Average Error: 33.9 → 9.0
Time: 1.5m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\frac{-1}{2}}{b_2} \le -1531330828834661.5:\\ \;\;\;\;\frac{\frac{c \cdot a}{\left(-b_2\right) - \sqrt{b_2 \cdot b_2 - a \cdot c}}}{a}\\ \mathbf{elif}\;\frac{\frac{-1}{2}}{b_2} \le 2.2890491399528697 \cdot 10^{-307}:\\ \;\;\;\;\frac{c}{\left(-b_2\right) + (\left(\frac{c}{b_2}\right) \cdot \left(\frac{1}{2} \cdot a\right) + \left(-b_2\right))_*}\\ \mathbf{elif}\;\frac{\frac{-1}{2}}{b_2} \le 1.2922456505044055 \cdot 10^{-155}:\\ \;\;\;\;c \cdot \frac{\frac{1}{2}}{b_2} - \left(\frac{b_2}{a} + \frac{b_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 4 regimes

  2. if (/ -1/2 b_2) < -1531330828834661.5

    1. Initial program 24.1

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

    3. Applied flip-+24.2

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

    4. Applied simplify17.7

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

    if -1531330828834661.5 < (/ -1/2 b_2) < 2.2890491399528697e-307

    1. Initial program 54.8

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

    3. Applied flip-+54.9

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

    4. Applied simplify26.7

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

    5. Taylor expanded around inf 17.7

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

    6. Applied simplify6.7

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

    if 2.2890491399528697e-307 < (/ -1/2 b_2) < 1.2922456505044055e-155

    1. Initial program 60.9

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

    2. Taylor expanded around -inf 10.5

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

    3. Applied simplify2.6

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

    if 1.2922456505044055e-155 < (/ -1/2 b_2)

    1. Initial program 8.4

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
  3. Recombined 4 regimes into one program.

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed 2018208 +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))