Average Error: 33.4 → 7.3
Time: 1.7m
Precision: 64
Internal Precision: 2688
\[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -1.324034465589458 \cdot 10^{+149}:\\ \;\;\;\;-2 \cdot \frac{b/2}{a}\\ \mathbf{if}\;b/2 \le 4.9585500140491613 \cdot 10^{-225}:\\ \;\;\;\;\left(\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;b/2 \le 1.1498155483847317 \cdot 10^{+81}:\\ \;\;\;\;\frac{c \cdot a}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}} \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-b/2\right) + b/2}{a} - \frac{\frac{1}{2}}{\frac{b/2}{c}}\\ \end{array}\]

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 < -1.324034465589458e+149

    1. Initial program 58.8

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]

    2. Taylor expanded around -inf 0

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

    if -1.324034465589458e+149 < b/2 < 4.9585500140491613e-225

    1. Initial program 8.8

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

    3. Applied div-inv9.0

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

    if 4.9585500140491613e-225 < b/2 < 1.1498155483847317e+81

    1. Initial program 33.9

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

    3. Applied flip-+34.0

      \[\leadsto \frac{\color{blue}{\frac{\left(-b/2\right) \cdot \left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c} \cdot \sqrt{b/2 \cdot b/2 - a \cdot c}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}}{a}\]

    4. Applied simplify15.3

      \[\leadsto \frac{\frac{\color{blue}{c \cdot a}}{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}}{a}\]
    5. Using strategy rm

    6. Applied div-inv15.3

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

    if 1.1498155483847317e+81 < b/2

    1. Initial program 58.3

      \[\frac{\left(-b/2\right) + \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]

    2. Taylor expanded around inf 40.0

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

    3. Applied simplify0.7

      \[\leadsto \color{blue}{\frac{\left(-b/2\right) + b/2}{a} - \frac{\frac{1}{2}}{\frac{b/2}{c}}}\]
  3. Recombined 4 regimes into one program.
  4. Removed slow pow expressions.

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(FPCore (a b/2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))