Average Error: 33.3 → 10.3
Time: 32.1s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -7.641765040703004 \cdot 10^{+148}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b_2 \le 2.34789448631425 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -7.641765040703004e+148

    1. Initial program 59.0

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

    2. Initial simplification59.0

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

    3. Taylor expanded around -inf 59.0

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

    4. Taylor expanded around -inf 2.0

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

    5. Simplified2.0

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

    if -7.641765040703004e+148 < b_2 < 2.34789448631425e-125

    1. Initial program 11.0

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

    2. Initial simplification11.0

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

    3. Taylor expanded around -inf 11.0

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

    if 2.34789448631425e-125 < b_2

    1. Initial program 50.4

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

    2. Initial simplification50.4

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

    3. Taylor expanded around -inf 50.4

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

    4. Taylor expanded around inf 11.8

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -7.641765040703004 \cdot 10^{+148}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \mathbf{elif}\;b_2 \le 2.34789448631425 \cdot 10^{-125}:\\ \;\;\;\;\frac{\sqrt{{b_2}^{2} - c \cdot a} - b_2}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Runtime

Time bar (total: 32.1s)Debug logProfile

herbie shell --seed 2018349 +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))