Average Error: 33.6 → 9.9
Time: 1.3m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le -2.9459803625918725 \cdot 10^{+87}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le 1.819213761444647 \cdot 10^{-307}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \mathbf{if}\;\frac{-\frac{1}{2}}{b_2} \le 1.8419002657732345 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot c}{b_2} - \left(\frac{b_2}{a} + \frac{b_2}{a}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if (/ (- 1/2) b_2) < -2.9459803625918725e+87 or 1.8419002657732345e-155 < (/ (- 1/2) b_2)

    1. Initial program 11.8

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

    if -2.9459803625918725e+87 < (/ (- 1/2) b_2) < 1.819213761444647e-307

    1. Initial program 52.2

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

    2. Taylor expanded around inf 20.3

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

    3. Applied simplify9.8

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

    if 1.819213761444647e-307 < (/ (- 1/2) b_2) < 1.8419002657732345e-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 simplify2.1

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

Runtime

Time bar (total: 1.3m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' 
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))