Average Error: 33.1 → 10.5
Time: 42.1s
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -1.419901435117473 \cdot 10^{+81}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.6504134482900307 \cdot 10^{-73}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \left(\sqrt[3]{-b_2}\right) + \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b_2}{c} \cdot -2}\\ \end{array}\]

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 < -1.419901435117473e+81

    1. Initial program 41.8

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

    2. Taylor expanded around -inf 5.0

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

    if -1.419901435117473e+81 < b_2 < 2.6504134482900307e-73

    1. Initial program 13.3

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

    3. Applied add-cube-cbrt13.5

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \sqrt[3]{-b_2}} + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    4. Applied fma-def13.5

      \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \left(\sqrt[3]{-b_2}\right) + \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}}{a}\]

    if 2.6504134482900307e-73 < b_2

    1. Initial program 51.9

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

    3. Applied clear-num51.9

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

    4. Taylor expanded around 0 9.6

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

  4. Final simplification10.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -1.419901435117473 \cdot 10^{+81}:\\ \;\;\;\;-2 \cdot \frac{b_2}{a}\\ \mathbf{elif}\;b_2 \le 2.6504134482900307 \cdot 10^{-73}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{-b_2} \cdot \sqrt[3]{-b_2}\right) \cdot \left(\sqrt[3]{-b_2}\right) + \left(\sqrt{b_2 \cdot b_2 - a \cdot c}\right))_*}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b_2}{c} \cdot -2}\\ \end{array}\]

Runtime

Time bar (total: 42.1s)Debug logProfile

herbie shell --seed 2018216 +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))