\ \mathbf{elif}\;b_2 \le 1.3348675093554875 \cdot 10^{-65}:\\ \;\;\;\;\frac{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}{\frac{a}{\sqrt{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Runtime

Time bar (total: 26.0s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))