Average Error: 33.8 → 10.3
Time: 40.0s
Precision: 64
Internal Precision: 3456
\[\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b/2 \le -2.437255775893752 \cdot 10^{-64}:\\ \;\;\;\;\frac{c}{b/2} \cdot \frac{-1}{2}\\ \mathbf{if}\;b/2 \le 1.0828682995093866 \cdot 10^{+26}:\\ \;\;\;\;\frac{\left(-b/2\right) - \sqrt{b/2 \cdot b/2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \frac{c}{b/2} + \frac{\left(-b/2\right) - b/2}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b/2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b/2 < -2.437255775893752e-64

    1. Initial program 53.3

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

    2. Taylor expanded around -inf 18.9

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

    3. Applied simplify8.3

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

    if -2.437255775893752e-64 < b/2 < 1.0828682995093866e+26

    1. Initial program 14.5

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

    if 1.0828682995093866e+26 < b/2

    1. Initial program 34.0

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

    3. Applied clear-num34.1

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

    4. Taylor expanded around inf 10.9

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

    5. Applied simplify6.5

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

Runtime

Time bar (total: 40.0s)Debug logProfile

herbie shell --seed '#(1064269945 2896236262 301053905 1701069080 1701464310 1614783279)' 
(FPCore (a b/2 c)
  :name "NMSE problem 3.2.1"
  (/ (- (- b/2) (sqrt (- (* b/2 b/2) (* a c)))) a))