Average Error: 32.9 → 9.5
Time: 40.0s
Precision: 64
Internal Precision: 128
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;-b_2 \le -1.643868493513513 \cdot 10^{+116}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;-b_2 \le -9.030475850962892 \cdot 10^{-102}:\\ \;\;\;\;\frac{-c \cdot a}{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right) \cdot a}\\ \mathbf{elif}\;-b_2 \le 4.238510332548367 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

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) < -1.643868493513513e+116

    1. Initial program 59.8

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

    2. Initial simplification59.8

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

    3. Taylor expanded around inf 2.0

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

    if -1.643868493513513e+116 < (- b_2) < -9.030475850962892e-102

    1. Initial program 41.9

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

    2. Initial simplification41.9

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

    4. Applied flip--42.0

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

    5. Applied associate-/l/44.6

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

    6. Simplified16.8

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

    if -9.030475850962892e-102 < (- b_2) < 4.238510332548367e+104

    1. Initial program 11.6

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

    2. Initial simplification11.6

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

    4. Applied *-un-lft-identity11.6

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

    5. Applied associate-/l*11.7

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

    if 4.238510332548367e+104 < (- b_2)

    1. Initial program 45.9

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

    2. Initial simplification45.9

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

    3. Taylor expanded around -inf 3.7

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

    4. Simplified3.7

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;-b_2 \le -1.643868493513513 \cdot 10^{+116}:\\ \;\;\;\;\frac{c}{b_2} \cdot \frac{-1}{2}\\ \mathbf{elif}\;-b_2 \le -9.030475850962892 \cdot 10^{-102}:\\ \;\;\;\;\frac{-c \cdot a}{\left(\sqrt{b_2 \cdot b_2 - c \cdot a} + b_2\right) \cdot a}\\ \mathbf{elif}\;-b_2 \le 4.238510332548367 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{a}{\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2}}\\ \mathbf{else}:\\ \;\;\;\;(-2 \cdot \left(\frac{b_2}{a}\right) + \left(\frac{c}{\frac{b_2}{\frac{1}{2}}}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 40.0s)Debug logProfile

herbie shell --seed 2018277 +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))